Latin Squares¶

A latin square of order \(n\) is an \(n \times n\) array such that each symbol \(s \in \{ 0, 1, \dots, n-1\}\) appears precisely once in each row, and precisely once in each column. A partial latin square of order \(n\) is an \(n \times n\) array such that each symbol \(s \in \{ 0, 1, \dots, n-1\}\) appears at most once in each row, and at most once in each column. Empty cells are denoted by \(-1\). A latin square \(L\) is a completion of a partial latin square \(P\) if \(P \subseteq L\). If \(P\) completes to just \(L\) then \(P\) has unique completion.

A latin bitrade \((T_1,\, T_2)\) is a pair of partial latin squares such that:

\(\{ (i,\,j) \mid (i,\,j,\,k) \in T_1 \text{ for some symbol }k \} = \{ (i,\,j) \mid (i,\,j,\,k') \in T_2 \text{ for some symbol }k' \};\)
for each \((i,\,j,\,k) \in T_1\) and \((i,\,j,\,k') \in T_2\), \(k \neq k'\);
the symbols appearing in row \(i\) of \(T_1\) are the same as those of row \(i\) of \(T_2\); the symbols appearing in column \(j\) of \(T_1\) are the same as those of column \(j\) of \(T_2\).

Intuitively speaking, a bitrade gives the difference between two latin squares, so if \((T_1,\, T_2)\) is a bitrade for the pair of latin squares \((L_1,\, L_2)\), then \(L1 = (L2 \setminus T_1) \cup T_2\) and \(L2 = (L1 \setminus T_2) \cup T_1\).

This file contains

LatinSquare class definition;
some named latin squares (back circulant, forward circulant, abelian \(2\)-group);
methods is_partial_latin_square() and is_latin_square() to test if a LatinSquare object satisfies the definition of a latin square or partial latin square, respectively;
tests for completion and unique completion (these use the C++ implementation of Knuth’s dancing links algorithm to solve the problem as a instance of \(0-1\) matrix exact cover);
functions for calculating the \(\tau_i\) representation of a bitrade and the genus of the associated hypermap embedding;
Markov chain of Jacobson and Matthews (1996) for generating latin squares uniformly at random (provides a generator interface);
a few examples of \(\tau_i\) representations of bitrades constructed from the action of a group on itself by right multiplication, functions for converting to a pair of LatinSquare objects.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B = back_circulant(5)
sage: B
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]
[4 0 1 2 3]
sage: B.is_latin_square()
True
sage: B[0, 1] = 0
sage: B.is_latin_square()
False

sage: (a, b, c, G) = alternating_group_bitrade_generators(1)
sage: (T1, T2) = bitrade_from_group(a, b, c, G)
sage: T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]
sage: T2
[ 1 -1  0  3]
[-1  0  2  1]
[ 2  1  3 -1]
[ 0  3 -1  2]
sage: T1.nr_filled_cells()
12
sage: genus(T1, T2)
1

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B = back_circulant(Integer(5))
>>> B
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]
[4 0 1 2 3]
>>> B.is_latin_square()
True
>>> B[Integer(0), Integer(1)] = Integer(0)
>>> B.is_latin_square()
False

>>> (a, b, c, G) = alternating_group_bitrade_generators(Integer(1))
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]
>>> T2
[ 1 -1  0  3]
[-1  0  2  1]
[ 2  1  3 -1]
[ 0  3 -1  2]
>>> T1.nr_filled_cells()
12
>>> genus(T1, T2)
1

Sage Live

from sage.combinat.matrices.latin import *
B = back_circulant(5)
B
B.is_latin_square()
B[0, 1] = 0
B.is_latin_square()
(a, b, c, G) = alternating_group_bitrade_generators(1)
(T1, T2) = bitrade_from_group(a, b, c, G)
T1
T2
T1.nr_filled_cells()
genus(T1, T2)

Todo

Latin squares with symbols from a ring instead of the integers \(\{ 0, 1, \dots, n-1 \}\).
Isotopism testing of latin squares and bitrades via graph isomorphism (nauty?).
Combinatorial constructions for bitrades.

AUTHORS:

Carlo Hamalainen (2008-03-23): initial version

class sage.combinat.matrices.latin.LatinSquare(*args)[source]¶

Bases: object

Latin squares.

This class implements a latin square of order n with rows and columns indexed by the set 0, 1, …, n-1 and symbols from the same set. The underlying latin square is a matrix(ZZ, n, n). If L is a latin square, then the cell at row r, column c is empty if and only if L[r, c] < 0. In this way we allow partial latin squares and can speak of completions to latin squares, etc.

There are two ways to declare a latin square:

Empty latin square of order n:

Sage

sage: n = 3
sage: L = LatinSquare(n)
sage: L
[-1 -1 -1]
[-1 -1 -1]
[-1 -1 -1]

Python

>>> from sage.all import *
>>> n = Integer(3)
>>> L = LatinSquare(n)
>>> L
[-1 -1 -1]
[-1 -1 -1]
[-1 -1 -1]

Sage Live

n = 3
L = LatinSquare(n)
L

Latin square from a matrix:

Sage

sage: M = matrix(ZZ, [[0, 1], [2, 3]])
sage: LatinSquare(M)
[0 1]
[2 3]

Python

>>> from sage.all import *
>>> M = matrix(ZZ, [[Integer(0), Integer(1)], [Integer(2), Integer(3)]])
>>> LatinSquare(M)
[0 1]
[2 3]

Sage Live

M = matrix(ZZ, [[0, 1], [2, 3]])
LatinSquare(M)

actual_row_col_sym_sizes()[source]¶

Bitrades sometimes end up in partial latin squares with unused rows, columns, or symbols. This function works out the actual number of used rows, columns, and symbols.

Warning

We assume that the unused rows/columns occur in the lower right of self, and that the used symbols are in the range {0, 1, …, m} (no holes in that list).

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B = back_circulant(3)
sage: B[0,2] = B[1,2] = B[2,2] = -1
sage: B[0,0] = B[2,1] = -1
sage: B
[-1  1 -1]
[ 1  2 -1]
[ 2 -1 -1]
sage: B.actual_row_col_sym_sizes()
(3, 2, 2)

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B = back_circulant(Integer(3))
>>> B[Integer(0),Integer(2)] = B[Integer(1),Integer(2)] = B[Integer(2),Integer(2)] = -Integer(1)
>>> B[Integer(0),Integer(0)] = B[Integer(2),Integer(1)] = -Integer(1)
>>> B
[-1  1 -1]
[ 1  2 -1]
[ 2 -1 -1]
>>> B.actual_row_col_sym_sizes()
(3, 2, 2)

Sage Live

from sage.combinat.matrices.latin import *
B = back_circulant(3)
B[0,2] = B[1,2] = B[2,2] = -1
B[0,0] = B[2,1] = -1
B
B.actual_row_col_sym_sizes()

apply_isotopism(row_perm, col_perm, sym_perm)[source]¶

An isotopism is a permutation of the rows, columns, and symbols of a partial latin square self. Use isotopism() to convert a tuple (indexed from 0) to a Permutation object.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B = back_circulant(5)
sage: B
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]
[4 0 1 2 3]
sage: alpha = isotopism((0,1,2,3,4))
sage: beta  = isotopism((1,0,2,3,4))
sage: gamma = isotopism((2,1,0,3,4))
sage: B.apply_isotopism(alpha, beta, gamma)
[3 4 2 0 1]
[0 2 3 1 4]
[1 3 0 4 2]
[4 0 1 2 3]
[2 1 4 3 0]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B = back_circulant(Integer(5))
>>> B
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]
[4 0 1 2 3]
>>> alpha = isotopism((Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)))
>>> beta  = isotopism((Integer(1),Integer(0),Integer(2),Integer(3),Integer(4)))
>>> gamma = isotopism((Integer(2),Integer(1),Integer(0),Integer(3),Integer(4)))
>>> B.apply_isotopism(alpha, beta, gamma)
[3 4 2 0 1]
[0 2 3 1 4]
[1 3 0 4 2]
[4 0 1 2 3]
[2 1 4 3 0]

Sage Live

from sage.combinat.matrices.latin import *
B = back_circulant(5)
B
alpha = isotopism((0,1,2,3,4))
beta  = isotopism((1,0,2,3,4))
gamma = isotopism((2,1,0,3,4))
B.apply_isotopism(alpha, beta, gamma)

clear_cells()[source]¶

Mark every cell in self as being empty.

EXAMPLES:

Sage

sage: A = LatinSquare(matrix(ZZ, [[0, 1], [2, 3]]))
sage: A.clear_cells()
sage: A
[-1 -1]
[-1 -1]

Python

>>> from sage.all import *
>>> A = LatinSquare(matrix(ZZ, [[Integer(0), Integer(1)], [Integer(2), Integer(3)]]))
>>> A.clear_cells()
>>> A
[-1 -1]
[-1 -1]

Sage Live

A = LatinSquare(matrix(ZZ, [[0, 1], [2, 3]]))
A.clear_cells()
A

column(x)[source]¶

Return column x of the latin square.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: back_circulant(3).column(0)
(0, 1, 2)

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> back_circulant(Integer(3)).column(Integer(0))
(0, 1, 2)

Sage Live

from sage.combinat.matrices.latin import *
back_circulant(3).column(0)

contained_in(Q)[source]¶

Return True if self is a subset of \(Q\).

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: P = elementary_abelian_2group(2)
sage: P[0, 0] = -1
sage: P.contained_in(elementary_abelian_2group(2))
True
sage: back_circulant(4).contained_in(elementary_abelian_2group(2))
False

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> P = elementary_abelian_2group(Integer(2))
>>> P[Integer(0), Integer(0)] = -Integer(1)
>>> P.contained_in(elementary_abelian_2group(Integer(2)))
True
>>> back_circulant(Integer(4)).contained_in(elementary_abelian_2group(Integer(2)))
False

Sage Live

from sage.combinat.matrices.latin import *
P = elementary_abelian_2group(2)
P[0, 0] = -1
P.contained_in(elementary_abelian_2group(2))
back_circulant(4).contained_in(elementary_abelian_2group(2))

disjoint_mate_dlxcpp_rows_and_map(allow_subtrade)[source]¶

Internal function for find_disjoint_mates.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B = back_circulant(4)
sage: B.disjoint_mate_dlxcpp_rows_and_map(allow_subtrade = True)
([[0, 16, 32],
  [1, 17, 32],
  [2, 18, 32],
  [3, 19, 32],
  [4, 16, 33],
  [5, 17, 33],
  [6, 18, 33],
  [7, 19, 33],
  [8, 16, 34],
  [9, 17, 34],
  [10, 18, 34],
  [11, 19, 34],
  [12, 16, 35],
  [13, 17, 35],
  [14, 18, 35],
  [15, 19, 35],
  [0, 20, 36],
  [1, 21, 36],
  [2, 22, 36],
  [3, 23, 36],
  [4, 20, 37],
  [5, 21, 37],
  [6, 22, 37],
  [7, 23, 37],
  [8, 20, 38],
  [9, 21, 38],
  [10, 22, 38],
  [11, 23, 38],
  [12, 20, 39],
  [13, 21, 39],
  [14, 22, 39],
  [15, 23, 39],
  [0, 24, 40],
  [1, 25, 40],
  [2, 26, 40],
  [3, 27, 40],
  [4, 24, 41],
  [5, 25, 41],
  [6, 26, 41],
  [7, 27, 41],
  [8, 24, 42],
  [9, 25, 42],
  [10, 26, 42],
  [11, 27, 42],
  [12, 24, 43],
  [13, 25, 43],
  [14, 26, 43],
  [15, 27, 43],
  [0, 28, 44],
  [1, 29, 44],
  [2, 30, 44],
  [3, 31, 44],
  [4, 28, 45],
  [5, 29, 45],
  [6, 30, 45],
  [7, 31, 45],
  [8, 28, 46],
  [9, 29, 46],
  [10, 30, 46],
  [11, 31, 46],
  [12, 28, 47],
  [13, 29, 47],
  [14, 30, 47],
  [15, 31, 47]],
 {(0, 16, 32): (0, 0, 0),
  (0, 20, 36): (1, 0, 0),
  (0, 24, 40): (2, 0, 0),
  (0, 28, 44): (3, 0, 0),
  (1, 17, 32): (0, 0, 1),
  (1, 21, 36): (1, 0, 1),
  (1, 25, 40): (2, 0, 1),
  (1, 29, 44): (3, 0, 1),
  (2, 18, 32): (0, 0, 2),
  (2, 22, 36): (1, 0, 2),
  (2, 26, 40): (2, 0, 2),
  (2, 30, 44): (3, 0, 2),
  (3, 19, 32): (0, 0, 3),
  (3, 23, 36): (1, 0, 3),
  (3, 27, 40): (2, 0, 3),
  (3, 31, 44): (3, 0, 3),
  (4, 16, 33): (0, 1, 0),
  (4, 20, 37): (1, 1, 0),
  (4, 24, 41): (2, 1, 0),
  (4, 28, 45): (3, 1, 0),
  (5, 17, 33): (0, 1, 1),
  (5, 21, 37): (1, 1, 1),
  (5, 25, 41): (2, 1, 1),
  (5, 29, 45): (3, 1, 1),
  (6, 18, 33): (0, 1, 2),
  (6, 22, 37): (1, 1, 2),
  (6, 26, 41): (2, 1, 2),
  (6, 30, 45): (3, 1, 2),
  (7, 19, 33): (0, 1, 3),
  (7, 23, 37): (1, 1, 3),
  (7, 27, 41): (2, 1, 3),
  (7, 31, 45): (3, 1, 3),
  (8, 16, 34): (0, 2, 0),
  (8, 20, 38): (1, 2, 0),
  (8, 24, 42): (2, 2, 0),
  (8, 28, 46): (3, 2, 0),
  (9, 17, 34): (0, 2, 1),
  (9, 21, 38): (1, 2, 1),
  (9, 25, 42): (2, 2, 1),
  (9, 29, 46): (3, 2, 1),
  (10, 18, 34): (0, 2, 2),
  (10, 22, 38): (1, 2, 2),
  (10, 26, 42): (2, 2, 2),
  (10, 30, 46): (3, 2, 2),
  (11, 19, 34): (0, 2, 3),
  (11, 23, 38): (1, 2, 3),
  (11, 27, 42): (2, 2, 3),
  (11, 31, 46): (3, 2, 3),
  (12, 16, 35): (0, 3, 0),
  (12, 20, 39): (1, 3, 0),
  (12, 24, 43): (2, 3, 0),
  (12, 28, 47): (3, 3, 0),
  (13, 17, 35): (0, 3, 1),
  (13, 21, 39): (1, 3, 1),
  (13, 25, 43): (2, 3, 1),
  (13, 29, 47): (3, 3, 1),
  (14, 18, 35): (0, 3, 2),
  (14, 22, 39): (1, 3, 2),
  (14, 26, 43): (2, 3, 2),
  (14, 30, 47): (3, 3, 2),
  (15, 19, 35): (0, 3, 3),
  (15, 23, 39): (1, 3, 3),
  (15, 27, 43): (2, 3, 3),
  (15, 31, 47): (3, 3, 3)})

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B = back_circulant(Integer(4))
>>> B.disjoint_mate_dlxcpp_rows_and_map(allow_subtrade = True)
([[0, 16, 32],
  [1, 17, 32],
  [2, 18, 32],
  [3, 19, 32],
  [4, 16, 33],
  [5, 17, 33],
  [6, 18, 33],
  [7, 19, 33],
  [8, 16, 34],
  [9, 17, 34],
  [10, 18, 34],
  [11, 19, 34],
  [12, 16, 35],
  [13, 17, 35],
  [14, 18, 35],
  [15, 19, 35],
  [0, 20, 36],
  [1, 21, 36],
  [2, 22, 36],
  [3, 23, 36],
  [4, 20, 37],
  [5, 21, 37],
  [6, 22, 37],
  [7, 23, 37],
  [8, 20, 38],
  [9, 21, 38],
  [10, 22, 38],
  [11, 23, 38],
  [12, 20, 39],
  [13, 21, 39],
  [14, 22, 39],
  [15, 23, 39],
  [0, 24, 40],
  [1, 25, 40],
  [2, 26, 40],
  [3, 27, 40],
  [4, 24, 41],
  [5, 25, 41],
  [6, 26, 41],
  [7, 27, 41],
  [8, 24, 42],
  [9, 25, 42],
  [10, 26, 42],
  [11, 27, 42],
  [12, 24, 43],
  [13, 25, 43],
  [14, 26, 43],
  [15, 27, 43],
  [0, 28, 44],
  [1, 29, 44],
  [2, 30, 44],
  [3, 31, 44],
  [4, 28, 45],
  [5, 29, 45],
  [6, 30, 45],
  [7, 31, 45],
  [8, 28, 46],
  [9, 29, 46],
  [10, 30, 46],
  [11, 31, 46],
  [12, 28, 47],
  [13, 29, 47],
  [14, 30, 47],
  [15, 31, 47]],
 {(0, 16, 32): (0, 0, 0),
  (0, 20, 36): (1, 0, 0),
  (0, 24, 40): (2, 0, 0),
  (0, 28, 44): (3, 0, 0),
  (1, 17, 32): (0, 0, 1),
  (1, 21, 36): (1, 0, 1),
  (1, 25, 40): (2, 0, 1),
  (1, 29, 44): (3, 0, 1),
  (2, 18, 32): (0, 0, 2),
  (2, 22, 36): (1, 0, 2),
  (2, 26, 40): (2, 0, 2),
  (2, 30, 44): (3, 0, 2),
  (3, 19, 32): (0, 0, 3),
  (3, 23, 36): (1, 0, 3),
  (3, 27, 40): (2, 0, 3),
  (3, 31, 44): (3, 0, 3),
  (4, 16, 33): (0, 1, 0),
  (4, 20, 37): (1, 1, 0),
  (4, 24, 41): (2, 1, 0),
  (4, 28, 45): (3, 1, 0),
  (5, 17, 33): (0, 1, 1),
  (5, 21, 37): (1, 1, 1),
  (5, 25, 41): (2, 1, 1),
  (5, 29, 45): (3, 1, 1),
  (6, 18, 33): (0, 1, 2),
  (6, 22, 37): (1, 1, 2),
  (6, 26, 41): (2, 1, 2),
  (6, 30, 45): (3, 1, 2),
  (7, 19, 33): (0, 1, 3),
  (7, 23, 37): (1, 1, 3),
  (7, 27, 41): (2, 1, 3),
  (7, 31, 45): (3, 1, 3),
  (8, 16, 34): (0, 2, 0),
  (8, 20, 38): (1, 2, 0),
  (8, 24, 42): (2, 2, 0),
  (8, 28, 46): (3, 2, 0),
  (9, 17, 34): (0, 2, 1),
  (9, 21, 38): (1, 2, 1),
  (9, 25, 42): (2, 2, 1),
  (9, 29, 46): (3, 2, 1),
  (10, 18, 34): (0, 2, 2),
  (10, 22, 38): (1, 2, 2),
  (10, 26, 42): (2, 2, 2),
  (10, 30, 46): (3, 2, 2),
  (11, 19, 34): (0, 2, 3),
  (11, 23, 38): (1, 2, 3),
  (11, 27, 42): (2, 2, 3),
  (11, 31, 46): (3, 2, 3),
  (12, 16, 35): (0, 3, 0),
  (12, 20, 39): (1, 3, 0),
  (12, 24, 43): (2, 3, 0),
  (12, 28, 47): (3, 3, 0),
  (13, 17, 35): (0, 3, 1),
  (13, 21, 39): (1, 3, 1),
  (13, 25, 43): (2, 3, 1),
  (13, 29, 47): (3, 3, 1),
  (14, 18, 35): (0, 3, 2),
  (14, 22, 39): (1, 3, 2),
  (14, 26, 43): (2, 3, 2),
  (14, 30, 47): (3, 3, 2),
  (15, 19, 35): (0, 3, 3),
  (15, 23, 39): (1, 3, 3),
  (15, 27, 43): (2, 3, 3),
  (15, 31, 47): (3, 3, 3)})

Sage Live

from sage.combinat.matrices.latin import *
B = back_circulant(4)
B.disjoint_mate_dlxcpp_rows_and_map(allow_subtrade = True)

dlxcpp_has_unique_completion()[source]¶

Check if the partial latin square self of order n can be embedded in precisely one latin square of order n.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: back_circulant(2).dlxcpp_has_unique_completion()
True
sage: P = LatinSquare(2)
sage: P.dlxcpp_has_unique_completion()
False
sage: P[0, 0] = 0
sage: P.dlxcpp_has_unique_completion()
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> back_circulant(Integer(2)).dlxcpp_has_unique_completion()
True
>>> P = LatinSquare(Integer(2))
>>> P.dlxcpp_has_unique_completion()
False
>>> P[Integer(0), Integer(0)] = Integer(0)
>>> P.dlxcpp_has_unique_completion()
True

Sage Live

from sage.combinat.matrices.latin import *
back_circulant(2).dlxcpp_has_unique_completion()
P = LatinSquare(2)
P.dlxcpp_has_unique_completion()
P[0, 0] = 0
P.dlxcpp_has_unique_completion()

dumps()[source]¶

Since the latin square class does not hold any other private variables we just call dumps on self.square:

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: back_circulant(2) == loads(dumps(back_circulant(2)))
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> back_circulant(Integer(2)) == loads(dumps(back_circulant(Integer(2))))
True

Sage Live

from sage.combinat.matrices.latin import *
back_circulant(2) == loads(dumps(back_circulant(2)))

filled_cells_map()[source]¶

Number the filled cells of self with integers from {1, 2, 3, …}.

INPUT:

self – partial latin square self (empty cells have negative values)

OUTPUT:

A dictionary cells_map where cells_map[(i,j)] = m means that (i,j) is the m-th filled cell in P, while cells_map[m] = (i,j).

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: (a, b, c, G) = alternating_group_bitrade_generators(1)
sage: (T1, T2) = bitrade_from_group(a, b, c, G)
sage: D = T1.filled_cells_map()
sage: {i: v for i,v in D.items() if i in ZZ}
{1: (0, 0),
 2: (0, 2),
 3: (0, 3),
 4: (1, 1),
 5: (1, 2),
 6: (1, 3),
 7: (2, 0),
 8: (2, 1),
 9: (2, 2),
 10: (3, 0),
 11: (3, 1),
 12: (3, 3)}
sage: {i: v for i,v in D.items() if i not in ZZ}
{(0, 0): 1,
 (0, 2): 2,
 (0, 3): 3,
 (1, 1): 4,
 (1, 2): 5,
 (1, 3): 6,
 (2, 0): 7,
 (2, 1): 8,
 (2, 2): 9,
 (3, 0): 10,
 (3, 1): 11,
 (3, 3): 12}

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> (a, b, c, G) = alternating_group_bitrade_generators(Integer(1))
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> D = T1.filled_cells_map()
>>> {i: v for i,v in D.items() if i in ZZ}
{1: (0, 0),
 2: (0, 2),
 3: (0, 3),
 4: (1, 1),
 5: (1, 2),
 6: (1, 3),
 7: (2, 0),
 8: (2, 1),
 9: (2, 2),
 10: (3, 0),
 11: (3, 1),
 12: (3, 3)}
>>> {i: v for i,v in D.items() if i not in ZZ}
{(0, 0): 1,
 (0, 2): 2,
 (0, 3): 3,
 (1, 1): 4,
 (1, 2): 5,
 (1, 3): 6,
 (2, 0): 7,
 (2, 1): 8,
 (2, 2): 9,
 (3, 0): 10,
 (3, 1): 11,
 (3, 3): 12}

Sage Live

from sage.combinat.matrices.latin import *
(a, b, c, G) = alternating_group_bitrade_generators(1)
(T1, T2) = bitrade_from_group(a, b, c, G)
D = T1.filled_cells_map()
{i: v for i,v in D.items() if i in ZZ}
{i: v for i,v in D.items() if i not in ZZ}

find_disjoint_mates(nr_to_find=None, allow_subtrade=False)[source]¶

Warning

If allow_subtrade is True then we may return a partial latin square that is not disjoint to self. In that case, use bitrade(P, Q) to get an actual bitrade.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B = back_circulant(4)
sage: g = B.find_disjoint_mates(allow_subtrade = True)
sage: B1 = next(g)
sage: B0, B1 = bitrade(B, B1)
sage: assert is_bitrade(B0, B1)
sage: print(B0)
[-1  1  2 -1]
[-1  2 -1  0]
[-1 -1 -1 -1]
[-1  0  1  2]
sage: print(B1)
[-1  2  1 -1]
[-1  0 -1  2]
[-1 -1 -1 -1]
[-1  1  2  0]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B = back_circulant(Integer(4))
>>> g = B.find_disjoint_mates(allow_subtrade = True)
>>> B1 = next(g)
>>> B0, B1 = bitrade(B, B1)
>>> assert is_bitrade(B0, B1)
>>> print(B0)
[-1  1  2 -1]
[-1  2 -1  0]
[-1 -1 -1 -1]
[-1  0  1  2]
>>> print(B1)
[-1  2  1 -1]
[-1  0 -1  2]
[-1 -1 -1 -1]
[-1  1  2  0]

Sage Live

from sage.combinat.matrices.latin import *
B = back_circulant(4)
g = B.find_disjoint_mates(allow_subtrade = True)
B1 = next(g)
B0, B1 = bitrade(B, B1)
assert is_bitrade(B0, B1)
print(B0)
print(B1)

gcs()[source]¶

A greedy critical set of a latin square self is found by successively removing elements in a row-wise (bottom-up) manner, checking for unique completion at each step.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: A = elementary_abelian_2group(3)
sage: G = A.gcs()
sage: A
[0 1 2 3 4 5 6 7]
[1 0 3 2 5 4 7 6]
[2 3 0 1 6 7 4 5]
[3 2 1 0 7 6 5 4]
[4 5 6 7 0 1 2 3]
[5 4 7 6 1 0 3 2]
[6 7 4 5 2 3 0 1]
[7 6 5 4 3 2 1 0]
sage: G
[ 0  1  2  3  4  5  6 -1]
[ 1  0  3  2  5  4 -1 -1]
[ 2  3  0  1  6 -1  4 -1]
[ 3  2  1  0 -1 -1 -1 -1]
[ 4  5  6 -1  0  1  2 -1]
[ 5  4 -1 -1  1  0 -1 -1]
[ 6 -1  4 -1  2 -1  0 -1]
[-1 -1 -1 -1 -1 -1 -1 -1]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> A = elementary_abelian_2group(Integer(3))
>>> G = A.gcs()
>>> A
[0 1 2 3 4 5 6 7]
[1 0 3 2 5 4 7 6]
[2 3 0 1 6 7 4 5]
[3 2 1 0 7 6 5 4]
[4 5 6 7 0 1 2 3]
[5 4 7 6 1 0 3 2]
[6 7 4 5 2 3 0 1]
[7 6 5 4 3 2 1 0]
>>> G
[ 0  1  2  3  4  5  6 -1]
[ 1  0  3  2  5  4 -1 -1]
[ 2  3  0  1  6 -1  4 -1]
[ 3  2  1  0 -1 -1 -1 -1]
[ 4  5  6 -1  0  1  2 -1]
[ 5  4 -1 -1  1  0 -1 -1]
[ 6 -1  4 -1  2 -1  0 -1]
[-1 -1 -1 -1 -1 -1 -1 -1]

Sage Live

from sage.combinat.matrices.latin import *
A = elementary_abelian_2group(3)
G = A.gcs()
A
G

is_completable()[source]¶

Return True if the partial latin square can be completed to a latin square.

EXAMPLES:

The following partial latin square has no completion because there is nowhere that we can place the symbol 0 in the third row:

Sage

sage: B = LatinSquare(3)

Python

>>> from sage.all import *
>>> B = LatinSquare(Integer(3))

Sage Live

B = LatinSquare(3)

Sage

sage: B[0, 0] = 0
sage: B[1, 1] = 0
sage: B[2, 2] = 1

Python

>>> from sage.all import *
>>> B[Integer(0), Integer(0)] = Integer(0)
>>> B[Integer(1), Integer(1)] = Integer(0)
>>> B[Integer(2), Integer(2)] = Integer(1)

Sage Live

B[0, 0] = 0
B[1, 1] = 0
B[2, 2] = 1

Python

>>> from sage.all import *
>>> B[Integer(0), Integer(0)] = Integer(0)
>>> B[Integer(1), Integer(1)] = Integer(0)
>>> B[Integer(2), Integer(2)] = Integer(1)

Sage Live

B[0, 0] = 0
B[1, 1] = 0
B[2, 2] = 1

Sage

sage: B
[ 0 -1 -1]
[-1  0 -1]
[-1 -1  1]

Python

>>> from sage.all import *
>>> B
[ 0 -1 -1]
[-1  0 -1]
[-1 -1  1]

Sage Live

Python

>>> from sage.all import *
>>> B
[ 0 -1 -1]
[-1  0 -1]
[-1 -1  1]

Sage Live

Sage

sage: B.is_completable()
False

Python

>>> from sage.all import *
>>> B.is_completable()
False

Sage Live

B.is_completable()

Python

>>> from sage.all import *
>>> B.is_completable()
False

Sage Live

B.is_completable()

Sage

sage: B[2, 2] = 0
sage: B.is_completable()
True

Python

>>> from sage.all import *
>>> B[Integer(2), Integer(2)] = Integer(0)
>>> B.is_completable()
True

Sage Live

B[2, 2] = 0
B.is_completable()

Python

>>> from sage.all import *
>>> B[Integer(2), Integer(2)] = Integer(0)
>>> B.is_completable()
True

Sage Live

B[2, 2] = 0
B.is_completable()

is_empty_column(c)[source]¶

Check if column c of the partial latin square self is empty.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: L = back_circulant(4)
sage: L.is_empty_column(0)
False
sage: L[0,0] = L[1,0] = L[2,0] = L[3,0] = -1
sage: L.is_empty_column(0)
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> L = back_circulant(Integer(4))
>>> L.is_empty_column(Integer(0))
False
>>> L[Integer(0),Integer(0)] = L[Integer(1),Integer(0)] = L[Integer(2),Integer(0)] = L[Integer(3),Integer(0)] = -Integer(1)
>>> L.is_empty_column(Integer(0))
True

Sage Live

from sage.combinat.matrices.latin import *
L = back_circulant(4)
L.is_empty_column(0)
L[0,0] = L[1,0] = L[2,0] = L[3,0] = -1
L.is_empty_column(0)

is_empty_row(r)[source]¶

Check if row r of the partial latin square self is empty.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: L = back_circulant(4)
sage: L.is_empty_row(0)
False
sage: L[0,0] = L[0,1] = L[0,2] = L[0,3] = -1
sage: L.is_empty_row(0)
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> L = back_circulant(Integer(4))
>>> L.is_empty_row(Integer(0))
False
>>> L[Integer(0),Integer(0)] = L[Integer(0),Integer(1)] = L[Integer(0),Integer(2)] = L[Integer(0),Integer(3)] = -Integer(1)
>>> L.is_empty_row(Integer(0))
True

Sage Live

from sage.combinat.matrices.latin import *
L = back_circulant(4)
L.is_empty_row(0)
L[0,0] = L[0,1] = L[0,2] = L[0,3] = -1
L.is_empty_row(0)

is_latin_square()[source]¶

self is a latin square if it is an n by n matrix, and each symbol in [0, 1, …, n-1] appears exactly once in each row, and exactly once in each column.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: elementary_abelian_2group(4).is_latin_square()
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> elementary_abelian_2group(Integer(4)).is_latin_square()
True

Sage Live

from sage.combinat.matrices.latin import *
elementary_abelian_2group(4).is_latin_square()

Sage

sage: forward_circulant(7).is_latin_square()
True

Python

>>> from sage.all import *
>>> forward_circulant(Integer(7)).is_latin_square()
True

Sage Live

forward_circulant(7).is_latin_square()

Python

>>> from sage.all import *
>>> forward_circulant(Integer(7)).is_latin_square()
True

Sage Live

forward_circulant(7).is_latin_square()

is_partial_latin_square()[source]¶

self is a partial latin square if it is an n by n matrix, and each symbol in [0, 1, …, n-1] appears at most once in each row, and at most once in each column.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: LatinSquare(4).is_partial_latin_square()
True
sage: back_circulant(3).gcs().is_partial_latin_square()
True
sage: back_circulant(6).is_partial_latin_square()
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> LatinSquare(Integer(4)).is_partial_latin_square()
True
>>> back_circulant(Integer(3)).gcs().is_partial_latin_square()
True
>>> back_circulant(Integer(6)).is_partial_latin_square()
True

Sage Live

from sage.combinat.matrices.latin import *
LatinSquare(4).is_partial_latin_square()
back_circulant(3).gcs().is_partial_latin_square()
back_circulant(6).is_partial_latin_square()

is_uniquely_completable()[source]¶

Return True if the partial latin square self has exactly one completion to a latin square. This is just a wrapper for the current best-known algorithm, Dancing Links by Knuth. See dancing_links.spyx

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: back_circulant(4).gcs().is_uniquely_completable()
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> back_circulant(Integer(4)).gcs().is_uniquely_completable()
True

Sage Live

from sage.combinat.matrices.latin import *
back_circulant(4).gcs().is_uniquely_completable()

Sage

sage: G = elementary_abelian_2group(3).gcs()
sage: G.is_uniquely_completable()
True

Python

>>> from sage.all import *
>>> G = elementary_abelian_2group(Integer(3)).gcs()
>>> G.is_uniquely_completable()
True

Sage Live

G = elementary_abelian_2group(3).gcs()
G.is_uniquely_completable()

Python

>>> from sage.all import *
>>> G = elementary_abelian_2group(Integer(3)).gcs()
>>> G.is_uniquely_completable()
True

Sage Live

G = elementary_abelian_2group(3).gcs()
G.is_uniquely_completable()

Sage

sage: G[0, 0] = -1
sage: G.is_uniquely_completable()
False

Python

>>> from sage.all import *
>>> G[Integer(0), Integer(0)] = -Integer(1)
>>> G.is_uniquely_completable()
False

Sage Live

G[0, 0] = -1
G.is_uniquely_completable()

Python

>>> from sage.all import *
>>> G[Integer(0), Integer(0)] = -Integer(1)
>>> G.is_uniquely_completable()
False

Sage Live

G[0, 0] = -1
G.is_uniquely_completable()

latex()[source]¶

Return LaTeX code for the latin square.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: print(back_circulant(3).latex())
\begin{array}{|c|c|c|}\hline 0 & 1 & 2\\\hline 1 & 2 & 0\\\hline 2 & 0 & 1\\\hline\end{array}

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> print(back_circulant(Integer(3)).latex())
\begin{array}{|c|c|c|}\hline 0 & 1 & 2\\\hline 1 & 2 & 0\\\hline 2 & 0 & 1\\\hline\end{array}

Sage Live

from sage.combinat.matrices.latin import *
print(back_circulant(3).latex())

list()[source]¶

Convert the latin square into a list, in a row-wise manner.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: back_circulant(3).list()
[0, 1, 2, 1, 2, 0, 2, 0, 1]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> back_circulant(Integer(3)).list()
[0, 1, 2, 1, 2, 0, 2, 0, 1]

Sage Live

from sage.combinat.matrices.latin import *
back_circulant(3).list()

ncols()[source]¶

Number of columns in the latin square.

EXAMPLES:

Sage

sage: LatinSquare(3).ncols()
3

Python

>>> from sage.all import *
>>> LatinSquare(Integer(3)).ncols()
3

Sage Live

LatinSquare(3).ncols()

nr_distinct_symbols()[source]¶

Return the number of distinct symbols in the partial latin square self.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: back_circulant(5).nr_distinct_symbols()
5
sage: L = LatinSquare(10)
sage: L.nr_distinct_symbols()
0
sage: L[0, 0] = 0
sage: L[0, 1] = 1
sage: L.nr_distinct_symbols()
2

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> back_circulant(Integer(5)).nr_distinct_symbols()
5
>>> L = LatinSquare(Integer(10))
>>> L.nr_distinct_symbols()
0
>>> L[Integer(0), Integer(0)] = Integer(0)
>>> L[Integer(0), Integer(1)] = Integer(1)
>>> L.nr_distinct_symbols()
2

Sage Live

from sage.combinat.matrices.latin import *
back_circulant(5).nr_distinct_symbols()
L = LatinSquare(10)
L.nr_distinct_symbols()
L[0, 0] = 0
L[0, 1] = 1
L.nr_distinct_symbols()

nr_filled_cells()[source]¶

Return the number of filled cells (i.e. cells with a positive value) in the partial latin square self.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: LatinSquare(matrix([[0, -1], [-1, 0]])).nr_filled_cells()
2

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> LatinSquare(matrix([[Integer(0), -Integer(1)], [-Integer(1), Integer(0)]])).nr_filled_cells()
2

Sage Live

from sage.combinat.matrices.latin import *
LatinSquare(matrix([[0, -1], [-1, 0]])).nr_filled_cells()

nrows()[source]¶

Number of rows in the latin square.

EXAMPLES:

Sage

sage: LatinSquare(3).nrows()
3

Python

>>> from sage.all import *
>>> LatinSquare(Integer(3)).nrows()
3

Sage Live

LatinSquare(3).nrows()

permissable_values(r, c)[source]¶

Find all values that do not appear in row r and column c of the latin square self. If self[r, c] is filled then we return the empty list.

INPUT:

self – LatinSquare
r – integer; row of the latin square
c – integer; column of the latin square

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: L = back_circulant(5)
sage: L[0, 0] = -1
sage: L.permissable_values(0, 0)
[0]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> L = back_circulant(Integer(5))
>>> L[Integer(0), Integer(0)] = -Integer(1)
>>> L.permissable_values(Integer(0), Integer(0))
[0]

Sage Live

from sage.combinat.matrices.latin import *
L = back_circulant(5)
L[0, 0] = -1
L.permissable_values(0, 0)

random_empty_cell()[source]¶

Find an empty cell of self, uniformly at random.

INPUT:

self – LatinSquare

OUTPUT:

[r, c] – cell such that self[r, c] is empty, or returns None if self is a (full) latin square

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: P = back_circulant(2)
sage: P[1,1] = -1
sage: P.random_empty_cell()
[1, 1]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> P = back_circulant(Integer(2))
>>> P[Integer(1),Integer(1)] = -Integer(1)
>>> P.random_empty_cell()
[1, 1]

Sage Live

from sage.combinat.matrices.latin import *
P = back_circulant(2)
P[1,1] = -1
P.random_empty_cell()

row(x)[source]¶

Return row x of the latin square.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: back_circulant(3).row(0)
(0, 1, 2)

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> back_circulant(Integer(3)).row(Integer(0))
(0, 1, 2)

Sage Live

from sage.combinat.matrices.latin import *
back_circulant(3).row(0)

set_immutable()[source]¶

A latin square is immutable if the underlying matrix is immutable.

EXAMPLES:

Sage

sage: L = LatinSquare(matrix(ZZ, [[0, 1], [2, 3]]))
sage: L.set_immutable()
sage: {L : 0}   # this would fail without set_immutable()
{[0 1]
[2 3]: 0}

Python

>>> from sage.all import *
>>> L = LatinSquare(matrix(ZZ, [[Integer(0), Integer(1)], [Integer(2), Integer(3)]]))
>>> L.set_immutable()
>>> {L : Integer(0)}   # this would fail without set_immutable()
{[0 1]
[2 3]: 0}

Sage Live

L = LatinSquare(matrix(ZZ, [[0, 1], [2, 3]]))
L.set_immutable()
{L : 0}   # this would fail without set_immutable()

top_left_empty_cell()[source]¶

Return the least [r, c] such that self[r, c] is an empty cell. If all cells are filled then we return None.

INPUT:

self – LatinSquare

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B = back_circulant(5)
sage: B[3, 4] = -1
sage: B.top_left_empty_cell()
[3, 4]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B = back_circulant(Integer(5))
>>> B[Integer(3), Integer(4)] = -Integer(1)
>>> B.top_left_empty_cell()
[3, 4]

Sage Live

from sage.combinat.matrices.latin import *
B = back_circulant(5)
B[3, 4] = -1
B.top_left_empty_cell()

vals_in_col(c)[source]¶

Return a dictionary with key e if and only if column c of self has the symbol e.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B = back_circulant(3)
sage: B[0, 0] = -1
sage: back_circulant(3).vals_in_col(0)
{0: True, 1: True, 2: True}

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B = back_circulant(Integer(3))
>>> B[Integer(0), Integer(0)] = -Integer(1)
>>> back_circulant(Integer(3)).vals_in_col(Integer(0))
{0: True, 1: True, 2: True}

Sage Live

from sage.combinat.matrices.latin import *
B = back_circulant(3)
B[0, 0] = -1
back_circulant(3).vals_in_col(0)

vals_in_row(r)[source]¶

Return a dictionary with key e if and only if row r of self has the symbol e.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B = back_circulant(3)
sage: B[0, 0] = -1
sage: back_circulant(3).vals_in_row(0)
{0: True, 1: True, 2: True}

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B = back_circulant(Integer(3))
>>> B[Integer(0), Integer(0)] = -Integer(1)
>>> back_circulant(Integer(3)).vals_in_row(Integer(0))
{0: True, 1: True, 2: True}

Sage Live

from sage.combinat.matrices.latin import *
B = back_circulant(3)
B[0, 0] = -1
back_circulant(3).vals_in_row(0)

sage.combinat.matrices.latin.LatinSquare_generator(L_start, check_assertions=False)[source]¶

Generator for a sequence of uniformly distributed latin squares, given L_start as the initial latin square.

This code implements the Markov chain algorithm of Jacobson and Matthews (1996), see below for the BibTex entry. This generator will never throw the StopIteration exception, so it provides an infinite sequence of latin squares.

EXAMPLES:

Use the back circulant latin square of order 4 as the initial square and print the next two latin squares given by the Markov chain:

Sage

sage: from sage.combinat.matrices.latin import *
sage: g = LatinSquare_generator(back_circulant(4))
sage: next(g).is_latin_square()
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> g = LatinSquare_generator(back_circulant(Integer(4)))
>>> next(g).is_latin_square()
True

Sage Live

from sage.combinat.matrices.latin import *
g = LatinSquare_generator(back_circulant(4))
next(g).is_latin_square()

REFERENCES:

[JacMat96]

Mark T. Jacobson and Peter Matthews, “Generating uniformly distributed random Latin squares”, Journal of Combinatorial Designs, 4 (1996)

sage.combinat.matrices.latin.alternating_group_bitrade_generators(m)[source]¶

Construct generators a, b, c for the alternating group on 3m+1 points, such that a*b*c = 1.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: a, b, c, G = alternating_group_bitrade_generators(1)
sage: (a, b, c, G)
((1,2,3), (1,4,2), (2,4,3), Permutation Group with generators [(1,2,3), (1,4,2)])
sage: a*b*c
()

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> a, b, c, G = alternating_group_bitrade_generators(Integer(1))
>>> (a, b, c, G)
((1,2,3), (1,4,2), (2,4,3), Permutation Group with generators [(1,2,3), (1,4,2)])
>>> a*b*c
()

Sage Live

from sage.combinat.matrices.latin import *
a, b, c, G = alternating_group_bitrade_generators(1)
(a, b, c, G)
a*b*c

Sage

sage: (T1, T2) = bitrade_from_group(a, b, c, G)
sage: T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]
sage: T2
[ 1 -1  0  3]
[-1  0  2  1]
[ 2  1  3 -1]
[ 0  3 -1  2]

Python

>>> from sage.all import *
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]
>>> T2
[ 1 -1  0  3]
[-1  0  2  1]
[ 2  1  3 -1]
[ 0  3 -1  2]

Sage Live

(T1, T2) = bitrade_from_group(a, b, c, G)
T1
T2

Python

>>> from sage.all import *
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]
>>> T2
[ 1 -1  0  3]
[-1  0  2  1]
[ 2  1  3 -1]
[ 0  3 -1  2]

Sage Live

(T1, T2) = bitrade_from_group(a, b, c, G)
T1
T2

sage.combinat.matrices.latin.back_circulant(n)[source]¶

The back-circulant latin square of order n is the Cayley table for (Z_n, +), the integers under addition modulo n.

INPUT:

n – integer; order of the latin square

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: back_circulant(5)
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]
[4 0 1 2 3]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> back_circulant(Integer(5))
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]
[4 0 1 2 3]

Sage Live

from sage.combinat.matrices.latin import *
back_circulant(5)

sage.combinat.matrices.latin.beta1(rce, T1, T2)[source]¶

Find the unique (x, c, e) in T2 such that (r, c, e) is in T1.

INPUT:

rce – tuple (or list) (r, c, e) in T1
T1, T2 – latin bitrade

OUTPUT: (x, c, e) in T2

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: is_bitrade(T1, T2)
True
sage: beta1([0, 0, 0], T1, T2)
(1, 0, 0)

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_bitrade(T1, T2)
True
>>> beta1([Integer(0), Integer(0), Integer(0)], T1, T2)
(1, 0, 0)

Sage Live

from sage.combinat.matrices.latin import *
T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_bitrade(T1, T2)
beta1([0, 0, 0], T1, T2)

sage.combinat.matrices.latin.beta2(rce, T1, T2)[source]¶

Find the unique (r, x, e) in T2 such that (r, c, e) is in T1.

INPUT:

rce – tuple (or list) (r, c, e) in T1
T1, T2 – latin bitrade

OUTPUT: (r, x, e) in T2

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: is_bitrade(T1, T2)
True
sage: beta2([0, 0, 0], T1, T2)
(0, 1, 0)

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_bitrade(T1, T2)
True
>>> beta2([Integer(0), Integer(0), Integer(0)], T1, T2)
(0, 1, 0)

Sage Live

from sage.combinat.matrices.latin import *
T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_bitrade(T1, T2)
beta2([0, 0, 0], T1, T2)

sage.combinat.matrices.latin.beta3(rce, T1, T2)[source]¶

Find the unique (r, c, x) in T2 such that (r, c, e) is in T1.

INPUT:

rce – tuple (or list) (r, c, e) in T1
T1, T2 – latin bitrade

OUTPUT: (r, c, x) in T2.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: is_bitrade(T1, T2)
True
sage: beta3([0, 0, 0], T1, T2)
(0, 0, 4)

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_bitrade(T1, T2)
True
>>> beta3([Integer(0), Integer(0), Integer(0)], T1, T2)
(0, 0, 4)

Sage Live

from sage.combinat.matrices.latin import *
T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_bitrade(T1, T2)
beta3([0, 0, 0], T1, T2)

sage.combinat.matrices.latin.bitrade(T1, T2)[source]¶

Form the bitrade (Q1, Q2) from (T1, T2) by setting empty the cells (r, c) such that T1[r, c] == T2[r, c].

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: B1 = back_circulant(5)
sage: alpha = isotopism((0,1,2,3,4))
sage: beta  = isotopism((1,0,2,3,4))
sage: gamma = isotopism((2,1,0,3,4))
sage: B2 = B1.apply_isotopism(alpha, beta, gamma)
sage: T1, T2 = bitrade(B1, B2)
sage: T1
[ 0  1 -1  3  4]
[ 1 -1 -1  4  0]
[ 2 -1  4  0  1]
[ 3  4  0  1  2]
[ 4  0  1  2  3]
sage: T2
[ 3  4 -1  0  1]
[ 0 -1 -1  1  4]
[ 1 -1  0  4  2]
[ 4  0  1  2  3]
[ 2  1  4  3  0]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> B1 = back_circulant(Integer(5))
>>> alpha = isotopism((Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)))
>>> beta  = isotopism((Integer(1),Integer(0),Integer(2),Integer(3),Integer(4)))
>>> gamma = isotopism((Integer(2),Integer(1),Integer(0),Integer(3),Integer(4)))
>>> B2 = B1.apply_isotopism(alpha, beta, gamma)
>>> T1, T2 = bitrade(B1, B2)
>>> T1
[ 0  1 -1  3  4]
[ 1 -1 -1  4  0]
[ 2 -1  4  0  1]
[ 3  4  0  1  2]
[ 4  0  1  2  3]
>>> T2
[ 3  4 -1  0  1]
[ 0 -1 -1  1  4]
[ 1 -1  0  4  2]
[ 4  0  1  2  3]
[ 2  1  4  3  0]

Sage Live

from sage.combinat.matrices.latin import *
B1 = back_circulant(5)
alpha = isotopism((0,1,2,3,4))
beta  = isotopism((1,0,2,3,4))
gamma = isotopism((2,1,0,3,4))
B2 = B1.apply_isotopism(alpha, beta, gamma)
T1, T2 = bitrade(B1, B2)
T1
T2

sage.combinat.matrices.latin.bitrade_from_group(a, b, c, G)[source]¶

Given group elements a, b, c in G such that abc = 1 and the subgroups a, b, c intersect (pairwise) only in the identity, construct a bitrade (T1, T2) where rows, columns, and symbols correspond to cosets of a, b, and c, respectively.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: a, b, c, G = alternating_group_bitrade_generators(1)
sage: (T1, T2) = bitrade_from_group(a, b, c, G)
sage: T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]
sage: T2
[ 1 -1  0  3]
[-1  0  2  1]
[ 2  1  3 -1]
[ 0  3 -1  2]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> a, b, c, G = alternating_group_bitrade_generators(Integer(1))
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]
>>> T2
[ 1 -1  0  3]
[-1  0  2  1]
[ 2  1  3 -1]
[ 0  3 -1  2]

Sage Live

from sage.combinat.matrices.latin import *
a, b, c, G = alternating_group_bitrade_generators(1)
(T1, T2) = bitrade_from_group(a, b, c, G)
T1
T2

sage.combinat.matrices.latin.cells_map_as_square(cells_map, n)[source]¶

Return a LatinSquare with cells numbered from 1, 2, … to given the dictionary cells_map.

Note

The value n should be the maximum of the number of rows and columns of the original partial latin square

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: (a, b, c, G) = alternating_group_bitrade_generators(1)
sage: (T1, T2) = bitrade_from_group(a, b, c, G)
sage: T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> (a, b, c, G) = alternating_group_bitrade_generators(Integer(1))
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> T1
[ 0 -1  3  1]
[-1  1  0  2]
[ 1  3  2 -1]
[ 2  0 -1  3]

Sage Live

from sage.combinat.matrices.latin import *
(a, b, c, G) = alternating_group_bitrade_generators(1)
(T1, T2) = bitrade_from_group(a, b, c, G)
T1

There are 12 filled cells in T:

Sage

sage: cells_map_as_square(T1.filled_cells_map(), max(T1.nrows(), T1.ncols()))
[ 1 -1  2  3]
[-1  4  5  6]
[ 7  8  9 -1]
[10 11 -1 12]

Python

>>> from sage.all import *
>>> cells_map_as_square(T1.filled_cells_map(), max(T1.nrows(), T1.ncols()))
[ 1 -1  2  3]
[-1  4  5  6]
[ 7  8  9 -1]
[10 11 -1 12]

Sage Live

cells_map_as_square(T1.filled_cells_map(), max(T1.nrows(), T1.ncols()))

sage.combinat.matrices.latin.check_bitrade_generators(a, b, c)[source]¶

Three group elements a, b, c will generate a bitrade if a*b*c = 1 and the subgroups a, b, c intersect (pairwise) in just the identity.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: a, b, c, G = p3_group_bitrade_generators(3)
sage: check_bitrade_generators(a, b, c)
True
sage: check_bitrade_generators(a, b, libgap(gap('()')))
False

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> a, b, c, G = p3_group_bitrade_generators(Integer(3))
>>> check_bitrade_generators(a, b, c)
True
>>> check_bitrade_generators(a, b, libgap(gap('()')))
False

Sage Live

from sage.combinat.matrices.latin import *
a, b, c, G = p3_group_bitrade_generators(3)
check_bitrade_generators(a, b, c)
check_bitrade_generators(a, b, libgap(gap('()')))

sage.combinat.matrices.latin.coin()[source]¶

Simulate a fair coin (returns True or False) using ZZ.random_element(2).

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import coin
sage: x = coin()
sage: x == 0 or x == 1
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import coin
>>> x = coin()
>>> x == Integer(0) or x == Integer(1)
True

Sage Live

from sage.combinat.matrices.latin import coin
x = coin()
x == 0 or x == 1

sage.combinat.matrices.latin.column_containing_sym(L, r, x)[source]¶

Given an improper latin square L with L[r, c1] = L[r, c2] = x, return c1 or c2 with equal probability. This is an internal function and should only be used in LatinSquare_generator().

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: L = matrix([(1, 0, 2, 3), (0, 2, 3, 0), (2, 3, 0, 1), (3, 0, 1, 2)])
sage: L
[1 0 2 3]
[0 2 3 0]
[2 3 0 1]
[3 0 1 2]
sage: c = column_containing_sym(L, 1, 0)
sage: c == 0 or c == 3
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> L = matrix([(Integer(1), Integer(0), Integer(2), Integer(3)), (Integer(0), Integer(2), Integer(3), Integer(0)), (Integer(2), Integer(3), Integer(0), Integer(1)), (Integer(3), Integer(0), Integer(1), Integer(2))])
>>> L
[1 0 2 3]
[0 2 3 0]
[2 3 0 1]
[3 0 1 2]
>>> c = column_containing_sym(L, Integer(1), Integer(0))
>>> c == Integer(0) or c == Integer(3)
True

Sage Live

from sage.combinat.matrices.latin import *
L = matrix([(1, 0, 2, 3), (0, 2, 3, 0), (2, 3, 0, 1), (3, 0, 1, 2)])
L
c = column_containing_sym(L, 1, 0)
c == 0 or c == 3

sage.combinat.matrices.latin.direct_product(L1, L2, L3, L4)[source]¶

The ‘direct product’ of four latin squares L1, L2, L3, L4 of order n is the latin square of order 2n consisting of

-----------
| L1 | L2 |
-----------
| L3 | L4 |
-----------

where the subsquares L2 and L3 have entries offset by n.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: direct_product(back_circulant(4), back_circulant(4), elementary_abelian_2group(2), elementary_abelian_2group(2))
[0 1 2 3 4 5 6 7]
[1 2 3 0 5 6 7 4]
[2 3 0 1 6 7 4 5]
[3 0 1 2 7 4 5 6]
[4 5 6 7 0 1 2 3]
[5 4 7 6 1 0 3 2]
[6 7 4 5 2 3 0 1]
[7 6 5 4 3 2 1 0]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> direct_product(back_circulant(Integer(4)), back_circulant(Integer(4)), elementary_abelian_2group(Integer(2)), elementary_abelian_2group(Integer(2)))
[0 1 2 3 4 5 6 7]
[1 2 3 0 5 6 7 4]
[2 3 0 1 6 7 4 5]
[3 0 1 2 7 4 5 6]
[4 5 6 7 0 1 2 3]
[5 4 7 6 1 0 3 2]
[6 7 4 5 2 3 0 1]
[7 6 5 4 3 2 1 0]

Sage Live

from sage.combinat.matrices.latin import *
direct_product(back_circulant(4), back_circulant(4), elementary_abelian_2group(2), elementary_abelian_2group(2))

sage.combinat.matrices.latin.dlxcpp_find_completions(P, nr_to_find=None)[source]¶

Return a list of all latin squares L of the same order as P such that P is contained in L. The optional parameter nr_to_find limits the number of latin squares that are found.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: dlxcpp_find_completions(LatinSquare(2))
[[0 1]
[1 0], [1 0]
[0 1]]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> dlxcpp_find_completions(LatinSquare(Integer(2)))
[[0 1]
[1 0], [1 0]
[0 1]]

Sage Live

from sage.combinat.matrices.latin import *
dlxcpp_find_completions(LatinSquare(2))

Sage

sage: dlxcpp_find_completions(LatinSquare(2), 1)
[[0 1]
[1 0]]

Python

>>> from sage.all import *
>>> dlxcpp_find_completions(LatinSquare(Integer(2)), Integer(1))
[[0 1]
[1 0]]

Sage Live

dlxcpp_find_completions(LatinSquare(2), 1)

Python

>>> from sage.all import *
>>> dlxcpp_find_completions(LatinSquare(Integer(2)), Integer(1))
[[0 1]
[1 0]]

Sage Live

dlxcpp_find_completions(LatinSquare(2), 1)

sage.combinat.matrices.latin.dlxcpp_rows_and_map(P)[source]¶

Internal function for dlxcpp_find_completions. Given a partial latin square P we construct a list of rows of a 0-1 matrix M such that an exact cover of M corresponds to a completion of P to a latin square.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: dlxcpp_rows_and_map(LatinSquare(2))
([[0, 4, 8],
  [1, 5, 8],
  [2, 4, 9],
  [3, 5, 9],
  [0, 6, 10],
  [1, 7, 10],
  [2, 6, 11],
  [3, 7, 11]],
 {(0, 4, 8): (0, 0, 0),
  (0, 6, 10): (1, 0, 0),
  (1, 5, 8): (0, 0, 1),
  (1, 7, 10): (1, 0, 1),
  (2, 4, 9): (0, 1, 0),
  (2, 6, 11): (1, 1, 0),
  (3, 5, 9): (0, 1, 1),
  (3, 7, 11): (1, 1, 1)})

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> dlxcpp_rows_and_map(LatinSquare(Integer(2)))
([[0, 4, 8],
  [1, 5, 8],
  [2, 4, 9],
  [3, 5, 9],
  [0, 6, 10],
  [1, 7, 10],
  [2, 6, 11],
  [3, 7, 11]],
 {(0, 4, 8): (0, 0, 0),
  (0, 6, 10): (1, 0, 0),
  (1, 5, 8): (0, 0, 1),
  (1, 7, 10): (1, 0, 1),
  (2, 4, 9): (0, 1, 0),
  (2, 6, 11): (1, 1, 0),
  (3, 5, 9): (0, 1, 1),
  (3, 7, 11): (1, 1, 1)})

Sage Live

from sage.combinat.matrices.latin import *
dlxcpp_rows_and_map(LatinSquare(2))

sage.combinat.matrices.latin.elementary_abelian_2group(s)[source]¶

Return the latin square based on the Cayley table for the elementary abelian 2-group of order 2s.

INPUT:

s – integer; order of the latin square will be 2s

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: elementary_abelian_2group(3)
[0 1 2 3 4 5 6 7]
[1 0 3 2 5 4 7 6]
[2 3 0 1 6 7 4 5]
[3 2 1 0 7 6 5 4]
[4 5 6 7 0 1 2 3]
[5 4 7 6 1 0 3 2]
[6 7 4 5 2 3 0 1]
[7 6 5 4 3 2 1 0]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> elementary_abelian_2group(Integer(3))
[0 1 2 3 4 5 6 7]
[1 0 3 2 5 4 7 6]
[2 3 0 1 6 7 4 5]
[3 2 1 0 7 6 5 4]
[4 5 6 7 0 1 2 3]
[5 4 7 6 1 0 3 2]
[6 7 4 5 2 3 0 1]
[7 6 5 4 3 2 1 0]

Sage Live

from sage.combinat.matrices.latin import *
elementary_abelian_2group(3)

sage.combinat.matrices.latin.forward_circulant(n)[source]¶

The forward-circulant latin square of order n is the Cayley table for the operation r + c = (n-c+r) mod n.

INPUT:

n – integer; order of the latin square

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: forward_circulant(5)
[0 4 3 2 1]
[1 0 4 3 2]
[2 1 0 4 3]
[3 2 1 0 4]
[4 3 2 1 0]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> forward_circulant(Integer(5))
[0 4 3 2 1]
[1 0 4 3 2]
[2 1 0 4 3]
[3 2 1 0 4]
[4 3 2 1 0]

Sage Live

from sage.combinat.matrices.latin import *
forward_circulant(5)

sage.combinat.matrices.latin.genus(T1, T2)[source]¶

Return the genus of hypermap embedding associated with the bitrade (T1, T2).

Informally, we compute the [tau_1, tau_2, tau_3] permutation representation of the bitrade. Each cycle of tau_1, tau_2, and tau_3 gives a rotation scheme for a black, white, and star vertex (respectively). The genus then comes from Euler’s formula.

For more details see Carlo Hamalainen: Partitioning 3-homogeneous latin bitrades. To appear in Geometriae Dedicata, available at arXiv 0710.0938

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: (a, b, c, G) = alternating_group_bitrade_generators(1)
sage: (T1, T2) = bitrade_from_group(a, b, c, G)
sage: genus(T1, T2)
1
sage: (a, b, c, G) = pq_group_bitrade_generators(3, 7)
sage: (T1, T2) = bitrade_from_group(a, b, c, G)
sage: genus(T1, T2)
3

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> (a, b, c, G) = alternating_group_bitrade_generators(Integer(1))
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> genus(T1, T2)
1
>>> (a, b, c, G) = pq_group_bitrade_generators(Integer(3), Integer(7))
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> genus(T1, T2)
3

Sage Live

from sage.combinat.matrices.latin import *
(a, b, c, G) = alternating_group_bitrade_generators(1)
(T1, T2) = bitrade_from_group(a, b, c, G)
genus(T1, T2)
(a, b, c, G) = pq_group_bitrade_generators(3, 7)
(T1, T2) = bitrade_from_group(a, b, c, G)
genus(T1, T2)

sage.combinat.matrices.latin.group_to_LatinSquare(G)[source]¶

Construct a latin square on the symbols [0, 1, …, n-1] for a group with an n by n Cayley table.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import group_to_LatinSquare

sage: group_to_LatinSquare(DihedralGroup(2))
[0 1 2 3]
[1 0 3 2]
[2 3 0 1]
[3 2 1 0]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import group_to_LatinSquare

>>> group_to_LatinSquare(DihedralGroup(Integer(2)))
[0 1 2 3]
[1 0 3 2]
[2 3 0 1]
[3 2 1 0]

Sage Live

from sage.combinat.matrices.latin import group_to_LatinSquare
group_to_LatinSquare(DihedralGroup(2))

Sage

sage: G = libgap.Group(PermutationGroupElement((1,2,3)))
sage: group_to_LatinSquare(G)
[0 1 2]
[1 2 0]
[2 0 1]

Python

>>> from sage.all import *
>>> G = libgap.Group(PermutationGroupElement((Integer(1),Integer(2),Integer(3))))
>>> group_to_LatinSquare(G)
[0 1 2]
[1 2 0]
[2 0 1]

Sage Live

G = libgap.Group(PermutationGroupElement((1,2,3)))
group_to_LatinSquare(G)

Python

>>> from sage.all import *
>>> G = libgap.Group(PermutationGroupElement((Integer(1),Integer(2),Integer(3))))
>>> group_to_LatinSquare(G)
[0 1 2]
[1 2 0]
[2 0 1]

Sage Live

G = libgap.Group(PermutationGroupElement((1,2,3)))
group_to_LatinSquare(G)

sage.combinat.matrices.latin.is_bitrade(T1, T2)[source]¶

Combinatorially, a pair (T1, T2) of partial latin squares is a bitrade if they are disjoint, have the same shape, and have row and column balance. For definitions of each of these terms see the relevant function in this file.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: is_bitrade(T1, T2)
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_bitrade(T1, T2)
True

Sage Live

from sage.combinat.matrices.latin import *
T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_bitrade(T1, T2)

sage.combinat.matrices.latin.is_disjoint(T1, T2)[source]¶

The partial latin squares T1 and T2 are disjoint if T1[r, c] != T2[r, c] or T1[r, c] == T2[r, c] == -1 for each cell [r, c].

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import is_disjoint, back_circulant, isotopism
sage: is_disjoint(back_circulant(2), back_circulant(2))
False

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import is_disjoint, back_circulant, isotopism
>>> is_disjoint(back_circulant(Integer(2)), back_circulant(Integer(2)))
False

Sage Live

from sage.combinat.matrices.latin import is_disjoint, back_circulant, isotopism
is_disjoint(back_circulant(2), back_circulant(2))

Sage

sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: is_disjoint(T1, T2)
True

Python

>>> from sage.all import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_disjoint(T1, T2)
True

Sage Live

T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_disjoint(T1, T2)

Python

>>> from sage.all import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_disjoint(T1, T2)
True

Sage Live

T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_disjoint(T1, T2)

sage.combinat.matrices.latin.is_primary_bitrade(a, b, c, G)[source]¶

A bitrade generated from elements a, b, c is primary if a, b, c = G.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: (a, b, c, G) = p3_group_bitrade_generators(5)
sage: is_primary_bitrade(a, b, c, G)
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> (a, b, c, G) = p3_group_bitrade_generators(Integer(5))
>>> is_primary_bitrade(a, b, c, G)
True

Sage Live

from sage.combinat.matrices.latin import *
(a, b, c, G) = p3_group_bitrade_generators(5)
is_primary_bitrade(a, b, c, G)

sage.combinat.matrices.latin.is_row_and_col_balanced(T1, T2)[source]¶

Partial latin squares T1 and T2 are balanced if the symbols appearing in row r of T1 are the same as the symbols appearing in row r of T2, for each r, and if the same condition holds on columns.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = matrix([[0,1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1]])
sage: T2 = matrix([[0,1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1]])
sage: is_row_and_col_balanced(T1, T2)
True
sage: T2 = matrix([[0,3,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1]])
sage: is_row_and_col_balanced(T1, T2)
False

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = matrix([[Integer(0),Integer(1),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)]])
>>> T2 = matrix([[Integer(0),Integer(1),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)]])
>>> is_row_and_col_balanced(T1, T2)
True
>>> T2 = matrix([[Integer(0),Integer(3),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)], [-Integer(1),-Integer(1),-Integer(1),-Integer(1)]])
>>> is_row_and_col_balanced(T1, T2)
False

Sage Live

from sage.combinat.matrices.latin import *
T1 = matrix([[0,1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1]])
T2 = matrix([[0,1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1]])
is_row_and_col_balanced(T1, T2)
T2 = matrix([[0,3,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1], [-1,-1,-1,-1]])
is_row_and_col_balanced(T1, T2)

sage.combinat.matrices.latin.is_same_shape(T1, T2)[source]¶

Two partial latin squares T1, T2 have the same shape if T1[r, c] = 0 if and only if T2[r, c] = 0.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: is_same_shape(elementary_abelian_2group(2), back_circulant(4))
True
sage: is_same_shape(LatinSquare(5), LatinSquare(5))
True
sage: is_same_shape(forward_circulant(5), LatinSquare(5))
False

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> is_same_shape(elementary_abelian_2group(Integer(2)), back_circulant(Integer(4)))
True
>>> is_same_shape(LatinSquare(Integer(5)), LatinSquare(Integer(5)))
True
>>> is_same_shape(forward_circulant(Integer(5)), LatinSquare(Integer(5)))
False

Sage Live

from sage.combinat.matrices.latin import *
is_same_shape(elementary_abelian_2group(2), back_circulant(4))
is_same_shape(LatinSquare(5), LatinSquare(5))
is_same_shape(forward_circulant(5), LatinSquare(5))

sage.combinat.matrices.latin.isotopism(p)[source]¶

Return a Permutation object that represents an isotopism (for rows, columns or symbols of a partial latin square).

Technically, all this function does is take as input a representation of a permutation of \(0,...,n-1\) and return a Permutation object defined on \(1,...,n\).

For a definition of isotopism, see the wikipedia section on isotopism.

INPUT:

According to the type of input (see examples below):

an integer \(n\) – the function returns the identity on \(1,...,n\)
a string representing a permutation in disjoint cycles notation, e.g. \((0,1,2)(3,4,5)\) – the corresponding permutation is returned, shifted by 1 to act on \(1,...,n\)
list/tuple of tuples – assumes disjoint cycle notation, see previous entry
a list of integers – the function adds \(1\) to each member of the list, and returns the corresponding permutation
a PermutationGroupElement p – returns a permutation describing p without any shift

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: isotopism(5) # identity on 5 points
[1, 2, 3, 4, 5]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> isotopism(Integer(5)) # identity on 5 points
[1, 2, 3, 4, 5]

Sage Live

from sage.combinat.matrices.latin import *
isotopism(5) # identity on 5 points

Sage

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: g = G.gen(0)
sage: isotopism(g)
[2, 3, 1, 5, 4]

Python

>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)'])
>>> g = G.gen(Integer(0))
>>> isotopism(g)
[2, 3, 1, 5, 4]

Sage Live

G = PermutationGroup(['(1,2,3)(4,5)'])
g = G.gen(0)
isotopism(g)

Python

>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)'])
>>> g = G.gen(Integer(0))
>>> isotopism(g)
[2, 3, 1, 5, 4]

Sage Live

G = PermutationGroup(['(1,2,3)(4,5)'])
g = G.gen(0)
isotopism(g)

Sage

sage: isotopism([0,3,2,1]) # 0 goes to 0, 1 goes to 3, etc.
[1, 4, 3, 2]

Python

>>> from sage.all import *
>>> isotopism([Integer(0),Integer(3),Integer(2),Integer(1)]) # 0 goes to 0, 1 goes to 3, etc.
[1, 4, 3, 2]

Sage Live

isotopism([0,3,2,1]) # 0 goes to 0, 1 goes to 3, etc.

Python

>>> from sage.all import *
>>> isotopism([Integer(0),Integer(3),Integer(2),Integer(1)]) # 0 goes to 0, 1 goes to 3, etc.
[1, 4, 3, 2]

Sage Live

isotopism([0,3,2,1]) # 0 goes to 0, 1 goes to 3, etc.

Sage

sage: isotopism( (0,1,2) ) # single cycle, presented as a tuple
[2, 3, 1]

Python

>>> from sage.all import *
>>> isotopism( (Integer(0),Integer(1),Integer(2)) ) # single cycle, presented as a tuple
[2, 3, 1]

Sage Live

isotopism( (0,1,2) ) # single cycle, presented as a tuple

Python

>>> from sage.all import *
>>> isotopism( (Integer(0),Integer(1),Integer(2)) ) # single cycle, presented as a tuple
[2, 3, 1]

Sage Live

isotopism( (0,1,2) ) # single cycle, presented as a tuple

Sage

sage: x = isotopism( ((0,1,2), (3,4)) ) # tuple of cycles
sage: x
[2, 3, 1, 5, 4]
sage: x.to_cycles()
[(1, 2, 3), (4, 5)]

Python

>>> from sage.all import *
>>> x = isotopism( ((Integer(0),Integer(1),Integer(2)), (Integer(3),Integer(4))) ) # tuple of cycles
>>> x
[2, 3, 1, 5, 4]
>>> x.to_cycles()
[(1, 2, 3), (4, 5)]

Sage Live

x = isotopism( ((0,1,2), (3,4)) ) # tuple of cycles
x
x.to_cycles()

Python

>>> from sage.all import *
>>> x = isotopism( ((Integer(0),Integer(1),Integer(2)), (Integer(3),Integer(4))) ) # tuple of cycles
>>> x
[2, 3, 1, 5, 4]
>>> x.to_cycles()
[(1, 2, 3), (4, 5)]

Sage Live

x = isotopism( ((0,1,2), (3,4)) ) # tuple of cycles
x
x.to_cycles()

sage.combinat.matrices.latin.next_conjugate(L)[source]¶

Permute L[r, c] = e to the conjugate L[c, e] = r.

We assume that L is an n by n matrix and has values in the range 0, 1, …, n-1.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: L = back_circulant(6)
sage: L
[0 1 2 3 4 5]
[1 2 3 4 5 0]
[2 3 4 5 0 1]
[3 4 5 0 1 2]
[4 5 0 1 2 3]
[5 0 1 2 3 4]
sage: next_conjugate(L)
[0 1 2 3 4 5]
[5 0 1 2 3 4]
[4 5 0 1 2 3]
[3 4 5 0 1 2]
[2 3 4 5 0 1]
[1 2 3 4 5 0]
sage: L == next_conjugate(next_conjugate(next_conjugate(L)))
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> L = back_circulant(Integer(6))
>>> L
[0 1 2 3 4 5]
[1 2 3 4 5 0]
[2 3 4 5 0 1]
[3 4 5 0 1 2]
[4 5 0 1 2 3]
[5 0 1 2 3 4]
>>> next_conjugate(L)
[0 1 2 3 4 5]
[5 0 1 2 3 4]
[4 5 0 1 2 3]
[3 4 5 0 1 2]
[2 3 4 5 0 1]
[1 2 3 4 5 0]
>>> L == next_conjugate(next_conjugate(next_conjugate(L)))
True

Sage Live

from sage.combinat.matrices.latin import *
L = back_circulant(6)
L
next_conjugate(L)
L == next_conjugate(next_conjugate(next_conjugate(L)))

sage.combinat.matrices.latin.p3_group_bitrade_generators(p)[source]¶

Generators for a group of order p3 where p is a prime.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: p3_group_bitrade_generators(3)  # random output
((2,6,7)(3,8,9),
 (1,2,3)(4,7,8)(5,6,9),
 (1,9,2)(3,7,4)(5,8,6),
 Permutation Group with generators [(2,6,7)(3,8,9), (1,2,3)(4,7,8)(5,6,9)])

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> p3_group_bitrade_generators(Integer(3))  # random output
((2,6,7)(3,8,9),
 (1,2,3)(4,7,8)(5,6,9),
 (1,9,2)(3,7,4)(5,8,6),
 Permutation Group with generators [(2,6,7)(3,8,9), (1,2,3)(4,7,8)(5,6,9)])

Sage Live

from sage.combinat.matrices.latin import *
p3_group_bitrade_generators(3)  # random output

sage.combinat.matrices.latin.pq_group_bitrade_generators(p, q)[source]¶

Generators for a group of order pq where p and q are primes such that (q % p) == 1.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: pq_group_bitrade_generators(3,7)
((2,3,5)(4,7,6), (1,2,3,4,5,6,7), (1,4,2)(3,5,6), Permutation Group with generators [(2,3,5)(4,7,6), (1,2,3,4,5,6,7)])

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> pq_group_bitrade_generators(Integer(3),Integer(7))
((2,3,5)(4,7,6), (1,2,3,4,5,6,7), (1,4,2)(3,5,6), Permutation Group with generators [(2,3,5)(4,7,6), (1,2,3,4,5,6,7)])

Sage Live

from sage.combinat.matrices.latin import *
pq_group_bitrade_generators(3,7)

sage.combinat.matrices.latin.row_containing_sym(L, c, x)[source]¶

Given an improper latin square L with L[r1, c] = L[r2, c] = x, return r1 or r2 with equal probability. This is an internal function and should only be used in LatinSquare_generator().

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: L = matrix([(0, 1, 0, 3), (3, 0, 2, 1), (1, 0, 3, 2), (2, 3, 1, 0)])
sage: L
[0 1 0 3]
[3 0 2 1]
[1 0 3 2]
[2 3 1 0]
sage: c = row_containing_sym(L, 1, 0)
sage: c == 1 or c == 2
True

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> L = matrix([(Integer(0), Integer(1), Integer(0), Integer(3)), (Integer(3), Integer(0), Integer(2), Integer(1)), (Integer(1), Integer(0), Integer(3), Integer(2)), (Integer(2), Integer(3), Integer(1), Integer(0))])
>>> L
[0 1 0 3]
[3 0 2 1]
[1 0 3 2]
[2 3 1 0]
>>> c = row_containing_sym(L, Integer(1), Integer(0))
>>> c == Integer(1) or c == Integer(2)
True

Sage Live

from sage.combinat.matrices.latin import *
L = matrix([(0, 1, 0, 3), (3, 0, 2, 1), (1, 0, 3, 2), (2, 3, 1, 0)])
L
c = row_containing_sym(L, 1, 0)
c == 1 or c == 2

sage.combinat.matrices.latin.tau1(T1, T2, cells_map)[source]¶

The definition of \(\tau_1\) is

\[\begin{split}\tau_1 : T1 \rightarrow T1 \\ \tau_1 = \beta_2^{-1} \beta_3\end{split}\]

where the composition is left to right and \(\beta_i : T2 \rightarrow T1\) changes just the \(i\)-th coordinate of a triple.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: is_bitrade(T1, T2)
True
sage: (cells_map, t1, t2, t3) = tau123(T1, T2)
sage: t1 = tau1(T1, T2, cells_map)
sage: t1
[2, 3, 4, 5, 1, 7, 8, 9, 10, 6, 12, 13, 14, 15, 11, 17, 18, 19, 20, 16, 22, 23, 24, 25, 21]
sage: t1.to_cycles()
[(1, 2, 3, 4, 5), (6, 7, 8, 9, 10), (11, 12, 13, 14, 15), (16, 17, 18, 19, 20), (21, 22, 23, 24, 25)]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_bitrade(T1, T2)
True
>>> (cells_map, t1, t2, t3) = tau123(T1, T2)
>>> t1 = tau1(T1, T2, cells_map)
>>> t1
[2, 3, 4, 5, 1, 7, 8, 9, 10, 6, 12, 13, 14, 15, 11, 17, 18, 19, 20, 16, 22, 23, 24, 25, 21]
>>> t1.to_cycles()
[(1, 2, 3, 4, 5), (6, 7, 8, 9, 10), (11, 12, 13, 14, 15), (16, 17, 18, 19, 20), (21, 22, 23, 24, 25)]

Sage Live

from sage.combinat.matrices.latin import *
T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_bitrade(T1, T2)
(cells_map, t1, t2, t3) = tau123(T1, T2)
t1 = tau1(T1, T2, cells_map)
t1
t1.to_cycles()

sage.combinat.matrices.latin.tau123(T1, T2)[source]¶

Compute the tau_i representation for a bitrade (T1, T2).

See the functions tau1, tau2, and tau3 for the mathematical definitions.

OUTPUT:

(cells_map, t1, t2, t3)

where cells_map is a map to/from the filled cells of T1, and t1, t2, t3 are the tau1, tau2, tau3 permutations.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: (a, b, c, G) = pq_group_bitrade_generators(3, 7)
sage: (T1, T2) = bitrade_from_group(a, b, c, G)
sage: T1
[ 0  1  3 -1 -1 -1 -1]
[ 1  2  4 -1 -1 -1 -1]
[ 2  3  5 -1 -1 -1 -1]
[ 3  4  6 -1 -1 -1 -1]
[ 4  5  0 -1 -1 -1 -1]
[ 5  6  1 -1 -1 -1 -1]
[ 6  0  2 -1 -1 -1 -1]
sage: T2
[ 1  3  0 -1 -1 -1 -1]
[ 2  4  1 -1 -1 -1 -1]
[ 3  5  2 -1 -1 -1 -1]
[ 4  6  3 -1 -1 -1 -1]
[ 5  0  4 -1 -1 -1 -1]
[ 6  1  5 -1 -1 -1 -1]
[ 0  2  6 -1 -1 -1 -1]
sage: (cells_map, t1, t2, t3) = tau123(T1, T2)
sage: D = cells_map
sage: {i: v for i,v in D.items() if i in ZZ}
{1: (0, 0),
 2: (0, 1),
 3: (0, 2),
 4: (1, 0),
 5: (1, 1),
 6: (1, 2),
 7: (2, 0),
 8: (2, 1),
 9: (2, 2),
 10: (3, 0),
 11: (3, 1),
 12: (3, 2),
 13: (4, 0),
 14: (4, 1),
 15: (4, 2),
 16: (5, 0),
 17: (5, 1),
 18: (5, 2),
 19: (6, 0),
 20: (6, 1),
 21: (6, 2)}
sage: {i: v for i,v in D.items() if i not in ZZ}
{(0, 0): 1,
 (0, 1): 2,
 (0, 2): 3,
 (1, 0): 4,
 (1, 1): 5,
 (1, 2): 6,
 (2, 0): 7,
 (2, 1): 8,
 (2, 2): 9,
 (3, 0): 10,
 (3, 1): 11,
 (3, 2): 12,
 (4, 0): 13,
 (4, 1): 14,
 (4, 2): 15,
 (5, 0): 16,
 (5, 1): 17,
 (5, 2): 18,
 (6, 0): 19,
 (6, 1): 20,
 (6, 2): 21}
sage: cells_map_as_square(cells_map, max(T1.nrows(), T1.ncols()))
[ 1  2  3 -1 -1 -1 -1]
[ 4  5  6 -1 -1 -1 -1]
[ 7  8  9 -1 -1 -1 -1]
[10 11 12 -1 -1 -1 -1]
[13 14 15 -1 -1 -1 -1]
[16 17 18 -1 -1 -1 -1]
[19 20 21 -1 -1 -1 -1]
sage: t1
[3, 1, 2, 6, 4, 5, 9, 7, 8, 12, 10, 11, 15, 13, 14, 18, 16, 17, 21, 19, 20]
sage: t2
[4, 8, 15, 7, 11, 18, 10, 14, 21, 13, 17, 3, 16, 20, 6, 19, 2, 9, 1, 5, 12]
sage: t3
[20, 18, 10, 2, 21, 13, 5, 3, 16, 8, 6, 19, 11, 9, 1, 14, 12, 4, 17, 15, 7]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> (a, b, c, G) = pq_group_bitrade_generators(Integer(3), Integer(7))
>>> (T1, T2) = bitrade_from_group(a, b, c, G)
>>> T1
[ 0  1  3 -1 -1 -1 -1]
[ 1  2  4 -1 -1 -1 -1]
[ 2  3  5 -1 -1 -1 -1]
[ 3  4  6 -1 -1 -1 -1]
[ 4  5  0 -1 -1 -1 -1]
[ 5  6  1 -1 -1 -1 -1]
[ 6  0  2 -1 -1 -1 -1]
>>> T2
[ 1  3  0 -1 -1 -1 -1]
[ 2  4  1 -1 -1 -1 -1]
[ 3  5  2 -1 -1 -1 -1]
[ 4  6  3 -1 -1 -1 -1]
[ 5  0  4 -1 -1 -1 -1]
[ 6  1  5 -1 -1 -1 -1]
[ 0  2  6 -1 -1 -1 -1]
>>> (cells_map, t1, t2, t3) = tau123(T1, T2)
>>> D = cells_map
>>> {i: v for i,v in D.items() if i in ZZ}
{1: (0, 0),
 2: (0, 1),
 3: (0, 2),
 4: (1, 0),
 5: (1, 1),
 6: (1, 2),
 7: (2, 0),
 8: (2, 1),
 9: (2, 2),
 10: (3, 0),
 11: (3, 1),
 12: (3, 2),
 13: (4, 0),
 14: (4, 1),
 15: (4, 2),
 16: (5, 0),
 17: (5, 1),
 18: (5, 2),
 19: (6, 0),
 20: (6, 1),
 21: (6, 2)}
>>> {i: v for i,v in D.items() if i not in ZZ}
{(0, 0): 1,
 (0, 1): 2,
 (0, 2): 3,
 (1, 0): 4,
 (1, 1): 5,
 (1, 2): 6,
 (2, 0): 7,
 (2, 1): 8,
 (2, 2): 9,
 (3, 0): 10,
 (3, 1): 11,
 (3, 2): 12,
 (4, 0): 13,
 (4, 1): 14,
 (4, 2): 15,
 (5, 0): 16,
 (5, 1): 17,
 (5, 2): 18,
 (6, 0): 19,
 (6, 1): 20,
 (6, 2): 21}
>>> cells_map_as_square(cells_map, max(T1.nrows(), T1.ncols()))
[ 1  2  3 -1 -1 -1 -1]
[ 4  5  6 -1 -1 -1 -1]
[ 7  8  9 -1 -1 -1 -1]
[10 11 12 -1 -1 -1 -1]
[13 14 15 -1 -1 -1 -1]
[16 17 18 -1 -1 -1 -1]
[19 20 21 -1 -1 -1 -1]
>>> t1
[3, 1, 2, 6, 4, 5, 9, 7, 8, 12, 10, 11, 15, 13, 14, 18, 16, 17, 21, 19, 20]
>>> t2
[4, 8, 15, 7, 11, 18, 10, 14, 21, 13, 17, 3, 16, 20, 6, 19, 2, 9, 1, 5, 12]
>>> t3
[20, 18, 10, 2, 21, 13, 5, 3, 16, 8, 6, 19, 11, 9, 1, 14, 12, 4, 17, 15, 7]

Sage Live

from sage.combinat.matrices.latin import *
(a, b, c, G) = pq_group_bitrade_generators(3, 7)
(T1, T2) = bitrade_from_group(a, b, c, G)
T1
T2
(cells_map, t1, t2, t3) = tau123(T1, T2)
D = cells_map
{i: v for i,v in D.items() if i in ZZ}
{i: v for i,v in D.items() if i not in ZZ}
cells_map_as_square(cells_map, max(T1.nrows(), T1.ncols()))
t1
t2
t3

Sage

sage: t1.to_cycles()
[(1, 3, 2), (4, 6, 5), (7, 9, 8), (10, 12, 11), (13, 15, 14), (16, 18, 17), (19, 21, 20)]
sage: t2.to_cycles()
[(1, 4, 7, 10, 13, 16, 19), (2, 8, 14, 20, 5, 11, 17), (3, 15, 6, 18, 9, 21, 12)]
sage: t3.to_cycles()
[(1, 20, 15), (2, 18, 4), (3, 10, 8), (5, 21, 7), (6, 13, 11), (9, 16, 14), (12, 19, 17)]

Python

>>> from sage.all import *
>>> t1.to_cycles()
[(1, 3, 2), (4, 6, 5), (7, 9, 8), (10, 12, 11), (13, 15, 14), (16, 18, 17), (19, 21, 20)]
>>> t2.to_cycles()
[(1, 4, 7, 10, 13, 16, 19), (2, 8, 14, 20, 5, 11, 17), (3, 15, 6, 18, 9, 21, 12)]
>>> t3.to_cycles()
[(1, 20, 15), (2, 18, 4), (3, 10, 8), (5, 21, 7), (6, 13, 11), (9, 16, 14), (12, 19, 17)]

Sage Live

t1.to_cycles()
t2.to_cycles()
t3.to_cycles()

Python

>>> from sage.all import *
>>> t1.to_cycles()
[(1, 3, 2), (4, 6, 5), (7, 9, 8), (10, 12, 11), (13, 15, 14), (16, 18, 17), (19, 21, 20)]
>>> t2.to_cycles()
[(1, 4, 7, 10, 13, 16, 19), (2, 8, 14, 20, 5, 11, 17), (3, 15, 6, 18, 9, 21, 12)]
>>> t3.to_cycles()
[(1, 20, 15), (2, 18, 4), (3, 10, 8), (5, 21, 7), (6, 13, 11), (9, 16, 14), (12, 19, 17)]

Sage Live

t1.to_cycles()
t2.to_cycles()
t3.to_cycles()

The product t1*t2*t3 is the identity, i.e. it fixes every point:

Sage

sage: len((t1*t2*t3).fixed_points()) == T1.nr_filled_cells()
True

Python

>>> from sage.all import *
>>> len((t1*t2*t3).fixed_points()) == T1.nr_filled_cells()
True

Sage Live

len((t1*t2*t3).fixed_points()) == T1.nr_filled_cells()

sage.combinat.matrices.latin.tau2(T1, T2, cells_map)[source]¶

The definition of \(\tau_2\) is

\[\begin{split}\tau_2 : T1 \rightarrow T1 \\ \tau_2 = \beta_3^{-1} \beta_1\end{split}\]

where the composition is left to right and \(\beta_i : T2 \rightarrow T1\) changes just the \(i\)-th coordinate of a triple.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: is_bitrade(T1, T2)
True
sage: (cells_map, t1, t2, t3) = tau123(T1, T2)
sage: t2 = tau2(T1, T2, cells_map)
sage: t2
[21, 22, 23, 24, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
sage: t2.to_cycles()
[(1, 21, 16, 11, 6), (2, 22, 17, 12, 7), (3, 23, 18, 13, 8), (4, 24, 19, 14, 9), (5, 25, 20, 15, 10)]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_bitrade(T1, T2)
True
>>> (cells_map, t1, t2, t3) = tau123(T1, T2)
>>> t2 = tau2(T1, T2, cells_map)
>>> t2
[21, 22, 23, 24, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
>>> t2.to_cycles()
[(1, 21, 16, 11, 6), (2, 22, 17, 12, 7), (3, 23, 18, 13, 8), (4, 24, 19, 14, 9), (5, 25, 20, 15, 10)]

Sage Live

from sage.combinat.matrices.latin import *
T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_bitrade(T1, T2)
(cells_map, t1, t2, t3) = tau123(T1, T2)
t2 = tau2(T1, T2, cells_map)
t2
t2.to_cycles()

sage.combinat.matrices.latin.tau3(T1, T2, cells_map)[source]¶

The definition of \(\tau_3\) is

\[\begin{split}\tau_3 : T1 \rightarrow T1 \\ \tau_3 = \beta_1^{-1} \beta_2\end{split}\]

where the composition is left to right and \(\beta_i : T2 \rightarrow T1\) changes just the \(i\)-th coordinate of a triple.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: is_bitrade(T1, T2)
True
sage: (cells_map, t1, t2, t3) = tau123(T1, T2)
sage: t3 = tau3(T1, T2, cells_map)
sage: t3
[10, 6, 7, 8, 9, 15, 11, 12, 13, 14, 20, 16, 17, 18, 19, 25, 21, 22, 23, 24, 5, 1, 2, 3, 4]
sage: t3.to_cycles()
[(1, 10, 14, 18, 22), (2, 6, 15, 19, 23), (3, 7, 11, 20, 24), (4, 8, 12, 16, 25), (5, 9, 13, 17, 21)]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> is_bitrade(T1, T2)
True
>>> (cells_map, t1, t2, t3) = tau123(T1, T2)
>>> t3 = tau3(T1, T2, cells_map)
>>> t3
[10, 6, 7, 8, 9, 15, 11, 12, 13, 14, 20, 16, 17, 18, 19, 25, 21, 22, 23, 24, 5, 1, 2, 3, 4]
>>> t3.to_cycles()
[(1, 10, 14, 18, 22), (2, 6, 15, 19, 23), (3, 7, 11, 20, 24), (4, 8, 12, 16, 25), (5, 9, 13, 17, 21)]

Sage Live

from sage.combinat.matrices.latin import *
T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
is_bitrade(T1, T2)
(cells_map, t1, t2, t3) = tau123(T1, T2)
t3 = tau3(T1, T2, cells_map)
t3
t3.to_cycles()

sage.combinat.matrices.latin.tau_to_bitrade(t1, t2, t3)[source]¶

Given permutations t1, t2, t3 that represent a latin bitrade, convert them to an explicit latin bitrade (T1, T2). The result is unique up to isotopism.

EXAMPLES:

Sage

sage: from sage.combinat.matrices.latin import *
sage: T1 = back_circulant(5)
sage: x = isotopism( (0,1,2,3,4) )
sage: y = isotopism(5) # identity
sage: z = isotopism(5) # identity
sage: T2 = T1.apply_isotopism(x, y, z)
sage: _, t1, t2, t3 = tau123(T1, T2)
sage: U1, U2 = tau_to_bitrade(t1, t2, t3)
sage: assert is_bitrade(U1, U2)
sage: U1
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]
[4 0 1 2 3]
sage: U2
[4 0 1 2 3]
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]

Python

>>> from sage.all import *
>>> from sage.combinat.matrices.latin import *
>>> T1 = back_circulant(Integer(5))
>>> x = isotopism( (Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)) )
>>> y = isotopism(Integer(5)) # identity
>>> z = isotopism(Integer(5)) # identity
>>> T2 = T1.apply_isotopism(x, y, z)
>>> _, t1, t2, t3 = tau123(T1, T2)
>>> U1, U2 = tau_to_bitrade(t1, t2, t3)
>>> assert is_bitrade(U1, U2)
>>> U1
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]
[4 0 1 2 3]
>>> U2
[4 0 1 2 3]
[0 1 2 3 4]
[1 2 3 4 0]
[2 3 4 0 1]
[3 4 0 1 2]

Sage Live

from sage.combinat.matrices.latin import *
T1 = back_circulant(5)
x = isotopism( (0,1,2,3,4) )
y = isotopism(5) # identity
z = isotopism(5) # identity
T2 = T1.apply_isotopism(x, y, z)
_, t1, t2, t3 = tau123(T1, T2)
U1, U2 = tau_to_bitrade(t1, t2, t3)
assert is_bitrade(U1, U2)
U1
U2