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This is documentation for Sage version 10.5.rc0 for development purpose.
Combinatorics
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Version 10.5.rc0 Reference Manual
  • Home - Combinatorics
  • Comprehensive Module List
    • Abstract Recursive Trees
    • Affine Permutations
    • Algebraic combinatorics
    • Combinatorics
    • Alternating Sign Matrices
    • Backtracking
    • Baxter permutations
    • A bijectionist’s toolkit
    • Binary Recurrence Sequences
    • Binary Trees
    • Blob Algebras
    • Cartesian Products
    • Enumerated sets of partitions, tableaux, …
    • Combinatorial Hopf algebras
    • Poirier-Reutenauer Hopf algebra of standard tableaux
    • Word Quasi-symmetric functions
    • Cluster algebras and quivers
    • ClusterSeed
    • mutation_class
    • Helper functions for mutation types of quivers
    • Quiver
    • Quiver mutation types
    • Cluster complex (or generalized dual associahedron)
    • Colored Permutations
    • Combinatorial Functions
    • Fast computation of combinatorial functions (Cython + mpz)
    • Combinations
    • Combinatorial maps
    • Integer compositions
    • Signed Compositions
    • Composition Tableaux
    • Constellations
    • Cores
    • Counting
    • Affine Crystals
    • Affine factorization crystal of type \(A\)
    • Affinization Crystals
    • Alcove paths
    • Crystals
    • Benkart-Kang-Kashiwara crystals for the general-linear Lie superalgebra
    • Catalog Of Crystals
    • Catalog Of Elementary Crystals
    • Catalog Of Crystal Models For \(B(\infty)\)
    • Catalog Of Crystal Models For Kirillov-Reshetikhin Crystals
    • An introduction to crystals
    • Direct Sum of Crystals
    • Elementary Crystals
    • Fast Rank Two Crystals
    • Fully commutative stable Grothendieck crystal
    • Crystals of Generalized Young Walls
    • Highest weight crystals
    • Induced Crystals
    • \(\mathcal{B}(\infty)\) Crystals of Tableaux in Nonexceptional Types and \(G_2\)
    • Crystals of Kac modules of the general-linear Lie superalgebra
    • Kirillov-Reshetikhin Crystals
    • Kyoto Path Model for Affine Highest Weight Crystals
    • Crystals of letters
    • Littelmann paths
    • Crystals of Modified Nakajima Monomials
    • Crystal of Bernstein-Zelevinsky Multisegments
    • Crystal Of Mirković-Vilonen (MV) Polytopes
    • \(\mathcal{B}(\infty)\) Crystal Of PBW Monomials
    • PBW Data
    • Polyhedral Realization of \(B(\infty)\)
    • Spin Crystals
    • Star-Crystal Structure On \(B(\infty)\)
    • Tensor Products of Crystals
    • Tensor Products of Crystal Elements
    • Cyclic sieving phenomenon
    • De Bruijn sequences
    • Degree sequences
    • Derangements
    • Descent Algebras
    • Combinatorial designs and incidence structures
    • Balanced Incomplete Block Designs (BIBD)
    • Resolvable Balanced Incomplete Block Design (RBIBD)
    • Group-Divisible Designs (GDD)
    • Block designs
    • Covering Arrays (CA)
    • Covering designs: coverings of \(t\)-element subsets of a \(v\)-set by \(k\)-sets
    • Database of small combinatorial designs
    • Catalog of designs
    • Cython functions for combinatorial designs
    • Difference families
    • Difference Matrices
    • Evenly distributed sets in finite fields
    • External Representations of Block Designs
    • Database of generalised quadrangles with spread
    • Incidence structures (i.e. hypergraphs, i.e. set systems)
    • Mutually Orthogonal Latin Squares (MOLS)
    • Orthogonal arrays (OA)
    • Orthogonal arrays (build recursive constructions)
    • Orthogonal arrays (find recursive constructions)
    • Steiner Quadruple Systems
    • Hypergraph isomorphic copy search
    • Two-graphs
    • Combinatorial diagrams
    • Diagram and Partition Algebras
    • Exact Cover Problem via Dancing Links
    • Dyck Words
    • Substitutions over unit cube faces (Rauzy fractals)
    • Enumerated sets and combinatorial objects
    • Tools for enumeration modulo the action of a permutation group
    • Compute Bell and Uppuluri-Carpenter numbers
    • Families
    • Brent Yorgey’s fast algorithm for integer vector (multiset) partitions.
    • Fully commutative elements of Coxeter groups
    • Finite state machines, automata, transducers
    • Common Automata and Transducers (Finite State Machines Generators)
    • Free Quasi-symmetric functions
    • Free modules
    • Free Dendriform Algebras
    • Free Pre-Lie Algebras
    • Fully packed loops
    • Gelfand-Tsetlin Patterns
    • Paths in Directed Acyclic Graphs
    • Gray codes
    • Growth diagrams and dual graded graphs
    • Grossman-Larson Hopf Algebras
    • Hall Polynomials
    • The Hillman-Grassl correspondence
    • Enumerated set of lists of integers with constraints: base classes
    • Enumerated set of lists of integers with constraints: front-end
    • Enumerated set of lists of integers with constraints, in inverse lexicographic order
    • Counting, generating, and manipulating nonnegative integer matrices
    • (Non-negative) Integer vectors
    • Weighted Integer Vectors
    • Integer vectors modulo the action of a permutation group
    • Tamari Interval-posets
    • Strong and weak tableaux
    • Kazhdan-Lusztig Polynomials
    • Key polynomials
    • Knutson-Tao Puzzles
    • Combinatorics on matrices
    • Dancing Links internal pyx code
    • Dancing links C++ wrapper
    • Hadamard matrices
    • Latin Squares
    • Miscellaneous
    • Ordered Multiset Partitions into Sets and the Minimaj Crystal
    • Non-commutative symmetric functions and quasi-symmetric functions
    • Common combinatorial tools
    • Generic code for bases
    • Non-Commutative Symmetric Functions
    • Quasisymmetric functions
    • Introduction to Quasisymmetric Functions
    • Symmetric functions in non-commuting variables
    • Bases for \(NCSym\)
    • Dual Symmetric Functions in Non-Commuting Variables
    • Symmetric Functions in Non-Commuting Variables
    • Necklaces
    • Non-Decreasing Parking Functions
    • \(\nu\)-Dyck Words
    • \(\nu\)-Tamari lattice
    • Ordered Rooted Trees
    • Output functions
    • Parallelogram Polyominoes
    • Parking Functions
    • Catalog of Path Tableaux
    • Dyck Paths
    • Frieze Patterns
    • Path Tableaux
    • Semistandard Tableaux
    • Plane Partitions
    • Integer partitions
    • Partition/Diagram Algebras
    • Kleshchev partitions
    • Partition Shifting Algebras
    • Partition tuples
    • Iterators over the partitions of an integer
    • Perfect matchings
    • Permutations
    • Permutations (Cython file)
    • Posets
    • Cartesian products of Posets
    • D-Complete Posets
    • Mobile posets
    • Elements of posets, lattices, semilattices, etc.
    • Forest Posets
    • Hasse diagrams of posets
    • Incidence Algebras
    • Finite lattices and semilattices
    • Linear Extensions of Posets
    • Möbius Algebras
    • Catalog of posets and lattices
    • Finite posets
    • \(q\)-Analogues
    • \(q\)-Bernoulli Numbers and Polynomials
    • Combinatorics quickref
    • Rankers
    • Recognizable Series
    • \(k\)-regular sequences
    • Restricted growth arrays
    • Ribbons
    • Ribbon Shaped Tableaux
    • Ribbon Tableaux
    • Rigged configurations
    • Abstract classes for the rigged configuration bijections
    • Bijection between rigged configurations for \(B(\infty)\) and marginally large tableaux
    • Bijection classes for type \(A_n^{(1)}\)
    • Bijection classes for type \(A_{2n}^{(2)\dagger}\)
    • Bijection classes for type \(A_{2n}^{(2)}\)
    • Bijection classes for type \(A_{2n-1}^{(2)}\).
    • Bijection classes for type \(B_n^{(1)}\)
    • Bijection classes for type \(C_n^{(1)}\)
    • Bijection classes for type \(D_n^{(1)}\)
    • Bijection classes for type \(D_{n+1}^{(2)}\)
    • Bijection classes for type \(D_4^{(3)}\)
    • Bijection between rigged configurations and KR tableaux
    • Kleber Trees
    • Kirillov-Reshetikhin Tableaux
    • Crystal of Rigged Configurations
    • Rigged Configurations of \(\mathcal{B}(\infty)\)
    • Rigged Configuration Elements
    • Rigged Configurations
    • Rigged Partitions
    • Tensor Product of Kirillov-Reshetikhin Tableaux
    • Tensor Product of Kirillov-Reshetikhin Tableaux Elements
    • Root Systems
    • Ambient lattices and ambient spaces
    • Associahedron
    • Braid Move Calculator
    • Braid Orbit
    • Branching Rules
    • Cartan matrices
    • Cartan types
    • Coxeter Groups
    • Coxeter Matrices
    • Coxeter Types
    • Dynkin diagrams
    • Hecke algebra representations
    • Integrable Representations of Affine Lie Algebras
    • Nonsymmetric Macdonald polynomials
    • Pieri Factors
    • Tutorial: visualizing root systems
    • Finite complex reflection groups
    • Finite real reflection groups
    • Group algebras of root lattice realizations
    • Root lattice realizations
    • Root lattices and root spaces
    • Root systems
    • Root system data for super type A
    • Root system data for type A
    • Root system data for (untwisted) type A affine
    • Root system data for type A infinity
    • Root system data for type B
    • Root system data for type BC affine
    • Root system data for (untwisted) type B affine
    • Root system data for type C
    • Root system data for (untwisted) type C affine
    • Root system data for type D
    • Root system data for (untwisted) type D affine
    • Root system data for type E
    • Root system data for (untwisted) type E affine
    • Root system data for type F
    • Root system data for (untwisted) type F affine
    • Root system data for type G
    • Root system data for (untwisted) type G affine
    • Root system data for type H
    • Root system data for type I
    • Root system data for type Q
    • Root system data for affine Cartan types
    • Root system data for dual Cartan types
    • Extended Affine Weyl Groups
    • Fundamental Group of an Extended Affine Weyl Group
    • Root system data for folded Cartan types
    • Root system data for Cartan types with marked nodes
    • Root system data for reducible Cartan types
    • Root system data for relabelled Cartan types
    • Weight lattice realizations
    • Weight lattices and weight spaces
    • Weyl Character Rings
    • Weyl Groups
    • Rooted (Unordered) Trees
    • Robinson-Schensted-Knuth correspondence
    • Schubert Polynomials
    • Set Partitions
    • Fast set partition iterators
    • Ordered Set Partitions
    • Symmetric Functions
    • Characters of the symmetric group as bases of the symmetric functions
    • Classical symmetric functions
    • Generic dual bases symmetric functions
    • Elementary symmetric functions
    • Hall-Littlewood Polynomials
    • Hecke Character Basis
    • Homogeneous symmetric functions
    • Jack Symmetric Functions
    • Quotient of symmetric function space by ideal generated by Hall-Littlewood symmetric functions
    • Kostka-Foulkes Polynomials
    • LLT symmetric functions
    • Macdonald Polynomials
    • Monomial symmetric functions
    • Multiplicative symmetric functions
    • \(k\)-Schur Functions
    • Non-symmetric Macdonald Polynomials
    • Orthogonal Symmetric Functions
    • Symmetric functions defined by orthogonality and triangularity
    • Power sum symmetric functions
    • Schur symmetric functions
    • Symplectic Symmetric Functions
    • Symmetric functions, with their multiple realizations
    • Symmetric Functions
    • Witt symmetric functions
    • Shard intersection order
    • Shifted primed tableaux
    • Shuffle product of iterables
    • Sidon sets and their generalizations, Sidon \(g\)-sets
    • Similarity class types of matrices with entries in a finite field
    • sine-Gordon Y-system plotter
    • Six Vertex Model
    • Skew Partitions
    • Skew Tableaux
    • Functions that compute some of the sequences in Sloane’s tables
    • Combinatorial species
    • Characteristic Species
    • Composition species
    • Cycle Species
    • Empty Species
    • Functorial composition species
    • Generating Series
    • Examples of Combinatorial Species
    • Linear-order Species
    • Miscellaneous Functions
    • Partition Species
    • Permutation species
    • Product species
    • Recursive Species
    • Set Species
    • Combinatorial Species
    • Species structures
    • Subset Species
    • Sum species
    • Specht Modules
    • Subsets
    • Subsets satisfying a hereditary property
    • Subsets whose elements satisfy a predicate pairwise
    • Subwords
    • Subword complex
    • Super Tableaux
    • Super Partitions
    • Symmetric Group Algebra
    • Representations of the Symmetric Group
    • T-sequences
    • Tableaux
    • Residue sequences of tableaux
    • TableauTuples
    • Generalized Tamari lattices
    • Tiling Solver
    • Transitive ideal closure tool
    • Combinatorial triangles for posets and fans
    • Tuples
    • Introduction to combinatorics in Sage
    • Vector Partitions
    • Abstract word (finite or infinite)
    • Combinatorics on words
    • Alphabet
    • Finite word
    • Infinite word
    • Lyndon words
    • Word morphisms/substitutions
    • Word paths
    • Shuffle product of words
    • Suffix Tries and Suffix Trees
    • Word classes
    • Fast word datatype using an array of unsigned char
    • Datatypes for finite words
    • Common words
    • Datatypes for words defined by iterators and callables
    • User-customizable options for words
    • Set of words
    • Yang-Baxter Graphs
    • C-Finite Sequences
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Combinatorial Hopf algebras¶

  • Symmetric Functions

  • Non-commutative symmetric functions and quasi-symmetric functions

  • Symmetric functions in non-commuting variables

  • Schubert Polynomials

  • Poirier-Reutenauer Hopf algebra of standard tableaux

  • Free Quasi-symmetric functions

  • Grossman-Larson Hopf Algebras

  • Word Quasi-symmetric functions

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Poirier-Reutenauer Hopf algebra of standard tableaux
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