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Combinatorics
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Introduction to combinatorics in Sage
Initial examples
Common enumerated sets
Constructions
Generic algorithms
Combinatorial Functions
Functions and classes
Functions that compute some of the sequences in Sloane’s tables
The On-Line Encyclopedia of Integer Sequences (OEIS)
Classes and methods
Compute Bell and Uppuluri-Carpenter numbers
Alternating Sign Matrices
Cartesian Products
Combinations
Signed Compositions
Integer compositions
Cores
De Bruijn sequences
Degree sequences
Definitions
Algorithms
About the implementation
References
Author
Tests
Derangements
Exact Cover Problem via Dancing Links
Dancing links C++ wrapper
Dyck Words
The family of generalized Tamari lattices
Substitutions over unit cube faces (Rauzy fractals)
Finite combinatorial classes
Hall Polynomials
Hadamard matrices
Tools for generating lists of integers in lexicographic order
Counting, generating, and manipulating non-negative integer matrices
(Non-negative) Integer vectors
Weighted Integer Vectors
Integer vectors modulo the action of a permutation group
Tools for enumeration modulo the action of a permutation group
Restricted growth arrays
Yang-Baxter Graphs
Gelfand-Tsetlin Patterns
Paths in Directed Acyclic Graphs
Latin Squares
Lyndon words
Necklaces
Non-Decreasing Parking Functions
Parking Functions
Partitions
Partition tuples
Permutations
What does this file define ?
Affine Permutations
Perfect matchings
q-Analogues
-Bernoulli Numbers
Robinson-Schensted-Knuth correspondence
Sidon sets and their generalizations, Sidon
-sets
Six Vertex Model
Ordered Set Partitions
Set Partitions
Abstract Recursive Trees
Ordered Rooted Trees
Binary trees
Similarity class types of matrices with entries in a finite field
Skew Partitions
Subsets
Subsets whose elements satisfy a predicate pairwise
Subwords
Tiling Solver
2d Easy Example
1d Easy Example
2d Puzzle allowing reflections
3d Puzzle
Donald Knuth example : the Y pentamino
Animation of Donald Knuth’s dancing links
5d Easy Example
Tuples
Vector Partitions
Cluster Algebras and Quivers
Quiver mutation types
Quiver
ClusterSeed
Binary Recurrence Sequences.
Combinatorial Algebras
Free modules
Combinatorial Algebras
Descent Algebras
Diagram and Partition Algebras
Symmetric Group Algebra
Representations of the Symmetric Group
Schubert Polynomials
Partition/Diagram Algebras
Iwahori-Hecke Algebras
Nil-Coxeter Algebra
Affine nilTemperley Lieb Algebra of type A
Partitions and Partition-like Objects
Partitions
Partition tuples
Skew Partitions
Tableaux and Tableaux-like Objects
Tableaux
Skew Tableaux
Ribbons
Ribbon Tableaux
TableauTuples
Strong and weak tableaux
Symmetric Functions
Symmetric Functions
Symmetric functions, with their multiple realizations
Classical symmetric functions.
Schur symmetric functions
Monomial symmetric functions
Multiplicative symmetric functions
Elementary symmetric functions
Homogeneous symmetric functions
Power sum symmetric functions
Generic dual bases symmetric functions
Symmetric functions defined by orthogonality and triangularity.
Kostka-Foulkes Polynomials
Hall-Littlewood Polynomials
Jack Symmetric Functions
-Schur Functions
Quotient of symmetric function space by ideal generated by Hall-Littlewood symmetric functions
LLT symmetric functions
Macdonald Polynomials
Non-symmetric Macdonald Polynomials
Witt symmetric functions
Non-Commutative Symmetric Functions and Quasi-Symmetric Functions
Introduction to Quasisymmetric Functions
Common combinatorial tools
Generic code for bases
Non-Commutative Symmetric Functions
Quasisymmetric functions
Symmetric Functions in Non-Commuting Variables
Bases for
.
Dual Symmetric Functions in Non-Commuting Variables
Symmetric Functions in Non-Commuting Variables
Root Systems
Cartan types
Dynkin diagrams
Cartan matrices
Coxeter matrices
Folded Cartan Types
Root systems
Tutorial: visualizing root systems
Root lattice realizations
Weight lattice realizations
Root lattices and root spaces
Weight lattices and weight spaces
Ambient lattices and ambient spaces
Coxeter Groups
Weyl Groups
Weyl Character Rings
Branching Rules
Root system data for affine Cartan types
Root system data for dual Cartan types
Root system data for reducible Cartan types
Root system data for relabelled Cartan types
Root system data for type A
Root system data for type B
Root system data for type C
Root system data for type D
Root system data for type E
Root system data for type F
Root system data for type G
Root system data for type H
Root system data for type I
Root system data for (untwisted) type A affine
Root system data for (untwisted) type B affine
Root system data for (untwisted) type C affine
Root system data for (untwisted) type D affine
Root system data for (untwisted) type E affine
Root system data for (untwisted) type F affine
Root system data for (untwisted) type G affine
Root system data for type BC affine
Kazhdan-Lusztig Polynomials
Knutson-Tao Puzzles
Acknowledgements
Crystals
Affine Crystals
Crystals
Direct Sum of Crystals
Crystals of letters
Elementary Crystals
Fast Rank Two Crystals
Highest weight crystals
Kirillov-Reshetikhin Crystals
Kyoto Path Model for Affine Highest Weight Crystals
Littelmann paths
Alcove paths
Spin Crystals
Tensor Products of Crystals
Crystals of Generalized Young Walls
Crystals of Tableaux in Nonexceptional Types and
Crystals of Modified Nakajima Monomials
Posets
Posets
Hasse diagrams of posets
Elements of posets, lattices, semilattices, etc.
Finite semilattices and lattices
Linear Extensions of Posets
A collection of posets and lattices.
Rigged Configurations
Rigged Configurations
Rigged Configuration Elements
Tensor Product of Kirillov-Reshetikhin Tableaux
Tensor Product of Kirillov-Reshetikhin Tableaux Elements
Kirillov-Reshetikhin Tableaux
Kleber Trees
Rigged Partitions
Rigged Configuration Bijections
Designs and Incidence Structures
Block designs.
Covering designs: coverings of
-element subsets of a
-set by
-sets
External Representations of Block Designs
Incidence structures.
Steiner Quadruple Systems
Catalog of designs
Combinatorial Species
Power Series
Basic Species
Operations on Species
Miscellaneous
Developer Tools
Backtracking
Output functions
Low-level permutations
Low-level splits
Low-level Combinations
Low-level multichoose
Tools
Rankers
Finite State Machines, Automata, Transducers
Examples
Words
Alphabet
Abstract word (finite or infinite)
Finite word
Infinite word
Word classes
A collection of constructors of common words
Combinatorial classes of words.
Shuffle product of words
Suffix Tries and Suffix Trees
Word morphisms/substitutions
Word paths
Pointwise addition of dictionaries
Miscellaneous
Combinatorial maps
Indices and Tables
¶
Index
Module Index
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