Contents Menu Expand Light mode Dark mode Auto light/dark, in light mode Auto light/dark, in dark mode Skip to content
Combinatorics
Light Logo Dark Logo
Version 10.8 Reference Manual
  • Home - Combinatorics
  • Comprehensive module list
    • A bijectionist’s toolkit
    • Abstract recursive trees
    • Affine permutations
    • Algebraic combinatorics
    • Alternating sign matrices
    • Backtracking
    • Baxter permutations
    • Bell and Uppuluri-Carpenter numbers
    • Binary recurrence sequences
    • Binary trees
    • Blob algebras
    • Boundedness of \(k\)-Regular Sequences
    • Brent Yorgey’s fast algorithm for integer vector (multiset) partitions
    • Cartesian products
    • Cluster complex (or generalized dual associahedron)
    • Colored permutations
    • Combinations
    • Combinatorial diagrams
    • Combinatorial functions
    • Combinatorial maps
    • Combinatorial triangles for posets and fans
    • Combinatorics
    • Introduction to combinatorics in Sage
    • Combinatorics quickref
    • Common automata and transducers (finite state machines generators)
    • Composition tableaux
    • Constellations
    • Cores
    • Counting
    • Counting, generating, and manipulating nonnegative integer matrices
    • Cyclic sieving phenomenon
    • De Bruijn sequences
    • Decorated permutations
    • Degree sequences
    • Derangements
    • Descent algebras
    • Diagram and partition algebras
    • Dyck words
    • Enumerated sets
    • Exact cover problem via dancing links
    • Families
    • Fast computation of combinatorial functions (Cython + mpz)
    • Fast set partition iterators
    • Finite state machines, automata, transducers
    • Free dendriform algebras
    • Free modules
    • Free pre-Lie algebras
    • Free quasi-symmetric functions
    • Fully commutative elements of Coxeter groups
    • Fully packed loops
    • Functions that compute some of the sequences in Sloane’s tables
    • Gelfand-Tsetlin patterns
    • Generalized Tamari lattices
    • Gray codes
    • Grossman-Larson Hopf algebras
    • Growth diagrams and dual graded graphs
    • Hall polynomials
    • Integer compositions
    • Integer partitions
    • Integer vectors modulo the action of a permutation group
    • Iterators over the partitions of an integer
    • Kazhdan-Lusztig polynomials
    • Key and Atom Polynomials
    • Kleshchev partitions
    • Knutson-Tao puzzles
    • Littlewood-Richardson tableaux
    • Miscellaneous
    • Necklaces
    • Non-decreasing parking functions
    • Nonnegative integer vectors
    • Ordered multiset partitions into sets and the minimaj crystal
    • Ordered rooted trees
    • Ordered set partitions
    • Output functions
    • Parallelogram polyominoes
    • Parking functions
    • Partition shifting algebras
    • Partition tuples
    • Partition/diagram algebras
    • Partitions, tableaux, and others
    • Paths in directed acyclic graphs
    • Perfect matchings
    • Permutations
    • Permutations (Cython file)
    • Plane partitions
    • Rankers
    • Recognizable series
    • Representations of the symmetric group
    • Residue sequences of tableaux
    • Restricted growth arrays
    • Ribbon shaped tableaux
    • Ribbon tableaux
    • Ribbons
    • Robinson-Schensted-Knuth correspondence
    • Rooted (unordered) trees
    • Schubert polynomials
    • Set partitions
    • Shard intersection order
    • Shifted primed tableaux
    • Shuffle product of iterables
    • Sidon sets and their generalizations, Sidon \(g\)-sets
    • Signed compositions
    • Similarity class types of matrices with entries in a finite field
    • Six vertex model
    • Skew partitions
    • Skew tableaux
    • Specht modules
    • Steinhaus-Johnson-Trotter algorithm
    • Strong and weak tableaux
    • Subsets
    • Subsets satisfying a hereditary property
    • Subsets whose elements satisfy a predicate pairwise
    • Substitutions over unit cube faces (Rauzy fractals)
    • Subword complex
    • Subword complex: auxiliary Cython functions
    • Subwords
    • Super partitions
    • Super tableaux
    • Symmetric group algebra
    • T-sequences
    • TableauTuples
    • Tableaux
    • Tamari Interval-posets
    • The Hillman-Grassl correspondence
    • Tiling solver
    • Tools for enumeration modulo the action of a permutation group
    • Transitive ideal closure tool
    • Tuples
    • Vector partitions
    • Weighted integer vectors
    • Yang-Baxter Graphs
    • \(\nu\)-Dyck words
    • \(\nu\)-Tamari lattice
    • \(k\)-regular sequences
    • \(q\)-Bernoulli numbers and polynomials
    • \(q\)-analogues
    • sine-Gordon Y-system plotter
    • Combinatorial Hopf algebras
    • Poirier-Reutenauer Hopf algebra of standard tableaux
    • Word quasi-symmetric functions
    • Cluster algebras and quivers
    • Cluster seeds
    • Helper functions for mutation types of quivers
    • Interactive display of quivers
    • Quiver
    • Quiver mutation types
    • mutation_class
    • Affine crystals
    • Affine factorization crystal of type \(A\)
    • Affinization crystals
    • Alcove paths
    • An introduction to crystals
    • Benkart-Kang-Kashiwara crystals for the general-linear Lie superalgebra
    • Catalog of crystal models For \(B(\infty)\)
    • Catalog of crystal models for Kirillov-Reshetikhin crystals
    • Catalog of crystals
    • Catalog of elementary crystals
    • Crystal of Bernstein-Zelevinsky multisegments
    • Crystal of Mirković-Vilonen polytopes
    • Crystals
    • Crystals of Kac modules of the general-linear Lie superalgebra
    • Crystals of generalized Young walls
    • Crystals of letters
    • Crystals of modified Nakajima monomials
    • Direct sum of crystals
    • Elementary crystals
    • Fast rank two crystals
    • Fully commutative stable Grothendieck crystal
    • Highest weight crystals
    • Induced crystals
    • Kirillov-Reshetikhin crystals
    • Kyoto path model for affine highest weight crystals
    • Littelmann paths
    • PBW data
    • Polyhedral realization of \(B(\infty)\)
    • Spin crystals
    • Star-crystal structure on \(B(\infty)\)
    • Subcrystals
    • Tensor products of crystal elements
    • Tensor products of crystals
    • Virtual crystals
    • \(\mathcal{B}(\infty)\) crystal of PBW monomials
    • \(\mathcal{B}(\infty)\) crystals of tableaux in nonexceptional types and \(G_2\)
    • Balanced incomplete block designs (BIBD)
    • Block designs
    • Bounds on the number of mutually orthogonal Latin squares
    • Catalog of designs
    • Combinatorial designs and incidence structures
    • Covering arrays
    • Covering designs: coverings of \(t\)-element subsets of a \(v\)-set by \(k\)-sets
    • Cython functions for combinatorial designs
    • Database of generalised quadrangles with spread
    • Database of small combinatorial designs
    • Difference families
    • Difference matrices
    • Evenly distributed sets in finite fields
    • External representations of block designs
    • Group-divisible designs (GDD)
    • Hypergraph isomorphic copy search
    • 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)
    • Resolvable balanced incomplete block design (RBIBD)
    • Steiner quadruple systems
    • Two-graphs
    • Enumerated set of lists of integers with constraints, in inverse lexicographic order
    • Enumerated set of lists of integers with constraints: base classes
    • Enumerated set of lists of integers with constraints: front-end
    • Lists of nonnegative integers with constraints.
    • Combinatorics on matrices
    • Dancing Links internal pyx code
    • Dancing links C++ wrapper
    • Hadamard matrices
    • Latin squares
    • Common combinatorial tools
    • Generic code for bases
    • Introduction to quasisymmetric functions
    • Noncommutative symmetric functions
    • Noncommutative symmetric functions and quasi-symmetric functions
    • Quasisymmetric functions
    • Bases for NCSym
    • Dual symmetric functions in non-commuting variables
    • Symmetric functions in non-commuting variables
    • Symmetric functions in non-commuting variables
    • Catalog of path tableaux
    • Dyck paths
    • Frieze patterns
    • Path tableaux
    • Path tableaux
    • Semistandard tableaux
    • Bubble and Shuffle lattices
    • Cartesian products of posets
    • Catalog of posets and lattices
    • D-complete posets
    • Elements of posets, lattices, semilattices, etc.
    • Fast linear extension iterator
    • Finite lattices and semilattices
    • Finite posets
    • Forest posets
    • Hasse diagrams of posets
    • Hochschild lattices
    • Incidence algebras
    • Linear extensions of posets
    • Mobile posets
    • Möbius algebras
    • Posets
    • Some fast computations for finite posets
    • Some fast computations for finite posets using FLINT matrices
    • Abstract classes for the rigged configuration bijections
    • Bijection between rigged configurations and KR tableaux
    • Bijection between rigged configurations for \(B(\infty)\) and marginally large tableaux
    • Bijection classes for type \(A_n^{(1)}\)
    • Bijection classes for type \(A_{2n-1}^{(2)}\)
    • Bijection classes for type \(A_{2n}^{(2)\dagger}\)
    • Bijection classes for type \(A_{2n}^{(2)}\)
    • Bijection classes for type \(B_n^{(1)}\)
    • Bijection classes for type \(C_n^{(1)}\)
    • Bijection classes for type \(D_4^{(3)}\)
    • Bijection classes for type \(D_n^{(1)}\)
    • Bijection classes for type \(D_{n+1}^{(2)}\)
    • Bijection classes for type \(E_{6,7}^{(1)}\)
    • Crystal of rigged configurations
    • Kirillov-Reshetikhin tableaux
    • Kleber trees
    • Rigged configuration elements
    • Rigged configurations
    • Rigged configurations
    • Rigged configurations of \(\mathcal{B}(\infty)\)
    • Rigged partitions
    • Tensor product of Kirillov-Reshetikhin tableaux
    • Tensor product of Kirillov-Reshetikhin tableaux elements
    • 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
    • Extended affine Weyl groups
    • Finite complex reflection groups
    • Finite real reflection groups
    • Fundamental group of an extended affine Weyl group
    • Group algebras of root lattice realizations
    • Hecke algebra representations
    • Integrable representations of affine Lie algebras
    • Nonsymmetric Macdonald polynomials
    • Pieri factors
    • Reflection group elements
    • Reflection groups: auxiliary Cython functions
    • Root lattice realizations
    • Root lattices and root spaces
    • 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 Cartan types with marked nodes
    • Root system data for affine Cartan types
    • Root system data for dual Cartan types
    • Root system data for folded Cartan types
    • Root system data for reducible Cartan types
    • Root system data for relabelled Cartan types
    • Root system data for super type A
    • Root system data for type A
    • Root system data for type A infinity
    • Root system data for type B
    • Root system data for type BC affine
    • 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 type Q
    • Root systems
    • Root systems
    • Tutorial: visualizing root systems
    • Weight lattice realizations
    • Weight lattices and weight spaces
    • Weyl character rings
    • Weyl groups
    • Abreu-Nigro symmetric functions
    • Characters of the symmetric group as bases of the symmetric functions
    • Classical symmetric functions
    • Elementary symmetric functions
    • Generic dual bases symmetric functions
    • Hall-Littlewood polynomials
    • Hecke character basis
    • Homogeneous symmetric functions
    • Jack symmetric functions
    • Kostka-Foulkes polynomials
    • LLT symmetric functions
    • Macdonald polynomials
    • Monomial symmetric functions
    • Multiplicative symmetric functions
    • Non-symmetric Macdonald polynomials
    • Orthogonal symmetric functions
    • Power sum symmetric functions
    • Quotient of symmetric function space by ideal generated by Hall-Littlewood symmetric functions
    • Schur symmetric functions
    • Symmetric functions
    • Symmetric functions
    • Symmetric functions defined by orthogonality and triangularity
    • Symmetric functions, with their multiple realizations
    • Symplectic symmetric functions
    • Witt symmetric functions
    • \(k\)-Schur functions
    • Characteristic species
    • Combinatorial species
    • Combinatorial species
    • Composition species
    • Cycle species
    • Empty species
    • Examples of combinatorial species
    • Functorial composition species
    • Generating series
    • Linear-order species
    • Miscellaneous functions
    • Partition species
    • Permutation species
    • Product species
    • Recursive species
    • Set species
    • Species structures
    • Subset species
    • Sum species
    • Abstract word (finite or infinite)
    • Alphabet
    • Combinatorics on words
    • Common words
    • Datatypes for finite words
    • Datatypes for words defined by iterators and callables
    • Fast word datatype using an array of unsigned char
    • Finite word
    • Infinite word
    • Lyndon words
    • Morphic words
    • Set of words
    • Shuffle product of words
    • Suffix tries and suffix trees
    • User-customizable options for words
    • Word classes
    • Word morphisms/substitutions
    • Word paths
    • C-finite sequences
    • Lazy Combinatorial Species
Back to top
View this page
Edit this page

Families¶

This is a backward compatibility stub. Use sage.sets.family instead.

Next
Fast computation of combinatorial functions (Cython + mpz)
Previous
Exact cover problem via dancing links
Copyright © 2005--2025, The Sage Development Team
Made with Sphinx and @pradyunsg's Furo