- sage.combinat.all
- sage.combinat.alternating_sign_matrix: Alternating Sign Matrices
- sage.combinat.backtrack: Backtracking
- sage.combinat.cartesian_product: Cartesian Products
- sage.combinat.choose_nk: Alternating sign matrices
- sage.combinat.combinat: Combinatorial Functions.
- sage.combinat.combination: Combinations
- sage.combinat.combinatorial_algebra: Combinatorial Algebras
A combinatorial algebra is an algebra whose basis elements are indexed
by a combinatorial class.
- sage.combinat.composition: Compositions
A composition c of a nonnegative integer n is a list of positive integers with total sum n.
- sage.combinat.composition_signed: Signed Compositions
- sage.combinat.crystals
- sage.combinat.dlx: Exact Cover Problem via Dancing Links
- sage.combinat.dyck_word: Dyck Words...
- sage.combinat.expnums: File: sage/combinat/expnums.pyx (starting at line 1)
Compute Bell and Uppuluri-Carpenter numbers.
- sage.combinat.family: Families
- sage.combinat.finite_class: Finite combinatorial classes
- sage.combinat.free_module: Free modules
- sage.combinat.generator
- sage.combinat.graph_path: Paths in Directed Acyclic Graphs
- sage.combinat.integer_list: Tools for generating lists of integers in lexicographic order.
- sage.combinat.integer_vector: Integer vectors
- sage.combinat.integer_vector_weighted
- sage.combinat.kshape: k-shapes
The k-interior of a partition is the subpartition whose
cells have hook sizes greater than k.
- sage.combinat.lyndon_word: Lyndon words
- sage.combinat.matrices
- sage.combinat.misc: Miscellaneous
- sage.combinat.multichoose_nk
- sage.combinat.necklace: Necklaces
Algorithm from
A fast algorithm to generate necklaces with fixed content
Source Theoretical Computer Science archive
Volume 301 , Issue 1-3 (May 2003) table of contents
Pages: 477 - 489
Year of Publication: 2003
ISSN:0304-3975
Author
Joe Sawada Department of Computer Science, University of Toronto, 10 King's College Road, Toronto, Ont.
- sage.combinat.output
- sage.combinat.parking_non_decreasing_function: Non Decreasing Parking Functions
A non decreasing parking functions of size $n$ is a non decreasing
function $f$ from $\{1,\dots,n\}$ to itself such that for all $i$, one
has $f(i) \leq i$.
- sage.combinat.partition: Partitions
A partition $p$ of a nonnegative integer $n$ is a non-increasing list of
positive integers (the \emph{parts} of the partition) with total sum $n$.
- sage.combinat.partition_algebra: Partition/Diagram Algebras
- sage.combinat.partitions: File: sage/combinat/partitions.pyx (starting at line 1)
Number of partitions of integer
AUTHOR:
-- William Stein (2007-07-28): initial version
-- Jonathan Bober (2007-07-28): wrote the program partitions_c.cc
that does all the actual heavy lifting.
- sage.combinat.permutation: Permutations
The Permutations module.
- sage.combinat.permutation_nk
- sage.combinat.posets
- sage.combinat.q_analogues: q-Analogues
- sage.combinat.ranker
- sage.combinat.restricted_growth
- sage.combinat.ribbon: Ribbons
- sage.combinat.ribbon_tableau: Ribbon Tableaux
- sage.combinat.root_system
- sage.combinat.schubert_polynomial: Schubert Polynomials
- sage.combinat.set_partition: Set Partitions
A set partition s of a set set is a partition of set, into subsets called parts and represented as a set of sets.
- sage.combinat.set_partition_ordered: Ordered Set Partitions
An ordered set partition p of a set s is a partition of s, into subsets called parts and represented as a list of sets.
- sage.combinat.sf
- sage.combinat.sf.all
- sage.combinat.sf.classical: Classical symmetric functions.
- sage.combinat.sf.dual: Generic dual bases symmetric functions
- sage.combinat.sf.elementary: Elementary symmetric functions
- sage.combinat.sf.hall_littlewood: Hall-Littlewood Polynomials
- sage.combinat.sf.homogeneous: Homogenous symmetric functions
- sage.combinat.sf.jack: Jack Polynomials
- sage.combinat.sf.kfpoly: Kostka-Foulkes Polynomials
Based on the algorithms in John Stembridge's SF package for Maple which
can be found at http://www.math.lsa.umich.edu/~jrs/maple.html .
- sage.combinat.sf.kschur: k-Schur Functions
- sage.combinat.sf.llt: LLT Polynomials
- sage.combinat.sf.macdonald: Macdonald Polynomials -- under development.
- sage.combinat.sf.misc:
- sage.combinat.sf.monomial: Monomial symmetric functions
- sage.combinat.sf.multiplicative: Multiplicative symmetric functions
- sage.combinat.sf.ns_macdonald
- sage.combinat.sf.orthotriang: Symmetric functions defined by orthogonality and triangularity.
- sage.combinat.sf.powersum: Power-sum symmetric functions
- sage.combinat.sf.schur: Schur symmetric functions
- sage.combinat.sf.sfa: Symmetric Functions
AUTHOR: Mike Hansen, 2007-06-15
sage: s = SymmetricFunctionAlgebra(QQ, basis='schur')
sage: e = SymmetricFunctionAlgebra(QQ, basis='elementary')
sage: f1 = s([2,1]); f1
s[2, 1]
sage: f2 = e(f1); f2
e[2, 1] - e[3]
sage: f1 == f2
True
sage: f1.expand(3, alphabet=['x','y','z'])
x^2*y + x*y^2 + x^2*z + 2*x*y*z + y^2*z + x*z^2 + y*z^2
sage: f2.expand(3, alphabet=['x','y','z'])
x^2*y + x*y^2 + x^2*z + 2*x*y*z + y^2*z + x*z^2 + y*z^2
sage: m = SFAMonomial(QQ)
sage: m([3,1])
m[3, 1]
sage: m(4)
4*m[]
sage: m([4])
m[4]
sage: 3*m([3,1])-1/2*m([4])
3*m[3, 1] - 1/2*m[4]
Code needs to be added to coerce symmetric polynomials into symmetric functions.
- sage.combinat.skew_partition: Skew Partitions
A skew partition $skp$ of size $n$ is a pair of partitions $[p_1, p_2]$
where $p_1$ is a partition of the integer $n_1$, $p_2$ is a partition
of the integer $n_2$, $p_2$ is an inner partition of $p_1$, and
$n = n_1 - n_2$.
- sage.combinat.skew_tableau: Skew Tableaux
- sage.combinat.sloane_functions: Functions that compute some of the sequences in Sloane's tables
EXAMPLES:
Type sloane.[tab] to see a list of the sequences that are defined.
- sage.combinat.species
- sage.combinat.species.all
- sage.combinat.species.generating_series: Generating Series
This file makes a number of extensions ot lazy power series by endowing
them with some semantic content for how they're to be interpreted.
- sage.combinat.species.library
- sage.combinat.species.series: Lazy Power Series
This file provides an implementation of lazy univariate power series, which uses
the stream class for its internal data structure.
- sage.combinat.species.series_order: Series Order
This file provides some utility classes which are useful when working with
unknown, known, and infinite series orders for univarite power series.
- sage.combinat.species.species: Combinatorial Species
This file defines the main classes for working with combinatorial species, operations
on them, as well as some implementations of basic species required for other constructions.
- sage.combinat.species.stream: Streams or Infinite Arrays
This code is based on the work of Ralf Hemmecke and Martin Rubey's Aldor-Combinat, which
can be found at http://www.risc.uni-linz.ac.at/people/hemmecke/aldor/combinat/index.html.
- sage.combinat.split_nk
- sage.combinat.subset: Subsets
- sage.combinat.subword: Subwords
- sage.combinat.symmetric_group_algebra: Symmetric Group Algebra
- sage.combinat.tableau: Tableaux
- sage.combinat.tools: Tools
- sage.combinat.tuple: Tuples
- sage.combinat.word: Words
- sage.combinat.yamanouchi: \begin{itemize}
\item A right (respectively left) Yamanouchi word on a completely ordered alphabet, for instance [1,2,...,n], is a word math such that any right (respectively left) factor of math contains more entries math than math.
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