Common combinatorial tools

REFERENCES:

[NCSF](1, 2, 3, 4) Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon, Noncommutative Symmetric Functions, Adv. Math. 112 (1995), no. 2, 218-348.
[QSCHUR]Haglund, Luoto, Mason, van Willigenburg, Quasisymmetric Schur functions, J. Comb. Theory Ser. A 118 (2011), 463-490.
sage.combinat.ncsf_qsym.combinatorics.coeff_dab(I, J)

Return the number of standard composition tableaux of shape I with descent composition J.

INPUT:

  • I, J – compositions

OUTPUT:

  • An integer

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_dab
sage: coeff_dab(Composition([2,1]),Composition([2,1]))
1
sage: coeff_dab(Composition([1,1,2]),Composition([1,2,1]))
0
sage.combinat.ncsf_qsym.combinatorics.coeff_ell(J, I)

Returns the coefficient \ell_{J,I} as defined in [NCSF].

INPUT:

  • J – a composition
  • I – a composition refining J

OUTPUT:

  • integer

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_ell
sage: coeff_ell(Composition([1,1,1]), Composition([2,1]))
2
sage: coeff_ell(Composition([2,1]), Composition([3]))
2
sage.combinat.ncsf_qsym.combinatorics.coeff_lp(J, I)

Returns the coefficient lp_{J,I} as defined in [NCSF].

INPUT:

  • J – a composition
  • I – a composition refining J

OUTPUT:

  • integer

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_lp
sage: coeff_lp(Composition([1,1,1]), Composition([2,1]))
1
sage: coeff_lp(Composition([2,1]), Composition([3]))
1
sage.combinat.ncsf_qsym.combinatorics.coeff_pi(J, I)

Returns the coefficient \pi_{J,I} as defined in [NCSF].

INPUT:

  • J – a composition
  • I – a composition refining J

OUTPUT:

  • integer

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_pi
sage: coeff_pi(Composition([1,1,1]), Composition([2,1]))
2
sage: coeff_pi(Composition([2,1]), Composition([3]))
6
sage.combinat.ncsf_qsym.combinatorics.coeff_sp(J, I)

Returns the coefficient sp_{J,I} as defined in [NCSF].

INPUT:

  • J – a composition
  • I – a composition refining J

OUTPUT:

  • integer

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.combinatorics import coeff_sp
sage: coeff_sp(Composition([1,1,1]), Composition([2,1]))
2
sage: coeff_sp(Composition([2,1]), Composition([3]))
4
sage.combinat.ncsf_qsym.combinatorics.compositions_order(n)

Return the compositions of n ordered as defined in [QSCHUR].

Let S(\gamma) return the composition \gamma after sorting. For compositions \alpha and \beta, we order \alpha \rhd \beta if

  1. S(\alpha) > S(\beta) lexicographically, or
  2. S(\alpha) = S(\beta) and \alpha > \beta lexicographically.

INPUT:

  • n – a positive integer

OUTPUT:

  • A list of the compositions of n sorted into decreasing order by \rhd

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.combinatorics import compositions_order
sage: compositions_order(3)
[[3], [2, 1], [1, 2], [1, 1, 1]]
sage: compositions_order(4)
[[4], [3, 1], [1, 3], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]
sage.combinat.ncsf_qsym.combinatorics.m_to_s_stat(R, I, K)

Returns the statistic for the expansion of the Monomial basis element indexed by two compositions, as in formula (36) of Tevlin’s “Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product”.

INPUT:

  • R – A ring
  • I, K – compositions

OUTPUT:

  • An integer

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.combinatorics import m_to_s_stat
sage: m_to_s_stat(QQ,Composition([2,1]), Composition([1,1,1]))
-1
sage: m_to_s_stat(QQ,Composition([3]), Composition([1,2]))
-2
sage.combinat.ncsf_qsym.combinatorics.number_of_fCT(content_comp, shape_comp)

Returns the number of Immaculate tableau of shape shape_comp and content content_comp.

INPUT:

  • content_comp, shape_comp – compositions

OUTPUT:

  • An integer

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.combinatorics import number_of_fCT
sage: number_of_fCT(Composition([3,1]), Composition([1,3]))
0
sage: number_of_fCT(Composition([1,2,1]), Composition([1,3]))
1
sage: number_of_fCT(Composition([1,1,3,1]), Composition([2,1,3]))
2

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