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# 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. http://www.sciencedirect.com/science/article/pii/S0097316509001745 , Arxiv 0810.2489v2.

[Tev2007] | Lenny Tevlin, Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product, Arxiv 0712.2201v1.

`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)

  Return the coefficient of the complete non-commutative symmetric function \(S^K\) in the expansion of the monomial non-commutative symmetric function \(M^I\) with respect to the complete basis over the ring \(R\). This is the coefficient in formula (36) of Tevlin’s paper [Tev2007].

  INPUT:

  - `R` – A ring, supposed to be a \(\QQ\)-algebra
  - `I`, `K` – compositions

  OUTPUT:

  - The coefficient of \(S^K\) in the expansion of \(M^I\) in the complete basis of the non-commutative symmetric functions over `R`.

  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: m_to_s_stat(QQ, Composition([2,1,2]), Composition([2,1,2]))
      8/3

`sage.combinat.ncsf_qsym.combinatorics.``number_of_fCT`(content_comp, shape_comp)

  Return the number of Immaculate tableaux of shape `shape_comp` and content `content_comp`.

  See [BBSSZ2012], Definition 3.9, for the notion of an immaculate tableau.

  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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