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. |
Return the number of standard composition tableaux of shape with
descent composition
.
INPUT:
OUTPUT:
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
Returns the coefficient as defined in [NCSF].
INPUT:
OUTPUT:
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
Returns the coefficient as defined in [NCSF].
INPUT:
OUTPUT:
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
Returns the coefficient as defined in [NCSF].
INPUT:
OUTPUT:
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
Returns the coefficient as defined in [NCSF].
INPUT:
OUTPUT:
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
Return the compositions of ordered as defined in [QSCHUR].
Let return the composition
after sorting. For
compositions
and
, we order
if
INPUT:
OUTPUT:
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]]
Return the coefficient of the complete non-commutative symmetric
function in the expansion of the monomial non-commutative
symmetric function
with respect to the complete basis
over the ring
. This is the coefficient in formula (36) of
Tevlin’s paper [Tev2007].
INPUT:
OUTPUT:
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
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:
OUTPUT:
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