# 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.kfpoly.KostkaFoulkesPolynomial(mu, nu, t=None)

Returns the Kostka-Foulkes polynomial $$K_{\mu, \nu}(t)$$.

INPUT:

• mu, nu – partitions
• t – an optional parameter (default: None)

OUTPUT:

• the Koskta-Foulkes polynomial indexed by partitions mu and nu and evaluated at the parameter t. If t is None the resulting polynomial is in the polynomial ring $$ZZ['t']$$.

EXAMPLES:

sage: KostkaFoulkesPolynomial([2,2],[2,2])
1
sage: KostkaFoulkesPolynomial([2,2],[4])
0
sage: KostkaFoulkesPolynomial([2,2],[1,1,1,1])
t^4 + t^2
sage: KostkaFoulkesPolynomial([2,2],[2,1,1])
t
sage: q = PolynomialRing(QQ,'q').gen()
sage: KostkaFoulkesPolynomial([2,2],[2,1,1],q)
q

sage.combinat.sf.kfpoly.compat(n, mu, nu)

Generate all possible partitions of $$n$$ that can precede $$\mu,\nu$$ in a rigging sequence.

INPUT:

• n – a positive integer
• mu, nu – partitions

OUTPUT:

• a list of partitions

EXAMPLES:

sage: from sage.combinat.sf.kfpoly import *
sage: compat(4, [1], [2,1])
[[1, 1, 1, 1], [2, 1, 1], [2, 2], [3, 1], [4]]
sage: compat(3, [1], [2,1])
[[1, 1, 1], [2, 1], [3]]
sage: compat(2, [1], [])
[[2]]
sage: compat(3, [1], [])
[[2, 1], [3]]
sage: compat(3, [2], [1])
[[3]]
sage: compat(4, [1,1], [])
[[2, 2], [3, 1], [4]]
sage: compat(4, [2], [])
[[4]]

sage.combinat.sf.kfpoly.dom(mu, snu)

Returns True if sum(mu[:i+1]) >= snu[i] for all 0 <= i < len(snu); otherwise, it returns False.

INPUT:

• mu – a partition
• snu – a sequence of positive integers

OUTPUT:

• a boolean value

EXAMPLES:

sage: from sage.combinat.sf.kfpoly import *
sage: dom([3,2,1],[2,4,5])
True
sage: dom([3,2,1],[2,4,7])
False
sage: dom([3,2,1],[2,6,5])
False
sage: dom([3,2,1],[4,4,4])
False

sage.combinat.sf.kfpoly.kfpoly(mu, nu, t=None)

Returns the Kostka-Foulkes polynomial $$K_{\mu, \nu}(t)$$ by generating all rigging sequences for the shape $$\mu$$, and then selecting those of content $$\nu$$.

INPUT:

• mu, nu – partitions
• t – an optional parameter (default: None)

OUTPUT:

• the Koskta-Foulkes polynomial indexed by partitions mu and nu and evaluated at the parameter t. If t is None the resulting polynomial is in the polynomial ring $$\mathbb{Z}['t']$$.

EXAMPLES:

sage: from sage.combinat.sf.kfpoly import kfpoly
sage: kfpoly([2,2], [2,1,1])
t
sage: kfpoly([4], [2,1,1])
t^3
sage: kfpoly([4], [2,2])
t^2
sage: kfpoly([1,1,1,1], [2,2])
0

sage.combinat.sf.kfpoly.q_bin(a, b, t=None)

Returns the $$t$$-binomial coefficient $$[a+b,b]_t$$.

INPUT:

• a, b – two nonnegative integers

By definition $$[a+b,b]_t = (1-t)(1-t^2) \cdots (1-t^{a+b}) / ((1-t) \cdots (1-t^b) (1-t) \cdots (1-t^a))$$.

EXAMPLES:

sage: from sage.combinat.sf.kfpoly import q_bin
sage: t = PolynomialRing(ZZ, 't').gen()
sage: q_bin(4,2, t)
See http://trac.sagemath.org/14496 for details.
t^8 + t^7 + 2*t^6 + 2*t^5 + 3*t^4 + 2*t^3 + 2*t^2 + t + 1
sage: q_bin(4,3, t)
t^12 + t^11 + 2*t^10 + 3*t^9 + 4*t^8 + 4*t^7 + 5*t^6 + 4*t^5 + 4*t^4 + 3*t^3 + 2*t^2 + t + 1

sage.combinat.sf.kfpoly.riggings(part)

Generate all possible rigging sequences for a fixed sage.combinat.partition.

INPUT:

• part – a partition

OUTPUT:

• returns a list of riggings associated to the partition part

EXAMPLES:

sage: from sage.combinat.sf.kfpoly import *
sage: riggings([3])
[[[1, 1, 1]], [[2, 1]], [[3]]]
sage: riggings([2,1])
[[[2, 1], [1]], [[3], [1]]]
sage: riggings([1,1,1])
[[[3], [2], [1]]]
sage: riggings([2,2])
[[[2, 2], [1, 1]], [[3, 1], [1, 1]], [[4], [1, 1]], [[4], [2]]]
sage: riggings([2,2,2])
[[[3, 3], [2, 2], [1, 1]],
[[4, 2], [2, 2], [1, 1]],
[[5, 1], [2, 2], [1, 1]],
[[6], [2, 2], [1, 1]],
[[5, 1], [3, 1], [1, 1]],
[[6], [3, 1], [1, 1]],
[[6], [4], [2]]]

sage.combinat.sf.kfpoly.schur_to_hl(mu, t=None)

Returns a dictionary corresponding to $$s_\mu$$ in Hall-Littlewood $$P$$ basis.

INPUT:

• mu – a partition
• t – an optional parameter (default : the generator from $$\mathbb{Z}['t']$$ )

OUTPUT:

• a dictionary with the coefficients $$K_{\mu\nu}(t)$$ for $$\nu$$ smaller in dominance order than $$\mu$$

EXAMPLES:

sage: from sage.combinat.sf.kfpoly import *
sage: schur_to_hl([1,1,1])
{[1, 1, 1]: 1}
sage: a = schur_to_hl([2,1])
sage: for m,c in sorted(a.iteritems()): print m, c
[1, 1, 1] t^2 + t
[2, 1] 1
sage: a = schur_to_hl([3])
sage: for m,c in sorted(a.iteritems()): print m, c
[1, 1, 1] t^3
[2, 1] t
[3] 1
sage: a = schur_to_hl([4])
sage: for m,c in sorted(a.iteritems()): print m, c
[1, 1, 1, 1] t^6
[2, 1, 1] t^3
[2, 2] t^2
[3, 1] t
[4] 1
sage: a = schur_to_hl([3,1])
sage: for m,c in sorted(a.iteritems()): print m, c
[1, 1, 1, 1] t^5 + t^4 + t^3
[2, 1, 1] t^2 + t
[2, 2] t
[3, 1] 1
sage: a = schur_to_hl([2,2])
sage: for m,c in sorted(a.iteritems()): print m, c
[1, 1, 1, 1] t^4 + t^2
[2, 1, 1] t
[2, 2] 1
sage: a = schur_to_hl([2,1,1])
sage: for m,c in sorted(a.iteritems()): print m, c
[1, 1, 1, 1] t^3 + t^2 + t
[2, 1, 1] 1
sage: a = schur_to_hl([1,1,1,1])
sage: for m,c in sorted(a.iteritems()): print m, c
[1, 1, 1, 1] 1
sage: a = schur_to_hl([2,2,2])
sage: for m,c in sorted(a.iteritems()): print m, c
[1, 1, 1, 1, 1, 1] t^9 + t^7 + t^6 + t^5 + t^3
[2, 1, 1, 1, 1] t^4 + t^2
[2, 2, 1, 1] t
[2, 2, 2] 1

sage.combinat.sf.kfpoly.weight(rg, t=None)

Returns the weight of a rigging.

INPUT:

• rg – a rigging, a list of partitions
• t – an optional parameter, (default : uses the generator from $$ZZ['t']$$)

OUTPUT:

• a polynomial in the parameter t

EXAMPLES:

sage: from sage.combinat.sf.kfpoly import weight
sage: weight([[2,1], [1]])
1
sage: weight([[3], [1]])
t^2 + t
sage: weight([[2,1], [3]])
t^4
sage: weight([[2, 2], [1, 1]])
1
sage: weight([[3, 1], [1, 1]])
t
sage: weight([[4], [1, 1]], 2)
16
sage: weight([[4], [2]], t=2)
4


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Hall-Littlewood Polynomials