def Ht(mu, q=None, t=None, pi=None):
"""
Returns the symmetric Macdonald polynomial using the Haiman,
Haglund, and Loehr formula.
Note that if both `q` and `t` are specified, then they must have the
same parent.
REFERENCE:
- 'A combinatorial formula for non-symmetric Macdonald
polynomials'. Haiman, Haglund, and Loehr.
http://arxiv.org/abs/math/0601693
EXAMPLES::
sage: from sage.combinat.sf.ns_macdonald import Ht
sage: HHt = SymmetricFunctions(QQ['q','t'].fraction_field()).macdonald().Ht()
sage: Ht([0,0,1])
x0 + x1 + x2
sage: HHt([1]).expand(3)
x0 + x1 + x2
sage: Ht([0,0,2])
x0^2 + (q + 1)*x0*x1 + x1^2 + (q + 1)*x0*x2 + (q + 1)*x1*x2 + x2^2
sage: HHt([2]).expand(3)
x0^2 + (q + 1)*x0*x1 + x1^2 + (q + 1)*x0*x2 + (q + 1)*x1*x2 + x2^2
"""
P, q, t, n, R, x = _check_muqt(mu, q, t, pi)
res = 0
for a in n:
weight = a.weight()
res += q**a.maj() * t**a.inv() * prod(x[i]**weight[i]
for i in range(len(weight)))
return res
def _equation(self, var_mapping):
"""
Returns the right hand side of an algebraic equation satisfied by
this species. This is a utility function called by the
algebraic_equation_system method.
EXAMPLES::
sage: X = species.SingletonSpecies()
sage: S = X * X
sage: S.algebraic_equation_system()
[node0 - z^2]
"""
from sage.rings.all import prod
return prod(var_mapping[operand] for operand in self._state_info)
def __pow__(self, n):
"""
EXAMPLES::
sage: L = LazyPowerSeriesRing(QQ)
sage: f = L([1,1,0]) # 1+x
sage: g = f^3
sage: g.coefficients(4)
[1, 3, 3, 1]
::
sage: f^0
1
"""
if not isinstance(n, (int, Integer)) or n < 0:
raise ValueError, "n must be a nonnegative integer"
return prod([self]*n, self.parent().identity_element())
def E(mu, q=None, t=None, pi=None):
"""
Returns the non-symmetric Macdonald polynomial in type A
corresponding to a shape ``mu``, with basement permuted according to
``pi``.
Note that if both `q` and `t` are specified, then they must have
the same parent.
REFERENCE:
- 'A combinatorial formula for non-symmetric Macdonald
polynomials'. Haiman, Haglund, and Loehr.
http://arxiv.org/abs/math/0601693
EXAMPLES::
sage: from sage.combinat.sf.ns_macdonald import E
sage: E([0,0,0])
1
sage: E([1,0,0])
x0
sage: E([0,1,0])
((-t + 1)/(-q*t^2 + 1))*x0 + x1
sage: E([0,0,1])
((-t + 1)/(-q*t + 1))*x0 + ((-t + 1)/(-q*t + 1))*x1 + x2
sage: E([1,1,0])
x0*x1
sage: E([1,0,1])
((-t + 1)/(-q*t^2 + 1))*x0*x1 + x0*x2
sage: E([0,1,1])
((-t + 1)/(-q*t + 1))*x0*x1 + ((-t + 1)/(-q*t + 1))*x0*x2 + x1*x2
sage: E([2,0,0])
x0^2 + ((-q*t + q)/(-q*t + 1))*x0*x1 + ((-q*t + q)/(-q*t + 1))*x0*x2
sage: E([0,2,0])
((-t + 1)/(-q^2*t^2 + 1))*x0^2 + ((-q^2*t^3 + q^2*t^2 - q*t^2 + 2*q*t - q + t - 1)/(-q^3*t^3 + q^2*t^2 + q*t - 1))*x0*x1 + x1^2 + ((q*t^2 - 2*q*t + q)/(q^3*t^3 - q^2*t^2 - q*t + 1))*x0*x2 + ((-q*t + q)/(-q*t + 1))*x1*x2
"""
P, q, t, n, R, x = _check_muqt(mu, q, t, pi)
res = 0
for a in n:
weight = a.weight()
res += q**a.maj() * t**a.coinv() * a.coeff(q, t) * prod(
x[i]**weight[i] for i in range(len(weight)))
return res
def E_integral(mu, q=None, t=None, pi=None):
"""
Returns the integral form for the non-symmetric Macdonald
polynomial in type A corresponding to a shape mu.
Note that if both q and t are specified, then they must have the
same parent.
REFERENCE:
- 'A combinatorial formula for non-symmetric Macdonald
polynomials'. Haiman, Haglund, and Loehr.
http://arxiv.org/abs/math/0601693
EXAMPLES::
sage: from sage.combinat.sf.ns_macdonald import E_integral
sage: E_integral([0,0,0])
1
sage: E_integral([1,0,0])
(-t + 1)*x0
sage: E_integral([0,1,0])
(-q*t^2 + 1)*x0 + (-t + 1)*x1
sage: E_integral([0,0,1])
(-q*t + 1)*x0 + (-q*t + 1)*x1 + (-t + 1)*x2
sage: E_integral([1,1,0])
(t^2 - 2*t + 1)*x0*x1
sage: E_integral([1,0,1])
(q*t^3 - q*t^2 - t + 1)*x0*x1 + (t^2 - 2*t + 1)*x0*x2
sage: E_integral([0,1,1])
(q^2*t^3 + q*t^4 - q*t^3 - q*t^2 - q*t - t^2 + t + 1)*x0*x1 + (q*t^2 - q*t - t + 1)*x0*x2 + (t^2 - 2*t + 1)*x1*x2
sage: E_integral([2,0,0])
(t^2 - 2*t + 1)*x0^2 + (q^2*t^2 - q^2*t - q*t + q)*x0*x1 + (q^2*t^2 - q^2*t - q*t + q)*x0*x2
sage: E_integral([0,2,0])
(q^2*t^3 - q^2*t^2 - t + 1)*x0^2 + (q^4*t^3 - q^3*t^2 - q^2*t + q*t^2 - q*t + q - t + 1)*x0*x1 + (t^2 - 2*t + 1)*x1^2 + (q^4*t^3 - q^3*t^2 - q^2*t + q)*x0*x2 + (q^2*t^2 - q^2*t - q*t + q)*x1*x2
"""
P, q, t, n, R, x = _check_muqt(mu, q, t, pi)
res = 0
for a in n:
weight = a.weight()
res += q**a.maj() * t**a.coinv() * a.coeff_integral(q, t) * prod(
x[i]**weight[i] for i in range(len(weight)))
return res
def change_support(perm, support, change_perm=None):
"""
Changes the support of a permutation defined on [1, ..., n] to
support.
EXAMPLES::
sage: from sage.combinat.species.misc import change_support
sage: p = PermutationGroupElement((1,2,3)); p
(1,2,3)
sage: change_support(p, [3,4,5])
(3,4,5)
"""
if change_perm is None:
change_perm = prod([PermutationGroupElement((i+1,support[i])) for i in range(len(support)) if i+1 != support[i]], PermutationGroupElement([], SymmetricGroup(support)))
if isinstance(perm, PermutationGroup_generic):
return PermutationGroup([change_support(g, support, change_perm) for g in perm.gens()])
return change_perm*perm*~change_perm