def _s_to_self_base(self, part):
r"""
Returns a function which gives the coefficient of a partition
in the expansion of the Schur functions ``s(part)`` in the Hall-Littlewood
`P` basis.
INPUT:
- ``self`` -- an instance of the Hall-Littlewood `P` basis
- ``part`` -- a partition
OUTPUT:
- returns a function which accepts a partition ``part2`` and returns
the coefficient of ``P(part2)`` in ``s(part)``
This coefficient is the t-Kostka-Foulkes polynomial `K_{part,part2}(t)`
EXAMPLES::
sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P()
sage: f21 = HLP._s_to_self_base(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[0, 1, t^2 + t]
"""
t = QQt.gen()
zero = self.base_ring()(0)
res_dict = kfpoly.schur_to_hl(part, t)
f = lambda part2: res_dict.get(part2, zero)
return f
def _s_to_self_base(self, part):
r"""
Returns a function which gives the coefficient of a partition
in the expansion of the Schur functions ``s(part)`` in the Hall-Littlewood
`P` basis.
INPUT:
- ``self`` -- an instance of the Hall-Littlewood `P` basis
- ``part`` -- a partition
OUTPUT:
- returns a function which accepts a partition ``part2`` and returns
the coefficient of ``P(part2)`` in ``s(part)``
This coefficient is the t-Kostka-Foulkes polynomial `K_{part,part2}(t)`
EXAMPLES::
sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P()
sage: f21 = HLP._s_to_self_base(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[0, 1, t^2 + t]
"""
t = QQt.gen()
zero = self.base_ring()(0)
res_dict = kfpoly.schur_to_hl(part, t)
f = lambda part2: res_dict.get(part2,zero)
return f
def _s_to_self_base(self, part):
"""
Returns a function which gives the coefficient of a partition
in the expansion of the Schur functions ``s(part)`` in self.
EXAMPLES::
sage: P = HallLittlewoodP(QQ)
sage: f21 = P._s_to_self_base(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[0, 1, t^2 + t]
"""
Zt = ZZ['t']
t = Zt.gen()
zero = Zt(0)
res_dict = kfpoly.schur_to_hl(part, t)
f = lambda part2: res_dict.get(part2, zero)
return f
def _s_to_self_base(self, part):
"""
Returns a function which gives the coefficient of a partition
in the expansion of the Schur functions ``s(part)`` in self.
EXAMPLES::
sage: P = HallLittlewoodP(QQ)
sage: f21 = P._s_to_self_base(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[0, 1, t^2 + t]
"""
Zt = ZZ['t']
t = Zt.gen()
zero = Zt(0)
res_dict = kfpoly.schur_to_hl(part, t)
f = lambda part2: res_dict.get(part2,zero)
return f
