def _to_s(self, part):
r"""
Returns a function which gives the coefficient of a partition
in the Schur expansion of ``self(part)``.
INPUT:
- ``self`` -- an instance of the Hall-Littlewood `P` basis
- ``part`` -- a partition
OUTPUT:
- returns a function which accepts a second partition ``part2``
and returns the coefficient of the expansion of the `Qp`
in the Schur basis. This is the `t`-Kostka-Foulkes polynomial
`K_{part2,part}(t)`
EXAMPLES::
sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLQp = Sym.hall_littlewood().Qp()
sage: f21 = HLQp._to_s(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[t, 1, 0]
"""
t = QQt.gen()
if part == []:
return lambda part2: QQt.one()
res = hall_littlewood(part) # call to symmetrica (returns in variable x)
f = lambda part2: res.coefficient(part2).subs(x=t)
return f
def _to_s(self, part):
r"""
Returns a function which gives the coefficient of a partition
in the Schur expansion of ``self(part)``.
INPUT:
- ``self`` -- an instance of the Hall-Littlewood `P` basis
- ``part`` -- a partition
OUTPUT:
- returns a function which accepts a second partition ``part2``
and returns the coefficient of the expansion of the `Qp`
in the Schur basis. This is the `t`-Kostka-Foulkes polynomial
`K_{part2,part}(t)`
EXAMPLES::
sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLQp = Sym.hall_littlewood().Qp()
sage: f21 = HLQp._to_s(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[t, 1, 0]
"""
t = QQt.gen()
if part == []:
return lambda part2: QQt(1)
res = hall_littlewood(part) # call to symmetrica (returns in variable x)
f = lambda part2: res.coefficient(part2).subs(x=t)
return f
def _to_s(self, part):
"""
Returns a function which gives the coefficient of a partition
in the Schur expansion of ``self(part)``.
EXAMPLES::
sage: Qp = HallLittlewoodQp(QQ)
sage: f21 = Qp._to_s(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[t, 1, 0]
"""
Zt = ZZ['t']
t = Zt.gen()
zero = Zt(0)
if part == []:
return lambda part2: Zt(1)
res = hall_littlewood(part)
f = lambda part2: res.coefficient(part2).subs(x=t)
return f
def _to_s(self, part):
"""
Returns a function which gives the coefficient of a partition
in the Schur expansion of ``self(part)``.
EXAMPLES::
sage: Qp = HallLittlewoodQp(QQ)
sage: f21 = Qp._to_s(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[t, 1, 0]
"""
Zt = ZZ['t']
t = Zt.gen()
zero = Zt(0)
if part == []:
return lambda part2: Zt(1)
res = hall_littlewood(part)
f = lambda part2: res.coefficient(part2).subs(x=t)
return f