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