diff --git a/src/sage/combinat/set_partition.py b/src/sage/combinat/set_partition.py index 092e2f9dbb7..5078003c6f2 100644 --- a/src/sage/combinat/set_partition.py +++ b/src/sage/combinat/set_partition.py @@ -73,7 +73,11 @@ def _repr_(self): sage: S([[1,3],[2,4]]) {{1, 3}, {2, 4}} """ - return '{' + ', '.join('{' + repr(sorted(x))[1:-1] + '}' for x in self) + '}' + try: + s = [sorted(x) for x in self] + except TypeError: + s = [sorted(x, key=str) for x in self] + return '{' + ', '.join('{' + repr(x)[1:-1] + '}' for x in s) + '}' def __hash__(self): """ @@ -532,7 +536,7 @@ def pre_conjugate(sp): class SetPartition(AbstractSetPartition, - metaclass=InheritComparisonClasscallMetaclass): + metaclass=InheritComparisonClasscallMetaclass): r""" A partition of a set. @@ -620,7 +624,11 @@ def __init__(self, parent, s, check=True): {} """ self._latex_options = {} - ClonableArray.__init__(self, parent, sorted(map(frozenset, s), key=min), check=check) + try: + s = sorted(map(frozenset, s), key=min) + except TypeError: + s = sorted(map(frozenset, s), key=lambda b: min(str(b))) + ClonableArray.__init__(self, parent, s, check=check) def check(self): """ @@ -2821,7 +2829,11 @@ def __iter__(self): sage: SetPartitions(["a", "b"]).list() [{{'a', 'b'}}, {{'a'}, {'b'}}] """ - for sp in set_partition_iterator(sorted(self._set)): + try: + s = sorted(self._set) + except TypeError: + s = sorted(self._set, key=str) + for sp in set_partition_iterator(s): yield self.element_class(self, sp, check=False) def base_set(self): @@ -3179,7 +3191,11 @@ def __iter__(self): sage: SetPartitions(["a", "b", "c"], 2).list() [{{'a', 'c'}, {'b'}}, {{'a'}, {'b', 'c'}}, {{'a', 'b'}, {'c'}}] """ - for sp in set_partition_iterator_blocks(sorted(self._set), self._k): + try: + s = sorted(self._set) + except TypeError: + s = sorted(self._set, key=str) + for sp in set_partition_iterator_blocks(s, self._k): yield self.element_class(self, sp, check=False) def __contains__(self, x): diff --git a/src/sage/rings/lazy_species.py b/src/sage/rings/lazy_species.py index ca9fb675c89..33004a29f91 100644 --- a/src/sage/rings/lazy_species.py +++ b/src/sage/rings/lazy_species.py @@ -468,15 +468,13 @@ def __call__(self, *args): sage: from sage.rings.species import PolynomialSpecies sage: P = PolynomialSpecies(QQ, "X") sage: Gc = L(lambda n: sum(P(G.automorphism_group()) for G in graphs(n) if G.is_connected()) if n else 0) - sage: E = L(lambda n: SymmetricGroup(n)) - - sage: G = L(lambda n: sum(P(G.automorphism_group()) for G in graphs(n))) + sage: E = L.Sets() + sage: G = L.Graphs() sage: E(Gc) - G O^7 - sage: L = LazySpecies(QQ, "X") - sage: X = L(SymmetricGroup(1)) - sage: E = L(lambda n: SymmetricGroup(n)) + sage: L. = LazySpecies(QQ) + sage: E = L.Sets() sage: A = L.undefined(1) sage: A.define(X*E(A)) sage: A[5] @@ -487,8 +485,8 @@ def __call__(self, *args): sage: [sum(F[n].monomial_coefficients().values()) for n in range(1, 7)] [1, 3, 7, 19, 47, 130] sage: oeis(_) - 0: A001372: Number of unlabeled mappings (or mapping patterns) from n points to themselves; number of unlabeled endofunctions. - + 0: A001372: Number of unlabeled mappings (or mapping patterns) + from n points to themselves; number of unlabeled endofunctions. sage: R. = QQ[] sage: L = LazySpecies(R, "X") @@ -499,8 +497,7 @@ def __call__(self, *args): TESTS:: - sage: L = LazySpecies(QQ, "X") - sage: X = L(SymmetricGroup(1)) + sage: L. = LazySpecies(QQ) sage: E2 = L(SymmetricGroup(2)) sage: X(X + E2) X + E_2 + O^8 @@ -510,18 +507,13 @@ def __call__(self, *args): sage: (1+E2)(X) 1 + E_2 + O^7 - sage: L = LazySpecies(QQ, "X, Y") - sage: P = PolynomialSpecies(QQ, "X, Y") - sage: X = L(P(SymmetricGroup(1), {0: [1]})) - sage: Y = L(P(SymmetricGroup(1), {1: [1]})) + sage: L. = LazySpecies(QQ) sage: X(Y, 0) Y + O^8 sage: L1 = LazySpecies(QQ, "X") - sage: L = LazySpecies(QQ, "X, Y") - sage: P = PolynomialSpecies(QQ, "X, Y") - sage: X = L(P(SymmetricGroup(1), {0: [1]})) - sage: E = L1(lambda n: SymmetricGroup(n)) + sage: E = L1.Sets() + sage: L. = LazySpecies(QQ) sage: E(X) 1 + X + E_2(X) + E_3(X) + E_4(X) + E_5(X) + E_6(X) + O^7 """ @@ -530,10 +522,10 @@ def __call__(self, *args): class SumSpeciesElement(LazySpeciesElement): def __init__(self, left, right): - self._left = left - self._right = right F = super(LazySpeciesElement, type(left))._add_(left, right) super().__init__(F.parent(), F._coeff_stream) + self._left = left + self._right = right def structures(self, *labels): labels = label_sets(self.parent()._arity, labels) @@ -543,10 +535,10 @@ def structures(self, *labels): class ProductSpeciesElement(LazySpeciesElement): def __init__(self, left, right): - self._left = left - self._right = right F = super(LazySpeciesElement, type(left))._mul_(left, right) super().__init__(F.parent(), F._coeff_stream) + self._left = left + self._right = right def structures(self, *labels): """ @@ -652,6 +644,66 @@ def coefficient(n): coeff_stream = Stream_function(coefficient, P._sparse, sorder * gv) super().__init__(P, coeff_stream) + self._left = left + self._args = args + + def structures(self, *labels): + r""" + + EXAMPLES:: + + sage: from sage.rings.lazy_species import LazySpecies + sage: L = LazySpecies(QQ, "X") + sage: E = L.Sets() + sage: E1 = L.Sets(min=1) + sage: list(E(E1).structures([1,2,3])) + [({((1, 0), (2, 0), (3, 0))}, ({1, 2, 3},)), + ({((1, 0), (2, 0)), ((3, 0),)}, ({1, 2}, {3})), + ({((1, 0), (3, 0)), ((2, 0),)}, ({1, 3}, {2})), + ({((1, 0),), ((2, 0), (3, 0))}, ({1}, {2, 3})), + ({((1, 0),), ((2, 0),), ((3, 0),)}, ({1}, {2}, {3}))] + + sage: C = L.Cycles() + sage: L. = LazySpecies(QQ) + sage: sum(1 for s in C(X*Y).structures([1,2,3], [1,2,3])) + + sage: C(X*Y).generating_series()[6] + 1/3*X^3*Y^3 + + sage: sum(1 for s in E(X*Y).structures([1,2,3], ["a", "b", "c"])) + 6 + """ + F = self._left + G = self._args + m = len(G) # == F.parent()._arity + k = self.parent()._arity # == G[i].parent()._arity + labels = label_sets(k, labels) + # make label sets disjoint + U = [(e, i) for i, l in enumerate(labels) for e in l] + + def split_set(C): + C_split = defaultdict(list) + for e, i in C: + C_split[i].append(e) + return [C_split[i] for i in range(k)] + + Par_U = SetPartitions(U) + for pi in Par_U: + # Fix an arbitrary order of the blocks + pi_list = list(pi) + # Generate all functions chi from pi to {0, ..., m-1} + for chi in itertools.product(range(m), repeat=len(pi_list)): + chi_inv = defaultdict(list) + for b, i in zip(pi_list, chi): + chi_inv[i].append(b) + + # The set of structures is the Cartesian product of + # the structures in F[chi_inv[i] for i in range(m)] + # and for each set C in chi_inv[i] the set of + # structures in G_i[C] + F_s = F.structures(*[[tuple(b) for b in chi_inv[i]] for i in range(m)]) + G_s = [G[i].structures(*split_set(C)) for i in range(m) for C in chi_inv[i]] + yield from itertools.product(F_s, itertools.product(*G_s)) class LazySpecies(LazyCompletionGradedAlgebra): @@ -719,14 +771,14 @@ def __init__(self, base_ring, names, sparse): if self._arity == 1: self.Graphs = lambda: GraphSpecies(self) self.SetPartitions = lambda: SetPartitionSpecies(self) - self.Sets = lambda: SetSpecies(self) + self.Sets = lambda min=0: SetSpecies(self, min) self.Cycles = lambda: CycleSpecies(self) class SetSpecies(LazySpeciesElement): - def __init__(self, parent): + def __init__(self, parent, min): P = parent._laurent_poly_ring - S = parent(SymmetricGroup) + S = parent(lambda n: SymmetricGroup(n) if n >= min else 0) super().__init__(parent, S._coeff_stream) def structures(self, *labels): @@ -785,8 +837,8 @@ def isotypes(self, labels): class SetPartitionSpecies(LazySpeciesElement): def __init__(self, parent): P = parent._laurent_poly_ring - E = parent(SymmetricGroup) - E1 = parent(lambda n: SymmetricGroup(n) if n else 0) + E = parent.Sets() + E1 = parent.Sets(min=1) super().__init__(parent, E(E1)._coeff_stream) def isotypes(self, labels):