.. MODULEAUTHOR:: Nicolas M. Thiéry <nthiery at users.sf.net>
sage: %hide # not tested sage: pretty_print_default(False) # not tested
sage: s = Permutation([5,3,2,6,4,8,9,7,1]) sage: s [5, 3, 2, 6, 4, 8, 9, 7, 1] sage: (p,q) = s.robinson_schensted() sage: p.pp() 1 4 7 9 2 6 8 3 5 sage: p.row_stabilizer() Permutation Group with generators [(), (7,9), (6,8), (4,7), (2,6), (1,4)]
sage: Partitions(100000).cardinality() 27493510569...
Species:
sage: from sage.combinat.species.library import *
sage: o = var("o")
sage: BT = CombinatorialSpecies(min=1)
sage: Leaf = SingletonSpecies()
sage: BT.define(Leaf+(BT*BT))
sage: BT.isotypes([o]*5).list()
[o*(o*(o*(o*o))), o*(o*((o*o)*o)), o*((o*o)*(o*o)), o*((o*(o*o))*o), o*(((o*o)*o)*o), (o*o)*(o*(o*o)), (o*o)*((o*o)*o), (o*(o*o))*(o*o), ((o*o)*o)*(o*o), (o*(o*(o*o)))*o, (o*((o*o)*o))*o, ((o*o)*(o*o))*o, ((o*(o*o))*o)*o, (((o*o)*o)*o)*o]
sage: m = WordMorphism('a->acabb,b->bcacacbb,c->baba')
sage: m.fixed_point('a')
word: acabbbabaacabbbcacacbbbcacacbbbcacacbbac...
sage: A = random_matrix(ZZ,6,3,x=7) sage: L = LatticePolytope(A) sage: L.plot3d() Graphics3d Object sage: L.npoints() # should be cardinality! # random 28
This example used PALP and J-mol
sage: show(next(graphs(5, lambda G: G.size() <= 4)))
Usual bases:
sage: Sym = SymmetricFunctions(QQ); Sym Symmetric Functions over Rational Field sage: Sym.inject_shorthands() Defining ... sage: m(( ( h[2,1] * ( 1 + 3 * p[2,1]) ) + s[2](s[3]))) 3*m[1, 1, 1] + ...
Macdonald polynomials:
sage: Sym = SymmetricFunctions(FractionField(QQ['q','t'])) sage: J = Sym.macdonald().J() sage: P = Sym.macdonald().P() sage: Q = Sym.macdonald().Q() sage: J Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald J basis sage: P(J[2,2] + 3 * Q[3,1]) (...)*McdP[2, 2] + ...
sage: L = RootSystem(['A',2,1]).weight_space() sage: L.plot(alcove_walk=[0,2,0,1,2,1,2,0,2,1]) Graphics object consisting of 148 graphics primitives sage: W = WeylGroup(["B", 3]) sage: W.cayley_graph(side = "left").plot3d(color_by_label = True) Graphics3d Object
sage: W = WeylGroup(["B", 3]) sage: print(W.character_table()) # Thanks GAP! CT1 ... X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 -1 1 1 -1 X.3 1 1 -1 -1 1 -1 1 1 -1 1 X.4 1 1 -1 1 -1 1 -1 1 -1 -1 X.5 2 2 . . -2 . 1 -1 . -2 X.6 2 2 . . 2 . -1 -1 . 2 X.7 3 -1 -1 1 -1 -1 . . 1 3 X.8 3 -1 1 1 1 -1 . . -1 -3 X.9 3 -1 1 -1 -1 1 . . -1 3 X.10 3 -1 -1 -1 1 1 . . 1 -3 sage: type(W.character_table()) <class 'sage.interfaces.interface.AsciiArtString'> sage: G = W.gap(); G <matrix group of size 48 with 3 generators> sage: G.Ch # not tested sage: T = G.CharacterTable(); T CharacterTable( <matrix group of size 48 with 3 generators> ) sage: T.Irr()[4,4] -2
sage: W3 = CoxeterGroup(["B", 3], implementation="coxeter3") sage: KL = matrix([ [ W3.kazhdan_lusztig_polynomial(u,v) if u.bruhat_le(v) else 0 for u in W3 ] ....: for v in W3]) sage: show(KL) sage: W = WeylGroup(["C", 3, 1]) sage: W Weyl Group of type ['C', 3, 1] (as a matrix group acting on the root space) sage: W.from_reduced_word([1,2,3,0,3,0,3,2,1,3,3,2]).stanley_symmetric_function() 256*m[1, 1, 1, 1, 1, 1] + 128*m[2, 1, 1, 1, 1] + 64*m[2, 2, 1, 1] + 32*m[2, 2, 2] + 48*m[3, 1, 1, 1] + 24*m[3, 2, 1] + 8*m[3, 3] + 16*m[4, 1, 1] + 8*m[4, 2] + 4*m[5, 1]
sage: K = crystals.KirillovReshetikhin(['A',3,1], 2,2)
sage: G = K.digraph()
sage: G.set_latex_options(format = "dot2tex", edge_labels = True, color_by_label = {0:"black", 1:"blue", 2:"red", 3:"green"}, edge_options=lambda u_v_label:({"backward": u_v_label[2] == 0}))
sage: view(G, viewer="pdf", tightpage=True)