Demonstration: Combinatorics (short)

Counting

sage: Partitions(100000).cardinality()

Species:

sage: from sage.combinat.species.library import *
sage: o   = var("o")
sage: BT = CombinatorialSpecies()
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]

Words

sage: m = WordMorphism('a->acabb,b->bcacacbb,c->baba')
sage: m.fixed_point('a')
word: acabbbabaacabbbcacacbbbcacacbbbcacacbbac...

For more, see: sage.combinat.words.demo.

Lattice points of polytopes

sage: A=random_matrix(ZZ,3,6,x=7)
sage: L=LatticePolytope(A)
sage: L.plot3d()

sage: L.npoints()  # should be cardinality!
28

This example used PALP and J-mol

Graphs up to an isomorphism

sage: show(graphs(5, lambda G: G.size() <= 4))