A114716 Number of linear extensions of a 3 X 2 X n lattice.

1, 5, 2452, 4877756, 20071150430, 129586764260850, 1138355914222027660, 12513844842339741519760, 163186564770917385358723138, 2434438822161210367337209525489, 40488679486377745566571570522228550, 736610570835499716578578298705683198672

Offset: 0

Author

Mitch Harris, Dec 27 2005; corrected May 25 2006

Keywords

• nonn
• hard

References

• Stanley, R., Enumerative Combinatorics, Vol. 2, Proposition 7.10.3 and Vol. 1, Sec 3.5 Chains in Distributive Lattices.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..45

Crossrefs

Cf. A114717.

Programs

Maple

b:= proc(u, v, w, x, y, z) option remember;
      `if`({u, v, w, x, y, z}={0}, 1,
      `if`(u>v and u>x, b(u-1, v, w, x, y, z), 0)+
      `if`(v>w and v>y, b(u, v-1, w, x, y, z), 0)+
      `if`(w>z, b(u, v, w-1, x, y, z), 0)+
      `if`(x>y, b(u, v, w, x-1, y, z), 0)+
      `if`(y>z, b(u, v, w, x, y-1, z), 0)+
      `if`(z>0, b(u, v, w, x, y, z-1), 0))
    end:
a:= n-> b(n$6):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 26 2012

Mathematica

b[u_, v_, w_, x_, y_, z_] := b[u, v, w, x, y, z] =
If[Union[{u, v, w, x, y, z}] == {0}, 1,
If[u>v && u>x, b[u-1, v, w, x, y, z], 0] +
If[v>w && v>y, b[u, v-1, w, x, y, z], 0] +
If[w>z, b[u, v, w-1, x, y, z], 0] +
If[x>y, b[u, v, w, x-1, y, z], 0] +
If[y>z, b[u, v, w, x, y-1, z], 0] +
If[z>0, b[u, v, w, x, y, z-1], 0]];
a[n_] := b[n, n, n, n, n, n]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, May 29 2015, after Alois P. Heinz *)

Extensions

a(6)-a(11) from Alois P. Heinz, Apr 26 2012