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Number of linear extensions of a 3 X 2 X n lattice.
4

%I #24 Feb 25 2016 12:27:57

%S 1,5,2452,4877756,20071150430,129586764260850,1138355914222027660,

%T 12513844842339741519760,163186564770917385358723138,

%U 2434438822161210367337209525489,40488679486377745566571570522228550,736610570835499716578578298705683198672

%N Number of linear extensions of a 3 X 2 X n lattice.

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

%H Alois P. Heinz, <a href="/A114716/b114716.txt">Table of n, a(n) for n = 0..45</a>

%p b:= proc(u, v, w, x, y, z) option remember;

%p `if`({u, v, w, x, y, z}={0}, 1,

%p `if`(u>v and u>x, b(u-1, v, w, x, y, z), 0)+

%p `if`(v>w and v>y, b(u, v-1, w, x, y, z), 0)+

%p `if`(w>z, b(u, v, w-1, x, y, z), 0)+

%p `if`(x>y, b(u, v, w, x-1, y, z), 0)+

%p `if`(y>z, b(u, v, w, x, y-1, z), 0)+

%p `if`(z>0, b(u, v, w, x, y, z-1), 0))

%p end:

%p a:= n-> b(n$6):

%p seq(a(n), n=0..12); # _Alois P. Heinz_, Apr 26 2012

%t b[u_, v_, w_, x_, y_, z_] := b[u, v, w, x, y, z] =

%t If[Union[{u, v, w, x, y, z}] == {0}, 1,

%t If[u>v && u>x, b[u-1, v, w, x, y, z], 0] +

%t If[v>w && v>y, b[u, v-1, w, x, y, z], 0] +

%t If[w>z, b[u, v, w-1, x, y, z], 0] +

%t If[x>y, b[u, v, w, x-1, y, z], 0] +

%t If[y>z, b[u, v, w, x, y-1, z], 0] +

%t If[z>0, b[u, v, w, x, y, z-1], 0]];

%t 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_ *)

%Y Cf. A114717.

%K nonn,hard

%O 0,2

%A _Mitch Harris_, Dec 27 2005; corrected May 25 2006

%E a(6)-a(11) from _Alois P. Heinz_, Apr 26 2012