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

%I A114716 #24 Feb 25 2016 12:27:57
%S A114716 1,5,2452,4877756,20071150430,129586764260850,1138355914222027660,
%T A114716 12513844842339741519760,163186564770917385358723138,
%U A114716 2434438822161210367337209525489,40488679486377745566571570522228550,736610570835499716578578298705683198672
%N A114716 Number of linear extensions of a 3 X 2 X n lattice.
%D A114716 Stanley, R., Enumerative Combinatorics, Vol. 2, Proposition 7.10.3 and Vol. 1, Sec 3.5 Chains in Distributive Lattices.
%H A114716 Alois P. Heinz, <a href="/A114716/b114716.txt">Table of n, a(n) for n = 0..45</a>
%p A114716 b:= proc(u, v, w, x, y, z) option remember;
%p A114716       `if`({u, v, w, x, y, z}={0}, 1,
%p A114716       `if`(u>v and u>x, b(u-1, v, w, x, y, z), 0)+
%p A114716       `if`(v>w and v>y, b(u, v-1, w, x, y, z), 0)+
%p A114716       `if`(w>z, b(u, v, w-1, x, y, z), 0)+
%p A114716       `if`(x>y, b(u, v, w, x-1, y, z), 0)+
%p A114716       `if`(y>z, b(u, v, w, x, y-1, z), 0)+
%p A114716       `if`(z>0, b(u, v, w, x, y, z-1), 0))
%p A114716     end:
%p A114716 a:= n-> b(n$6):
%p A114716 seq(a(n), n=0..12);  # _Alois P. Heinz_, Apr 26 2012
%t A114716 b[u_, v_, w_, x_, y_, z_] := b[u, v, w, x, y, z] =
%t A114716 If[Union[{u, v, w, x, y, z}] == {0}, 1,
%t A114716 If[u>v && u>x, b[u-1, v, w, x, y, z], 0] +
%t A114716 If[v>w && v>y, b[u, v-1, w, x, y, z], 0] +
%t A114716 If[w>z, b[u, v, w-1, x, y, z], 0] +
%t A114716 If[x>y, b[u, v, w, x-1, y, z], 0] +
%t A114716 If[y>z, b[u, v, w, x, y-1, z], 0] +
%t A114716 If[z>0, b[u, v, w, x, y, z-1], 0]];
%t A114716 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 A114716 Cf. A114717.
%K A114716 nonn,hard
%O A114716 0,2
%A A114716 _Mitch Harris_, Dec 27 2005; corrected May 25 2006
%E A114716 a(6)-a(11) from _Alois P. Heinz_, Apr 26 2012