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A114714 Number of linear extensions of a 2 X 2 X n lattice.

%I A114714 #27 Oct 27 2023 21:55:42
%S A114714 1,2,48,2452,183958,17454844,1941406508,242201554680,32959299267334,
%T A114714 4801233680739724,738810565910888784,118929992674840615128,
%U A114714 19880920716640427983476,3431624482227380273056728,608880419873586515669564728,110654016191338341346670548240
%N A114714 Number of linear extensions of a 2 X 2 X n lattice.
%C A114714 The additional terms were found using dynamic programming to count the maximal chains in the distributive lattice of order-preserving functions from the chain of length n to J(P), where J is the lattice of downsets of the poset P = 2x2. - _Nick Krempel_, Jul 08 2010
%D A114714 Stanley, R., Enumerative Combinatorics, Vol. 2, Prop. 7.10.3 and Vol. 1, Sec 3.5, Chains in Distributive Lattices.
%H A114714 Alois P. Heinz, <a href="/A114714/b114714.txt">Table of n, a(n) for n = 0..180</a> (first 41 terms from Nick Krempel)
%p A114714 b:= proc(x, y, u, w) option remember;
%p A114714       `if`(x=0 and y=0 and u=0 and w=0, 1, `if`(x>y and x>u,
%p A114714        b(x-1, y, u, w), 0)+ `if`(y>w, b(x, y-1, u, w), 0)+
%p A114714       `if`(u>w, b(x, y, u-1, w), 0)+ `if`(w>0, b(x, y, u, w-1), 0))
%p A114714     end:
%p A114714 a:= n-> b(n$4):
%p A114714 seq(a(n), n=0..20);  # _Alois P. Heinz_, Apr 27 2012
%t A114714 b[x_, y_, u_, w_] := b[x, y, u, w] = If[x == 0 && y == 0 && u == 0 && w == 0, 1, If[x>y && x>u, b[x-1, y, u, w], 0] + If[y>w, b[x, y-1, u, w], 0] + If[u>w, b[x, y, u-1, w], 0] + If[w>0, b[x, y, u, w-1], 0]]; a[n_] := b[n, n, n, n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 29 2015, after _Alois P. Heinz_ *)
%Y A114714 Cf. A114717.
%K A114714 nonn,hard
%O A114714 0,2
%A A114714 _Mitch Harris_, Dec 27 2005
%E A114714 More terms from _Nick Krempel_, Jul 08 2010