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

%I #27 Oct 27 2023 21:55:42

%S 1,2,48,2452,183958,17454844,1941406508,242201554680,32959299267334,

%T 4801233680739724,738810565910888784,118929992674840615128,

%U 19880920716640427983476,3431624482227380273056728,608880419873586515669564728,110654016191338341346670548240

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

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

%H 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 b:= proc(x, y, u, w) option remember;

%p `if`(x=0 and y=0 and u=0 and w=0, 1, `if`(x>y and x>u,

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

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

%p end:

%p a:= n-> b(n$4):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Apr 27 2012

%t 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 Cf. A114717.

%K nonn,hard

%O 0,2

%A _Mitch Harris_, Dec 27 2005

%E More terms from _Nick Krempel_, Jul 08 2010