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A114715 Number A(n,m) of linear extensions of a 2 X n X m lattice; square array A(n,m), n>=1, m>=1, read by antidiagonals.

%I A114715 #24 Feb 13 2024 15:03:07
%S A114715 1,2,2,5,48,5,14,2452,2452,14,42,183958,4877756,183958,42,132,
%T A114715 17454844,20071150430,20071150430,17454844,132,429,1941406508,
%U A114715 129586764260850,6708527580006468,129586764260850,1941406508,429
%N A114715 Number A(n,m) of linear extensions of a 2 X n X m lattice; square array A(n,m), n>=1, m>=1, read by antidiagonals.
%D A114715 Stanley, R., Enumerative Combinatorics, Vol. 2, Proposition 7.10.3 and Vol. 1, Sec 3.5 Chains in Distributive Lattices.
%H A114715 Alois P. Heinz, <a href="/A114715/b114715.txt">Antidiagonals n = 1..12, flattened</a>
%F A114715 A(n,1) = A(1,n) = A000108(n).
%F A114715 A(n,2) = A(2,n) = A114714(n).
%F A114715 A(n,3) = A(3,n) = A114716(n).
%e A114715 Square array A(n,m) begins:
%e A114715    1,      2,           5,               14, ...
%e A114715    2,     48,        2452,           183958, ...
%e A114715    5,   2452,     4877756,      20071150430, ...
%e A114715   14, 183958, 20071150430, 6708527580006468, ...
%p A114715 b := proc(l) option remember; local n; n:= nops(l);
%p A114715       `if`({seq(l[i][], i=1..n)}={0}, 1, add(`if`(l[i][1]>l[i][2] and
%p A114715        l[i][1]>l[i+1][1], b(subsop(i=[l[i][1]-1, l[i][2]], l)), 0),
%p A114715        i=1..n-1)+ add(`if`(l[i][2]>l[i+1][2], b(subsop(i=[l[i][1],
%p A114715        l[i][2]-1], l)), 0), i=1..n-1)+ `if`(l[n][1]>l[n][2],
%p A114715        b(subsop(n=[l[n][1]-1, l[n][2]], l)), 0)+ `if`(l[n][2]>0,
%p A114715        b(subsop(n=[l[n][1], l[n][2]-1], l)), 0))
%p A114715      end:
%p A114715 A:= (n, m)-> `if`(m>=n, b([[m$2]$n]), b([[n$2]$m])):
%p A114715 seq(seq(A(n, d+1-n), n=1..d), d=1..8);  # _Alois P. Heinz_, Jun 29 2012
%t A114715 b[l_List] := b[l] = With[{n = Length[l]}, If[Union[Table[l[[i]], {i, 1, n}] // Flatten] == {0}, 1, Sum[If[l[[i, 1]] > l[[i, 2]] && l[[i, 1]] > l[[i+1, 1]], b[ReplacePart[l, i -> {l[[i, 1]]-1, l[[i, 2]]}]], 0], {i, 1, n-1}] + Sum[If[l[[i, 2]] > l[[i+1, 2]], b[ReplacePart[l, i -> {l[[i, 1]], l[[i, 2]]-1}]], 0], {i, 1, n-1}] + If[l[[n, 1]] > l[[n, 2]], b[ReplacePart[l, n -> {l[[n, 1]]-1, l[[n, 2]]} ]], 0] + If[l[[n, 2]] > 0, b[ReplacePart[l, n -> {l[[n, 1]], l[[n, 2]]-1}]], 0]]] ; A[n_, m_] := If[m >= n, b[Array[{m, m}&, n]], b[Array[{n, n}&, m]]]; Table[ Table[A[n, d+1-n], {n, 1, d}], {d, 1, 8}] // Flatten (* _Jean-François Alcover_, Mar 11 2015, after _Alois P. Heinz_ *)
%Y A114715 Cf. A000108, A114714, A114716, A114717.
%Y A114715 Main diagonal gives A370257.
%K A114715 nonn,tabl,hard
%O A114715 1,2
%A A114715 _Mitch Harris_, Dec 27 2005
%E A114715 Edited by _Alois P. Heinz_, Jun 29 2012