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%I #24 Feb 13 2024 15:03:07
%S 1,2,2,5,48,5,14,2452,2452,14,42,183958,4877756,183958,42,132,
%T 17454844,20071150430,20071150430,17454844,132,429,1941406508,
%U 129586764260850,6708527580006468,129586764260850,1941406508,429
%N 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 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="/A114715/b114715.txt">Antidiagonals n = 1..12, flattened</a>
%F A(n,1) = A(1,n) = A000108(n).
%F A(n,2) = A(2,n) = A114714(n).
%F A(n,3) = A(3,n) = A114716(n).
%e Square array A(n,m) begins:
%e 1, 2, 5, 14, ...
%e 2, 48, 2452, 183958, ...
%e 5, 2452, 4877756, 20071150430, ...
%e 14, 183958, 20071150430, 6708527580006468, ...
%p b := proc(l) option remember; local n; n:= nops(l);
%p `if`({seq(l[i][], i=1..n)}={0}, 1, add(`if`(l[i][1]>l[i][2] and
%p l[i][1]>l[i+1][1], b(subsop(i=[l[i][1]-1, l[i][2]], l)), 0),
%p i=1..n-1)+ add(`if`(l[i][2]>l[i+1][2], b(subsop(i=[l[i][1],
%p l[i][2]-1], l)), 0), i=1..n-1)+ `if`(l[n][1]>l[n][2],
%p b(subsop(n=[l[n][1]-1, l[n][2]], l)), 0)+ `if`(l[n][2]>0,
%p b(subsop(n=[l[n][1], l[n][2]-1], l)), 0))
%p end:
%p A:= (n, m)-> `if`(m>=n, b([[m$2]$n]), b([[n$2]$m])):
%p seq(seq(A(n, d+1-n), n=1..d), d=1..8); # _Alois P. Heinz_, Jun 29 2012
%t 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 Cf. A000108, A114714, A114716, A114717.
%Y Main diagonal gives A370257.
%K nonn,tabl,hard
%O 1,2
%A _Mitch Harris_, Dec 27 2005
%E Edited by _Alois P. Heinz_, Jun 29 2012
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