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A215087 Number of solid standard Young tableaux of n cells and height <= 2.

%I A215087 #22 Dec 01 2023 07:48:37
%S A215087 1,1,3,8,26,92,372,1566,7086,33550,167504,873226,4764614,26947632,
%T A215087 157926628,954523378,5945067490,38060781922,250345198424,
%U A215087 1688978186742,11679437620552,82652840640478,598018846154666,4418072084681592,33298670603875846,255782905412464810
%N A215087 Number of solid standard Young tableaux of n cells and height <= 2.
%H A215087 Alois P. Heinz, <a href="/A215087/b215087.txt">Table of n, a(n) for n = 0..40</a>
%H A215087 S. B. Ekhad, D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229v1 [math.CO], 2012
%H A215087 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%p A215087 b:= proc(n, l) option remember; `if`(n=0, 1,
%p A215087        b(n-1, [l[], [1]])+ add(`if`(i=1 or nops(l[i])<nops(l[i-1]),
%p A215087        b(n-1, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<2 and
%p A215087        (i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),
%p A215087        b(n-1, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),
%p A215087        j=1..nops(l[i])), i=1..nops(l)))
%p A215087     end:
%p A215087 a:= n-> b(n, []):
%p A215087 seq(a(n), n=0..20);
%t A215087 b[n_, l_] := b[n, l] = If[n == 0, 1, b[n - 1, Append[l, {1}]] + Sum[If[i == 1 || Length[l[[i]]] < Length[l[[i - 1]]], b[n - 1, ReplacePart[l, i -> Append[l[[i]], 1]]], 0] + Sum[If[l[[i, j]] < 2 && (i == 1 || l[[i, j]] < l[[i - 1, j]]) && (j == 1 || l[[i, j]] < l[[i, j - 1]]), b[n - 1, ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] + 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, Length[l]}]];
%t A215087 a[n_] := b[n, {}];
%t A215087 Table[Print[n, " ", a[n]]; a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 01 2023, after _Alois P. Heinz_ *)
%Y A215087 Column k=2 of A215086.
%K A215087 nonn
%O A215087 0,3
%A A215087 _Alois P. Heinz_, Aug 02 2012