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A290225 Number of solid standard Young tableaux of cylindrical shape lambda X n, where lambda ranges over all partitions of n.

%I A290225 #11 Jan 04 2023 21:08:48
%S A290225 1,1,4,276,1161678,620383261034,80434777704834144228,
%T A290225 3212151962391797592956111856142,
%U A290225 58141033434729590882944205957642581926272684,738506234630963222745737660670442498620046849638365979249010
%N A290225 Number of solid standard Young tableaux of cylindrical shape lambda X n, where lambda ranges over all partitions of n.
%H A290225 S. B. Ekhad and D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229 [math.CO], 2012.
%H A290225 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A290225 a(n) = A215204(n,n).
%p A290225 b:= proc(l) option remember; local m; m:= nops(l);
%p A290225       `if`({map(x-> x[], l)[]}minus{0}={}, 1, add(add(`if`(l[i][j]>
%p A290225       `if`(i=m or nops(l[i+1])<j, 0, l[i+1][j]) and l[i][j]>
%p A290225       `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
%p A290225        j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
%p A290225     end:
%p A290225 g:= proc(n, i, k, l) `if`(n=0 or i=1, b(map(x-> [k$x], [l[], 1$n])),
%p A290225        add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i))
%p A290225     end:
%p A290225 a:= n-> g(n$3, []):
%p A290225 seq(a(n), n=0..6);
%t A290225 b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {0} == {}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i + 1]]] < j, 0, l[[i + 1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j + 1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]];
%t A290225 g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}] & /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
%t A290225 a[n_] := g[n, n, n, {}];
%t A290225 Table[a[n], {n, 0, 7}] (* _Jean-François Alcover_, Dec 28 2022, after _Alois P. Heinz_ in A215204 *)
%Y A290225 Main diagonal of A215204.
%K A290225 nonn
%O A290225 0,3
%A A290225 _Alois P. Heinz_, Jul 24 2017