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A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A215204 #35 Sep 24 2022 13:13:03
%S A215204 1,1,1,1,1,2,1,1,2,3,1,1,4,4,5,1,1,10,26,10,7,1,1,28,276,258,26,11,1,
%T A215204 1,84,3740,14318,3346,76,15,1,1,264,58604,1161678,1214358,54108,232,
%U A215204 22,1,1,858,1010616,118316062,741215012,150910592,1054256,764,30
%N A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A215204 Alois P. Heinz, <a href="/A215204/b215204.txt">Antidiagonals n = 0..15, flattened</a>
%H A215204 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:1202.6229v1 [math.CO], 2012
%H A215204 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A215204 Square array A(n,k) begins:
%e A215204 :  1,  1,     1,         1,            1,                1, ...
%e A215204 :  1,  1,     1,         1,            1,                1, ...
%e A215204 :  2,  2,     4,        10,           28,               84, ...
%e A215204 :  3,  4,    26,       276,         3740,            58604, ...
%e A215204 :  5, 10,   258,     14318,      1161678,        118316062, ...
%e A215204 :  7, 26,  3346,   1214358,    741215012,     620383261034, ...
%e A215204 : 11, 76, 54108, 150910592, 840790914296, 7137345113624878, ...
%p A215204 b:= proc(l) option remember; local m; m:= nops(l);
%p A215204       `if`({map(x-> x[], l)[]}minus{0}={}, 1, add(add(`if`(l[i][j]>
%p A215204       `if`(i=m or nops(l[i+1])<j, 0, l[i+1][j]) and l[i][j]>
%p A215204       `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
%p A215204        j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
%p A215204     end:
%p A215204 g:= proc(n, i, k, l) `if`(n=0 or i=1, b(map(x-> [k$x], [l[], 1$n])),
%p A215204        add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i))
%p A215204     end:
%p A215204 A:= (n, k)-> g(n, n, k, []):
%p A215204 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A215204 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 A215204 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 A215204 A[n_, k_] := g[n, n, k, {}];
%t A215204 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Sep 24 2022, after _Alois P. Heinz_ *)
%Y A215204 Columns k=0-5 give: A000041, A000085, A215266, A290202, A290214, A290274.
%Y A215204 Rows n=0+1, 2-5 give: A000012, 2*A000108, 2*A005789 + A006335, 2*A005790 + 2*A213978 + A114714, 2*A005791 + 2*A215220 + 2*A213932 + A214638.
%Y A215204 Main diagonal gives A290225.
%K A215204 nonn,tabl
%O A215204 0,6
%A A215204 _Alois P. Heinz_, Aug 05 2012