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A215086 Number A(n,k) of solid standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A215086 #34 Oct 07 2018 08:35:35
%S A215086 1,1,0,1,1,0,1,1,2,0,1,1,3,4,0,1,1,3,8,10,0,1,1,3,9,26,26,0,1,1,3,9,
%T A215086 32,92,76,0,1,1,3,9,33,126,372,232,0,1,1,3,9,33,134,564,1566,764,0,1,
%U A215086 1,3,9,33,135,622,2700,7086,2620,0,1,1,3,9,33,135,632,3106,13802,33550,9496,0
%N A215086 Number A(n,k) of solid standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A215086 Alois P. Heinz, <a href="/A215086/b215086.txt">Antidiagonals n = 0..30, flattened</a>
%H A215086 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:1202.6229 [math.CO], 2012
%H A215086 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A215086 A(n,k) = Sum_{i=0..k} A214753(n,i).
%e A215086 Square array A(n,k) begins:
%e A215086   1,   1,    1,    1,    1,    1,    1,    1, ...
%e A215086   0,   1,    1,    1,    1,    1,    1,    1, ...
%e A215086   0,   2,    3,    3,    3,    3,    3,    3, ...
%e A215086   0,   4,    8,    9,    9,    9,    9,    9, ...
%e A215086   0,  10,   26,   32,   33,   33,   33,   33, ...
%e A215086   0,  26,   92,  126,  134,  135,  135,  135, ...
%e A215086   0,  76,  372,  564,  622,  632,  633,  633, ...
%e A215086   0, 232, 1566, 2700, 3106, 3194, 3206, 3207, ...
%p A215086 b:= proc(n, k, l) option remember; `if`(n=0, 1,
%p A215086        b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i])<nops(l[i-1]),
%p A215086        b(n-1, k, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<k and
%p A215086        (i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),
%p A215086        b(n-1, k, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),
%p A215086        j=1..nops(l[i])), i=1..nops(l)))
%p A215086     end:
%p A215086 A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
%p A215086 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A215086 b[n_, k_, l_] := b[n, k, l] = If[n==0, 1, b[n-1, k, Append[l, {1}]] + Sum[If[i==1 || Length[l[[i]]]<Length[l[[i-1]]], b[n-1, k, ReplacePart[l, i -> Append[l[[i]], 1]]], 0] + Sum[If[l[[i, j]]<k && (i==1 || l[[i, j]] < l[[i-1, j]]) && (j==1 || l[[i, j]]<l[[i, j-1]]), b[n-1, k, ReplacePart[l, i -> ReplacePart[ l[[i]], j -> l[[i, j]]+1]]], 0], {j, 1, Length[l[[i]]]} ], {i, 1, Length[l]}]]; A[n_, k_] := If[k==0, If[n==0, 1, 0], b[n, Min[n, k], {}]]; Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jan 26 2017, after _Alois P. Heinz_ *)
%Y A215086 Columns k=0-10 give: A000007, A000085, A215087, A320180, A320181, A320182, A320183, A320184, A320185, A320186, A320187.
%Y A215086 Rows n=0-1 give: A000012, A057427.
%Y A215086 Main diagonal gives A207542.
%Y A215086 Cf. A214753.
%K A215086 nonn,tabl
%O A215086 0,9
%A A215086 _Alois P. Heinz_, Aug 02 2012