# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a214753 Showing 1-1 of 1 %I A214753 #36 Oct 05 2018 19:32:54 %S A214753 1,0,1,0,2,1,0,4,4,1,0,10,16,6,1,0,26,66,34,8,1,0,76,296,192,58,10,1, %T A214753 0,232,1334,1134,406,88,12,1,0,764,6322,6716,2918,730,124,14,1,0,2620, %U A214753 30930,40872,20718,6118,1186,166,16,1,0,9496,158008,255308,149826,50056,11310,1796,214,18,1 %N A214753 Number T(n,k) of solid standard Young tableaux of n cells and height = k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. %H A214753 Alois P. Heinz, Rows n = 0..20, flattened %H A214753 S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012 %H A214753 Wikipedia, Young tableau %F A214753 T(n,k) = A215086(n,k) - A215086(n,k-1) for k>0, T(n,0) = A215086(n,0) = A000007(n). %e A214753 Triangle T(n,k) begins: %e A214753 1; %e A214753 0, 1; %e A214753 0, 2, 1; %e A214753 0, 4, 4, 1; %e A214753 0, 10, 16, 6, 1; %e A214753 0, 26, 66, 34, 8, 1; %e A214753 0, 76, 296, 192, 58, 10, 1; %e A214753 0, 232, 1334, 1134, 406, 88, 12, 1; %p A214753 b:= proc(n, k, l) option remember; `if`(n=0, 1, %p A214753 b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i]) `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])): %p A214753 T:= (n, k)-> A(n,k) -`if`(k=0, 0, A(n, k-1)): %p A214753 seq(seq(T(n, k), k=0..n), n=0..10); %t A214753 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]}]]; %t A214753 A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]]; %t A214753 T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]]; %t A214753 Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, May 23 2016, after _Alois P. Heinz_ *) %Y A214753 Columns k=0-10 give: A000007(n), A000085(n) for n>0, A273582, A273583, A273584, A273585, A273586, A273587, A273588, A273589, A273590. %Y A214753 Diagonal and lower diagonal give: A000012, A005843. %Y A214753 Row sums give: A207542. %Y A214753 T(2n,n) gives A273591. %Y A214753 Cf. A215086. %K A214753 nonn,tabl %O A214753 0,5 %A A214753 _Alois P. Heinz_, Aug 02 2012 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE