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A219273 Number of standard Young tableaux for all partitions of nonnegative integers into distinct parts with largest part <= n.

%I A219273 #23 Nov 02 2022 14:45:18
%S A219273 1,2,5,30,1099,369267,1299735768,55209313116171,32401252746609874301,
%T A219273 297072994236730724952120013,47538200124908778784793653003318415,
%U A219273 146779873670882872946054150750588724458499598,9581411392466176519699106616122834087912451590912289450
%N A219273 Number of standard Young tableaux for all partitions of nonnegative integers into distinct parts with largest part <= n.
%H A219273 Alois P. Heinz, <a href="/A219273/b219273.txt">Table of n, a(n) for n = 0..22</a>
%H A219273 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A219273 a(2) = 5:
%e A219273 +------+  +------+  +------+  +---+  +-+
%e A219273 | 1  2 |  | 1  3 |  | 1  2 |  | 1 |  +-+
%e A219273 | 3 .--+  | 2 .--+  +------+  +---+
%e A219273 +---+     +---+
%p A219273 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A219273       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A219273     end:
%p A219273 b:= (n, l)-> `if`(n<1, h(l), b(n-1, l) +b(n-1, [l[], n])):
%p A219273 a:= n-> b(n, []):
%p A219273 seq(a(n), n=0..12);
%t A219273 h[l_] := With[{n = Length[l]}, Total[l]!/Product[ Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
%t A219273 b[n_, l_] := If[n < 1, h[l], b[n - 1, l] + b[n - 1, Append[l, n]]];
%t A219273 a[n_] := b[n, {}];
%t A219273 Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Nov 02 2022, after _Alois P. Heinz_ *)
%Y A219273 Column sums of A219272.
%Y A219273 Partial sums of A219275.
%K A219273 nonn
%O A219273 0,2
%A A219273 _Alois P. Heinz_, Nov 17 2012