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

1, 2, 5, 30, 1099, 369267, 1299735768, 55209313116171, 32401252746609874301, 297072994236730724952120013, 47538200124908778784793653003318415, 146779873670882872946054150750588724458499598, 9581411392466176519699106616122834087912451590912289450

Offset: 0

Author

Alois P. Heinz, Nov 17 2012

Keywords

• nonn

Examples

			a(2) = 5:
+------+  +------+  +------+  +---+  +-+
| 1  2 |  | 1  3 |  | 1  2 |  | 1 |  +-+
| 3 .--+  | 2 .--+  +------+  +---+
+---+     +---+
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..22
• Wikipedia, Young tableau

Crossrefs

Column sums of A219272.

Partial sums of A219275.

Programs

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
b:= (n, l)-> `if`(n<1, h(l), b(n-1, l) +b(n-1, [l[], n])):
a:= n-> b(n, []):
seq(a(n), n=0..12);

Mathematica

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}]];
b[n_, l_] := If[n < 1, h[l], b[n - 1, l] + b[n - 1, Append[l, n]]];
a[n_] := b[n, {}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 02 2022, after Alois P. Heinz *)