A208447 Sum of the k-th powers of the numbers of standard Young tableaux over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 6, 10, 7, 1, 1, 2, 10, 24, 26, 11, 1, 1, 2, 18, 64, 120, 76, 15, 1, 1, 2, 34, 180, 596, 720, 232, 22, 1, 1, 2, 66, 520, 3060, 8056, 5040, 764, 30, 1, 1, 2, 130, 1524, 16076, 101160, 130432, 40320, 2620, 42

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..50

Wikipedia, Young tableau

Example

A(3,2) = 1^2 + 2^2 + 1^2 = 6 = 3! because 3 has partitions 111, 21, 3 with 1, 2, 1 standard Young tableaux, respectively:
  .111.    . 21 . . . . . . .    . 3 . . . .
  +---+    +------+  +------+    +---------+
  | 1 |    | 1  2 |  | 1  3 |    | 1  2  3 |
  | 2 |    | 3 .--+  | 2 .--+    +---------+
  | 3 |    +---+     +---+
  +---+
Square array A(n,k) begins:
   1,  1,   1,    1,      1,       1,        1, ...
   1,  1,   1,    1,      1,       1,        1, ...
   2,  2,   2,    2,      2,       2,        2, ...
   3,  4,   6,   10,     18,      34,       66, ...
   5, 10,  24,   64,    180,     520,     1524, ...
   7, 26, 120,  596,   3060,   16076,    86100, ...
  11, 76, 720, 8056, 101160, 1379176, 19902600, ...

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:
g:= proc(n, i, k, l) `if`(n=0, h(l)^k, `if`(i<1, 0, g(n, i-1, k, l)
       + `if`(i>n, 0, g(n-i, i, k, [l[], i]))))
    end:
A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

h[l_] := With[{n = Length[l]}, Sum[i, {i, 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}]]; g[n_, i_, k_, l_] := If[n == 0, h[l]^k, If[i < 1, 0, g[n, i-1, k, l] + If[i > n, 0, g[n-i, i, k, Append[l, i]]]]]; a [n_, k_] := If[n == 0, 1, g[n, n, k, {}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

Crossrefs

Columns 0-10 give: A000041, A000085, A000142, A130721, A129627, A218432, A218433, A218434, A218435, A218436, A218437.

Rows 0+1, 2, 3 give: A000012, A007395, A052548.

Main diagonal gives A319607.

Sequence in context: A152977 A360334 A259799 * A320750 A117935 A224698

Adjacent sequences: A208444 A208445 A208446 * A208448 A208449 A208450

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 26 2012

Status

approved