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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..50
- Wikipedia, Young tableau
Crossrefs
Programs
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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 *)