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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.

%I A208447 #32 Sep 24 2018 14:09:07
%S A208447 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,
%T A208447 18,64,120,76,15,1,1,2,34,180,596,720,232,22,1,1,2,66,520,3060,8056,
%U A208447 5040,764,30,1,1,2,130,1524,16076,101160,130432,40320,2620,42
%N 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.
%H A208447 Alois P. Heinz, <a href="/A208447/b208447.txt">Antidiagonals n = 0..50</a>
%H A208447 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A208447 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:
%e A208447   .111.    . 21 . . . . . . .    . 3 . . . .
%e A208447   +---+    +------+  +------+    +---------+
%e A208447   | 1 |    | 1  2 |  | 1  3 |    | 1  2  3 |
%e A208447   | 2 |    | 3 .--+  | 2 .--+    +---------+
%e A208447   | 3 |    +---+     +---+
%e A208447   +---+
%e A208447 Square array A(n,k) begins:
%e A208447    1,  1,   1,    1,      1,       1,        1, ...
%e A208447    1,  1,   1,    1,      1,       1,        1, ...
%e A208447    2,  2,   2,    2,      2,       2,        2, ...
%e A208447    3,  4,   6,   10,     18,      34,       66, ...
%e A208447    5, 10,  24,   64,    180,     520,     1524, ...
%e A208447    7, 26, 120,  596,   3060,   16076,    86100, ...
%e A208447   11, 76, 720, 8056, 101160, 1379176, 19902600, ...
%p A208447 h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
%p A208447        +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A208447     end:
%p A208447 g:= proc(n, i, k, l) `if`(n=0, h(l)^k, `if`(i<1, 0, g(n, i-1, k, l)
%p A208447        + `if`(i>n, 0, g(n-i, i, k, [l[], i]))))
%p A208447     end:
%p A208447 A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
%p A208447 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A208447 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 *)
%Y A208447 Columns 0-10 give: A000041, A000085, A000142, A130721, A129627, A218432, A218433, A218434, A218435, A218436, A218437.
%Y A208447 Rows 0+1, 2, 3 give: A000012, A007395, A052548.
%Y A208447 Main diagonal gives A319607.
%K A208447 nonn,tabl
%O A208447 0,6
%A A208447 _Alois P. Heinz_, Feb 26 2012