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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. 18
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 (list; table; graph; refs; listen; history; text; internal format)
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
Rows 0+1, 2, 3 give: A000012, A007395, A052548.
Main diagonal gives A319607.
Sequence in context: A152977 A360334 A259799 * A320750 A117935 A224698
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 26 2012
STATUS
approved

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Last modified April 12 20:17 EDT 2024. Contains 371636 sequences. (Running on oeis4.)