A271708 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with largest part k and Aut(p) = 1^j[1]*j[1]!*...*n^j[n]*j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

1, 0, 1, 0, 2, 2, 0, 6, 2, 3, 0, 24, 12, 3, 4, 0, 120, 20, 12, 4, 5, 0, 720, 112, 42, 16, 5, 6, 0, 5040, 336, 126, 44, 20, 6, 7, 0, 40320, 2112, 492, 188, 55, 24, 7, 8, 0, 362880, 11712, 2802, 640, 215, 66, 28, 8, 9, 0, 3628800, 92160, 16938, 3624, 830, 258, 77, 32, 9, 10

Offset: 0

Author

Peter Luschny, Apr 17 2016

Keywords

• nonn
• tabl

Comments

Also T(n,k) = Sum_{p in P(n,k)} Cen(p) where Cen(p) is the size of the centralizer of any permutation of cycle type p.

Examples

			Triangle starts:
[1]
[0, 1]
[0, 2, 2]
[0, 6, 2, 3]
[0, 24, 12, 3, 4]
[0, 120, 20, 12, 4, 5]
[0, 720, 112, 42, 16, 5, 6]
[0, 5040, 336, 126, 44, 20, 6, 7]
[0, 40320, 2112, 492, 188, 55, 24, 7, 8]
		

Crossrefs

Cf. A110143 (row sums), A126074.

Programs

Sage

def A271708(n,k):
    P = Partitions(n, max_part=k, inner=[k])
    return sum([p.aut() for p in P])
for n in (0..9): print([A271708(n,k) for k in (0..n)])