A271707 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 length 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, 3, 2, 6, 0, 4, 11, 4, 24, 0, 5, 10, 14, 12, 120, 0, 6, 31, 62, 34, 48, 720, 0, 7, 28, 60, 84, 120, 240, 5040, 0, 8, 66, 102, 490, 228, 552, 1440, 40320, 0, 9, 60, 299, 292, 708, 912, 3120, 10080, 362880, 0, 10, 120, 282, 722, 4396, 2136, 4752, 20880, 80640, 3628800

Offset: 0

Author

Peter Luschny, Apr 17 2016

Keywords

• nonn
• tabl

Comments

S(n,k) = Sum_{p in P(n,k)} n!/Aut(p) are the Stirling cycle numbers A132393.

Examples

			Triangle starts:
[1]
[0, 1]
[0, 2, 2]
[0, 3, 2, 6]
[0, 4, 11, 4, 24]
[0, 5, 10, 14, 12, 120]
[0, 6, 31, 62, 34, 48, 720]
[0, 7, 28, 60, 84, 120, 240, 5040]
		

Crossrefs

Cf. A110143 (row sums), A132393, A271708.

Programs

Sage

def A271707(n,k):
    P = Partitions(n, length=k)
    return sum(p.aut() for p in P)
for n in (0..10): print([A271707(n,k) for k in (0..n)])