A256589 Triangle read by rows: T(n,k) divided by (n-k+1)! is the expected value of number of possible subsets in a partition of a set of n elements with no subsets of cardinality smaller than k.

1, 3, 1, 11, 2, 1, 50, 8, 2, 1, 274, 36, 6, 2, 1, 1764, 200, 30, 6, 2, 1, 13068, 1300, 168, 24, 6, 2, 1, 109584, 9720, 1080, 144, 24, 6, 2, 1, 1026576, 82180, 8100, 960, 120, 24, 6, 2, 1

Offset: 1

Author

Martin Y. Champel, Apr 03 2015

Keywords

• nonn
• tabl

Comments

This sequence can be seen as a generalization of A233744 which is the particular case where minimum subset cardinality is 2 (k=2).

T(n,k) / A004736(n,k)! = 1 + 1 / ( A002024(n,k) - A002260(n,k) + 1) * (Sum of (T(n,p) / A004736(n,p)!) for p starting at A002260(n,k) up to A002024(n) - A002260(n) if 2 * A002260(n) <= A002024(n)).

The triangle below but including the diagonal is A166350 because there is only one possible partition of subsets of cardinality >= k in any set whose cardinality is between k and 2*k-1.

Examples

			The triangle T(n, k) starts:
n\k  1   2   3  4  5  6  ...
1:   1
2:   3   1
3:  11   2   1
4:  50   8   2  1
5: 274  36   6  2  1
6:1764 200  36  6  2  1
		

Links

• Martin Y. Champel, Table of n, a(n) for n = 1..10000

Crossrefs

Column 2 is A233744.

Cf. A002024, A002260, A004736, A026791, A166350, A135010.