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

1,2

Comments

A232744 can be found in the second column of this triangle.

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.

Links

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

Example

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

Crossrefs

Cf. A002024, A002260, A004736, A026791, A166350, A135010, A233744.

Sequence in context: A205483 A230262 A323854 * A229834 A120291 A099001

Adjacent sequences: A256586 A256587 A256588 * A256590 A256591 A256592

Keyword

nonn,tabl

Author

Martin Y. Champel, Apr 03 2015

Status

approved