A127125 Triangle read by rows: T(n,k) is the number of endofunctions on n objects where the multiset of loop sizes forms the k-th partition in Mathematica ordering.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 3, 3, 1, 2, 1, 1, 1, 1, 1, 1, 9, 6, 6, 3, 6, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 20, 16, 16, 9, 15, 7, 4, 6, 4, 7, 3, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 48, 37, 37, 23, 41, 18, 11, 18, 9, 18, 7, 4, 7, 7, 7, 7, 7, 3, 1, 2, 2, 2, 1, 3, 2, 1, 2, 2, 1, 1, 1

Offset: 1

Author

Franklin T. Adams-Watters, Jan 05 2007

Keywords

• nonn
• tabf

Comments

The number of loops is equal to the number of components, but the sizes may be smaller.

Examples

			For n = 3, the 7 endofunctions are (1,2,3) -> (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,2,3), (1,3,2) and (2,3,1). The loops are respectively 1, 1, 1|2, 12, 1|2|3, 1|23 and 123, corresponding to partitions [1], [1], [1^2], [2], [1^3], [2,1] and [3]. The partitions of 1 to 3 in Mathematica order are [1], [2], [1^2], [3], [2,1] and [1^3], so row 3 is 2, 1,1, 1,1,1.
The triangle starts:
1
1, 1 1
2, 1 1, 1 1 1
4, 3 3, 1 2 1, 1 1 1 1 1