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

1,5

Comments

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

Links

Table of n, a(n) for n=1..99.

Example

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

Crossrefs

Sequence in context: A062540 A173636 A115878 * A376799 A256671 A327156

Adjacent sequences: A127122 A127123 A127124 * A127126 A127127 A127128

Keyword

nonn,tabf

Author

Franklin T. Adams-Watters, Jan 05 2007

Status

approved