login
Number of endofunctions on a set, where the multiset of indegrees forms the n-th partition in Mathematica order (ignoring 0's).
1

%I #5 Mar 30 2012 17:35:17

%S 1,1,1,2,1,3,3,1,3,3,7,5,1,3,4,8,10,14,7,1,3,4,8,3,19,17,6,32,26,11,1,

%T 3,4,8,4,19,18,11,14,63,34,29,75,45,15,1,3,4,8,4,19,18,3,20,14,64,37,

%U 14,39,85,168,62,15,109,167,75,22,1,3,4,8,4,19,18,4,20,14,64,38,11,26,71

%N Number of endofunctions on a set, where the multiset of indegrees forms the n-th partition in Mathematica order (ignoring 0's).

%C Can be regarded as a triangle with one row for each size of partition.

%e 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). In the first, node 1 has indegree 3, the next 3 node 1 has indegree 2 and node 2 has indegree 1 (forming partition [2,1]) and the final 3 are permutations, each node having indegree 1. The partitions of 3 in Mathematica order are [3], [2,1], [1^3], so row 3 of the triangle is 1,3,3.

%e The triangle starts:

%e 1

%e 1

%e 1 2

%e 1 3 3

%e 1 3 3 7 5

%e 1 3 4 8 10 14 7

%K nonn,tabf

%O 0,4

%A _Franklin T. Adams-Watters_, Jan 05 2007