OFFSET
1,5
COMMENTS
The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. The row lengths of this irregular table are given by the partition function A000041.
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
LINKS
Marko Riedel, Table of n, a(n) for n = 1..271
Marko Riedel et. al, Partitioning a multiset into multisets of fixed sizes
Marko Riedel, Maple code for sequence using cycle index of the symmetric group and cycle index substitution.
Wikipedia, Young's lattice
FORMULA
With the partition given by Prod_{k=1}^l A_k^{tau_k} the closed form is [Prod_{k=1}^l A_k^{tau_k}] Prod_{k=1}^l Z(S_{tau_k}; Z(S_k; Prod_{k'=1}^l A_k')) where Z(S_k) is the cycle index of the symmetric group.
EXAMPLE
With the partition [1,1,2] or A_1 A_2 A_3^2 we get four multisets: {{A_1}, {A_2}, {A_3^2}}, {{A_1}, {A_3}, {A_2 A_3}}, {{A_2}, {A_3}, {A_1, A_3}} and {{A_3}, {A_3}, {A_1 A_2}}.
The initial list of the partitions is:
1;
1,1; 2;
1,1,1; 1,2; 3;
1,1,1,1; 1,1,2; 2+2; 1+3; 4;
The data then yields the following values:
1,
1, 1,
1, 2, 1,
1, 4, 2, 2.
CROSSREFS
KEYWORD
tabf,nonn
AUTHOR
Marko Riedel, Jul 31 2018
STATUS
approved