%I #41 Aug 01 2018 10:31:46
%S 1,1,1,1,2,1,1,4,2,2,1,1,7,6,4,3,2,1,1,11,17,5,8,12,2,4,3,2,1,1,16,41,
%T 23,15,39,14,8,8,12,4,4,3,2,1,1,22,87,86,17,26,108,81,27,18,16,40,15,
%U 17,3,8,12,4,4,3,2,1,1,29,167,263,109,42,263,342,78,81,115,10,31,116,87,60,39,11,16,40,15,17,5,8,12,4,4,3,2,1,1,37,296,695,509,73,64,578,1177,602,216,525,169,64,57,306,380,90,189,261,34,38,26,32,117,88,61,40,22,3,16,40,15,17,5,8,12,4,4,3,2,1
%N Multisets of multisets corresponding to integer partitions lambda, drawn from |lambda| symbols, where the sizes of the multisets are given by the elements of lambda as is the total number of occurrences of each symbol.
%C 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.
%D F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
%H Marko Riedel, <a href="/A244581/b244581.txt">Table of n, a(n) for n = 1..271</a>
%H Marko Riedel et. al, <a href="https://math.stackexchange.com/questions/2856255/">Partitioning a multiset into multisets of fixed sizes</a>
%H Marko Riedel, <a href="/A244581/a244581.maple.txt">Maple code for sequence using cycle index of the symmetric group and cycle index substitution.</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young%27s_lattice">Young's lattice</a>
%F 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.
%e 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}}.
%e The initial list of the partitions is:
%e 1;
%e 1,1; 2;
%e 1,1,1; 1,2; 3;
%e 1,1,1,1; 1,1,2; 2+2; 1+3; 4;
%e The data then yields the following values:
%e 1,
%e 1, 1,
%e 1, 2, 1,
%e 1, 4, 2, 2.
%K tabf,nonn
%O 1,5
%A _Marko Riedel_, Jul 31 2018