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%I #5 Aug 29 2018 16:52:20
%S 1,3,8,21,54,137,343,847,2075,5031,12109,28921,68633,161865,379655
%N Number of pairs (c, y) where c is an integer composition and y is an integer partition and y can be obtained from c by choosing a partition of each part, flattening, and sorting.
%C Also the number of combinatory separations of normal multisets of weight n with constant parts. A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}.
%e The a(3) = 8 combinatory separations:
%e 111<={111}
%e 111<={1,11}
%e 111<={1,1,1}
%e 112<={1,11}
%e 112<={1,1,1}
%e 122<={1,11}
%e 122<={1,1,1}
%e 123<={1,1,1}
%t Table[Sum[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@c]]],{c,Join@@Permutations/@IntegerPartitions[n]}],{n,30}]
%Y Cf. A000041, A007716, A011782, A034691, A255906, A265947, A269134.
%Y Cf. A317533, A317791, A318396, A318559, A318560, A318562, A318565.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Aug 29 2018
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