cp's OEIS Frontend

A318567 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.

%I A318567 #5 Aug 29 2018 16:52:20
%S A318567 1,3,8,21,54,137,343,847,2075,5031,12109,28921,68633,161865,379655
%N A318567 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 A318567 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 A318567 The a(3) = 8 combinatory separations:
%e A318567   111<={111}
%e A318567   111<={1,11}
%e A318567   111<={1,1,1}
%e A318567   112<={1,11}
%e A318567   112<={1,1,1}
%e A318567   122<={1,11}
%e A318567   122<={1,1,1}
%e A318567   123<={1,1,1}
%t A318567 Table[Sum[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@c]]],{c,Join@@Permutations/@IntegerPartitions[n]}],{n,30}]
%Y A318567 Cf. A000041, A007716, A011782, A034691, A255906, A265947, A269134.
%Y A318567 Cf. A317533, A317791, A318396, A318559, A318560, A318562, A318565.
%K A318567 nonn,more
%O A318567 1,2
%A A318567 _Gus Wiseman_, Aug 29 2018