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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. 3

%I

%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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Last modified December 6 06:34 EST 2019. Contains 329784 sequences. (Running on oeis4.)