proposed

approved

editing

proposed

Cf. A000041, A007716, A011782, A034691, A255906, A265947, A269134, A317533, A317791.

Cf. A317533, A317791, A318396, A318559, A318560, A318562, A318565.

allocated for Gus WisemanNumber 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.

1, 3, 8, 21, 54, 137, 343, 847, 2075, 5031, 12109, 28921, 68633, 161865, 379655

1,2

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}.

The a(3) = 8 combinatory separations:

111<={111}

111<={1,11}

111<={1,1,1}

112<={1,11}

112<={1,1,1}

122<={1,11}

122<={1,1,1}

123<={1,1,1}

Table[Sum[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@c]]], {c, Join@@Permutations/@IntegerPartitions[n]}], {n, 30}]

Cf. A000041, A007716, A011782, A255906, A265947, A269134, A317533, A317791.

allocated

nonn,more

Gus Wiseman, Aug 29 2018

approved

editing

allocated for Gus Wiseman

allocated

approved