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.

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

Offset

1,2

Comments

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}.

Links

Table of n, a(n) for n=1..15.

Example

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}

Mathematica

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

Crossrefs

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

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

Sequence in context: A038200 A291039 A030015 * A103446 A218482 A094723

Adjacent sequences: A318564 A318565 A318566 * A318568 A318569 A318570

Keyword

nonn,more

Author

Gus Wiseman, Aug 29 2018

Status

approved