A318563 Number of combinatory separations of strongly normal multisets of weight n.

1, 4, 10, 33, 85, 272, 730, 2197, 6133

Offset

1,2

Comments

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers 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}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

Links

Table of n, a(n) for n=1..9.

Example

The a(3) = 10 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={112}
  112<={1,11}
  112<={1,12}
  112<={1,1,1}
  123<={123}
  123<={1,12}
  123<={1,1,1}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]], i}, {i, Length[Union[m]]}];
Table[Length[Union@@Table[{m, Sort[normize/@#]}&/@mps[m], {m, strnorm[n]}]], {n, 7}]

Crossrefs

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

Cf. A317533, A317791, A318396, A318559, A318562, A318564, A318565, A318566.

Sequence in context: A165730 A052366 A309682 * A052367 A052372 A052373

Adjacent sequences: A318560 A318561 A318562 * A318564 A318565 A318566

Keyword

nonn,more

Author

Gus Wiseman, Aug 29 2018

Status

approved