A356943 Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.

1, 1, 4, 11, 37, 101, 328, 909, 2801

Offset

0,3

Comments

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.

Links

Table of n, a(n) for n=0..8.

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Example

The a(1) = 1 through a(3) = 11 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1,2}}    {{1,1,2}}
         {{1},{1}}  {{1,2,3}}
         {{1},{2}}  {{1},{1,1}}
                    {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,1}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}

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];
nogapQ[m_]:=Or[m=={}, Union[m]==Range[Min[m], Max[m]]];
Table[Length[Select[Join@@mps/@strnorm[n], And@@nogapQ/@#&]], {n, 0, 5}]

Crossrefs

A000041 counts integer partitions, strict A000009.

A000670 counts patterns, ranked by A333217, necklace A019536.

A011782 counts multisets covering an initial interval.

Gapless multisets are counted by A034296, ranked by A073491.

Other conditions: A035310, A063834, A330783, A356934, A356938, A356954.

Other types: A356233, A356941, A356942, A356944.

Cf. A055887, A072233, A270995, A304969, A349050, A349055.

Sequence in context: A300772 A238244 A247816 * A017939 A282702 A130494

Adjacent sequences: A356940 A356941 A356942 * A356944 A356945 A356946

Keyword

nonn,more

Author

Gus Wiseman, Sep 09 2022

Status

approved