A356938 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers with weakly decreasing multiplicities.

1, 1, 3, 7, 18, 41, 101, 228, 538, 1209

Offset

0,3

Comments

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

Links

Table of n, a(n) for n=0..9.

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

Example

The a(1) = 1 through a(4) = 18 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{1,2}}    {{1},{1,2,3}}
         {{1},{2}}  {{1},{2,3}}    {{1,2},{1,2}}
                    {{3},{1,2}}    {{1},{2,3,4}}
                    {{1},{1},{1}}  {{1,2},{3,4}}
                    {{1},{1},{2}}  {{4},{1,2,3}}
                    {{1},{2},{3}}  {{1},{1},{1,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{1,2}}
                                   {{1},{4},{2,3}}
                                   {{3},{4},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{3},{4}}

Mathematica

strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
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]]]];
chQ[y_]:=Or[Length[y]<=1, Union[Differences[y]]=={1}];
Table[Length[Select[Join@@mps/@strnorm[n], And@@chQ/@#&]], {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.

Intervals are counted by A000012, A001227, ranked by A073485.

Other conditions: A035310, A063834, A330783, A356934.

Other types: A107742, A356936, A356937, A356939, A356943, A356954.

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

Sequence in context: A026533 A131630 A305652 * A208771 A036884 A102291

Adjacent sequences: A356935 A356936 A356937 * A356939 A356940 A356941

Keyword

nonn,more

Author

Gus Wiseman, Sep 09 2022

Status

approved