A356934 Number of multisets of odd-size multisets whose multiset union is a size-n multiset covering an initial interval with weakly decreasing multiplicities.

1, 1, 2, 6, 17, 46, 166, 553, 2093

Offset

0,3

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(4) = 17 multiset partitions:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,1,2}}      {{1},{1,1,2}}
                    {{1,2,3}}      {{1},{1,2,2}}
                    {{1},{1},{1}}  {{1},{1,2,3}}
                    {{1},{1},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{2},{1,1,2}}
                                   {{2},{1,1,3}}
                                   {{2},{1,3,4}}
                                   {{3},{1,1,2}}
                                   {{3},{1,2,4}}
                                   {{4},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{3},{4}}

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];
Table[Length[Select[Join@@mps/@strnorm[n], OddQ[Times@@Length/@#]&]], {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.

Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.

Other conditions: A035310, A063834, A330783, A356938, A356943, A356954.

Other types: A050330, A356932, A356933, A356935.

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

Sequence in context: A102403 A278428 A344433 * A200379 A032638 A292229

Adjacent sequences: A356931 A356932 A356933 * A356935 A356936 A356937

Keyword

nonn,more

Author

Gus Wiseman, Sep 09 2022

Status

approved