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A356934
Number of multisets of odd-size multisets whose multiset union is a size-n multiset covering an initial interval with weakly decreasing multiplicities.
7
1, 1, 2, 6, 17, 46, 166, 553, 2093
OFFSET
0,3
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.
Sequence in context: A102403 A278428 A344433 * A200379 A032638 A292229
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 09 2022
STATUS
approved