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A356943
Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.
6
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.
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.
Sequence in context: A300772 A238244 A247816 * A017939 A282702 A130494
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 09 2022
STATUS
approved