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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.
7
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.
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.
Sequence in context: A026533 A131630 A305652 * A208771 A036884 A102291
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 09 2022
STATUS
approved