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A356937
Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers.
7
1, 1, 3, 9, 29, 94, 310, 1026, 3411, 11360, 37886, 126442, 422203, 1410189, 4711039, 15740098, 52593430, 175742438, 587266782, 1962469721, 6558071499, 21915580437, 73237274083, 244744474601, 817889464220, 2733235019732, 9133973730633, 30524096110942, 102006076541264
OFFSET
0,3
COMMENTS
An interval such as {3,4,5} is a set with all differences of adjacent elements equal to 1.
EXAMPLE
The a(1) = 1 through a(3) = 9 set multipartitions (multisets of sets):
{{1}} {{1,2}} {{1,2,3}}
{{1},{1}} {{1},{1,2}}
{{1},{2}} {{1},{2,3}}
{{2},{1,2}}
{{3},{1,2}}
{{1},{1},{1}}
{{1},{1},{2}}
{{1},{2},{2}}
{{1},{2},{3}}
MATHEMATICA
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
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/@allnorm[n], And@@chQ/@#&]], {n, 0, 5}]
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k) = {EulerT(vector(n, j, max(0, 1+k-j)))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n, k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A, x, 1))} \\ Andrew Howroyd, Jan 01 2023
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: A034691, A116540, A255906, A356933, A356942.
Other types: A107742, A356936, A356938, A356939.
Sequence in context: A024964 A061534 A300044 * A071728 A036550 A290897
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 08 2022
EXTENSIONS
Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023
STATUS
approved