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A356942 Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval. 7
1, 1, 4, 15, 61, 249, 1040, 4363, 18424, 78014, 331099, 1407080, 5985505, 25477399, 108493103, 462147381, 1969025286, 8390475609, 35757524184, 152398429323, 649555719160, 2768653475487, 11801369554033, 50304231997727, 214428538858889, 914039405714237 (list; graph; refs; listen; history; text; internal format)
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.
LINKS
EXAMPLE
The a(1) = 1 through a(3) = 14 multiset partitions:
{{1}} {{1,1}} {{1,1,1}}
{{1,2}} {{1,1,2}}
{{1},{1}} {{1,2,2}}
{{1},{2}} {{1,2,3}}
{{1},{1,1}}
{{1},{1,2}}
{{1},{2,2}}
{{1},{2,3}}
{{2},{1,1}}
{{2},{1,2}}
{{3},{1,2}}
{{1},{1},{1}}
{{1},{1},{2}}
{{1},{2},{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]]]];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
nogapQ[m_]:=Or[m=={}, Union[m]==Range[Min[m], Max[m]]];
Table[Length[Select[Join@@mps/@allnorm[n], And@@nogapQ/@#&]], {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, sum(i=1, min(k, j), (k-i+1)*binomial(j-1, i-1))))}
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.
Gapless multisets are counted by A034296, ranked by A073491.
Other conditions: A034691, A055887, A116540, A255906, A356933, A356937.
Other types of multiset partitions: A356233, A356941, A356943, A356944.
Sequence in context: A369838 A070071 A285363 * A151484 A275871 A007161
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 08 2022
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023
STATUS
approved

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Last modified May 19 19:45 EDT 2024. Contains 372703 sequences. (Running on oeis4.)