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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 Andrew Howroyd, Table of n, a(n) for n = 0..200 Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets 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. Cf. A063834, A072233, A270995, A304969, A349050, A349055, A356934. 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 Adjacent sequences: A356939 A356940 A356941 * A356943 A356944 A356945 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.)