A356942 Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval.

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

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: A383949 A070071 A285363 * A151484 A391169 A275871

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