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

Author

Gus Wiseman, Sep 08 2022

Keywords

• nonn

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.

Examples

			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}}
		

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

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.

Programs

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}]

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

Extensions

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023