A356937 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers.

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

Author

Gus Wiseman, Sep 08 2022

Keywords

• nonn

Comments

An interval such as {3,4,5} is a set with all differences of adjacent elements equal to 1.

Examples

			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}}
		

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. A055887, A063834, A270995, A304969, A349050, A349055.

Intervals are counted by A000012, A001227, ranked by A073485.

Other conditions: A034691, A116540, A255906, A356933, A356942.

Other types: A107742, A356936, A356938, A356939.

Programs

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}]

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

Extensions

Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023