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%I #10 Jan 01 2023 19:29:39
%S 1,1,4,15,61,249,1040,4363,18424,78014,331099,1407080,5985505,
%T 25477399,108493103,462147381,1969025286,8390475609,35757524184,
%U 152398429323,649555719160,2768653475487,11801369554033,50304231997727,214428538858889,914039405714237
%N Number of multisets of gapless multisets whose multiset union is a size-n multiset covering an initial interval.
%C 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.
%H Andrew Howroyd, <a href="/A356942/b356942.txt">Table of n, a(n) for n = 0..200</a>
%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vR-C_picqWlu0KOguRGWaPjhS2HY7m43aGXGDcolDh4Qtyy-pu2lkq5mbHAbiMSyQoiIESG2mCGtc2j/pub">Counting and ranking classes of multiset partitions related to gapless multisets</a>
%e The a(1) = 1 through a(3) = 14 multiset partitions:
%e {{1}} {{1,1}} {{1,1,1}}
%e {{1,2}} {{1,1,2}}
%e {{1},{1}} {{1,2,2}}
%e {{1},{2}} {{1,2,3}}
%e {{1},{1,1}}
%e {{1},{1,2}}
%e {{1},{2,2}}
%e {{1},{2,3}}
%e {{2},{1,1}}
%e {{2},{1,2}}
%e {{3},{1,2}}
%e {{1},{1},{1}}
%e {{1},{1},{2}}
%e {{1},{2},{2}}
%e {{1},{2},{3}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
%t Table[Length[Select[Join@@mps/@allnorm[n],And@@nogapQ/@#&]],{n,0,5}]
%o (PARI)
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o R(n,k) = {EulerT(vector(n, j, sum(i=1, min(k, j), (k-i+1)*binomial(j-1, i-1))))}
%o 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
%Y A000041 counts integer partitions, strict A000009.
%Y A000670 counts patterns, ranked by A333217, necklace A019536.
%Y A011782 counts multisets covering an initial interval.
%Y Cf. A063834, A072233, A270995, A304969, A349050, A349055, A356934.
%Y Gapless multisets are counted by A034296, ranked by A073491.
%Y Other conditions: A034691, A055887, A116540, A255906, A356933, A356937.
%Y Other types of multiset partitions: A356233, A356941, A356943, A356944.
%K nonn
%O 0,3
%A _Gus Wiseman_, Sep 08 2022
%E Terms a(9) and beyond from _Andrew Howroyd_, Jan 01 2023
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