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Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers.
7

%I #10 Jan 01 2023 19:29:49

%S 1,1,3,9,29,94,310,1026,3411,11360,37886,126442,422203,1410189,

%T 4711039,15740098,52593430,175742438,587266782,1962469721,6558071499,

%U 21915580437,73237274083,244744474601,817889464220,2733235019732,9133973730633,30524096110942,102006076541264

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

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

%H Andrew Howroyd, <a href="/A356937/b356937.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) = 9 set multipartitions (multisets of sets):

%e {{1}} {{1,2}} {{1,2,3}}

%e {{1},{1}} {{1},{1,2}}

%e {{1},{2}} {{1},{2,3}}

%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 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];

%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 chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];

%t Table[Length[Select[Join@@mps/@allnorm[n],And@@chQ/@#&]],{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, max(0, 1+k-j)))}

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

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

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

%Y Other types: A107742, A356936, A356938, A356939.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 08 2022

%E Terms a(10) and beyond from _Andrew Howroyd_, Jan 01 2023