cp's OEIS Frontend

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

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