login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.
6

%I #11 Jan 01 2023 17:58:57

%S 1,1,2,8,28,108,524,2608,14176,86576,550672,3782496,27843880,

%T 214071392,1751823600,15041687664,134843207240,1269731540864,

%U 12427331494304,126619822952928,1341762163389920,14712726577081248,167209881188545344,1963715680476759040,23794190474350155856

%N Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.

%H Andrew Howroyd, <a href="/A356933/b356933.txt">Table of n, a(n) for n = 0..500</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(4) = 28 multiset partitions:

%e {1}{111} {1}{112} {1}{123} {1}{234}

%e {1}{1}{1}{1} {1}{122} {1}{223} {2}{134}

%e {1}{222} {1}{233} {3}{124}

%e {2}{111} {2}{113} {4}{123}

%e {2}{112} {2}{123} {1}{2}{3}{4}

%e {2}{122} {2}{133}

%e {1}{1}{1}{2} {3}{112}

%e {1}{1}{2}{2} {3}{122}

%e {1}{2}{2}{2} {3}{123}

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

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

%e {1}{2}{3}{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 Table[Length[Select[Join@@mps/@allnorm[n],OddQ[Times@@Length/@#]&]],{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, if(j%2 == 1, binomial(j+k-1, j))))}

%o seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ _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, A072233, A270995, A304969, A349050, A349055.

%Y Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.

%Y Other conditions: A034691, A116540, A255906, A356937, A356942.

%Y Other types: A050330, A356932, A356934, A356935.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 08 2022

%E Terms a(9) and beyond from _Andrew Howroyd_, Jan 01 2023