login
A356933
Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.
6
1, 1, 2, 8, 28, 108, 524, 2608, 14176, 86576, 550672, 3782496, 27843880, 214071392, 1751823600, 15041687664, 134843207240, 1269731540864, 12427331494304, 126619822952928, 1341762163389920, 14712726577081248, 167209881188545344, 1963715680476759040, 23794190474350155856
OFFSET
0,3
EXAMPLE
The a(4) = 28 multiset partitions:
{1}{111} {1}{112} {1}{123} {1}{234}
{1}{1}{1}{1} {1}{122} {1}{223} {2}{134}
{1}{222} {1}{233} {3}{124}
{2}{111} {2}{113} {4}{123}
{2}{112} {2}{123} {1}{2}{3}{4}
{2}{122} {2}{133}
{1}{1}{1}{2} {3}{112}
{1}{1}{2}{2} {3}{122}
{1}{2}{2}{2} {3}{123}
{1}{1}{2}{3}
{1}{2}{2}{3}
{1}{2}{3}{3}
MATHEMATICA
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]]]];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n], OddQ[Times@@Length/@#]&]], {n, 0, 5}]
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k) = {EulerT(vector(n, j, if(j%2 == 1, binomial(j+k-1, j))))}
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
CROSSREFS
A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other conditions: A034691, A116540, A255906, A356937, A356942.
Other types: A050330, A356932, A356934, A356935.
Sequence in context: A292668 A122447 A150715 * A026528 A150716 A151299
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 08 2022
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023
STATUS
approved