OFFSET
1,3
COMMENTS
A multiset is normal if it spans an initial interval of positive integers.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..100
EXAMPLE
The a(3) = 8 multiset partitions with empty intersection:
{{2},{1,1}}
{{1},{2,2}}
{{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{1},{2}}
{{1},{2},{2}}
{{1},{2},{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_]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Length[Join@@Table[Select[mps[m], Intersection@@#=={}&], {m, allnorm[n]}]], {n, 6}]
PROG
(PARI)
P(n, k)={1/prod(i=1, n, (1 - x^i*y + O(x*x^n))^binomial(k+i-1, k-1))}
R(n, k)={my(p=P(n, k), q=p/(1-y+O(y*y^n))); Vec(sum(i=2, n, polcoef(p, i, y) + polcoef(q, i, y)*sum(j=1, n\i, (-1)^j*binomial(k, j)*x^(i*j))), -n)}
seq(n)={sum(k=2, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Feb 05 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 06 2018
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Feb 05 2021
STATUS
approved