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A317752
Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.
22
0, 1, 8, 49, 305, 1984, 13686, 100124, 776885, 6386677, 55532358, 509549386, 4921352952, 49899820572, 529807799836, 5876162077537, 67928460444139, 816764249684450, 10195486840926032, 131896905499007474, 1765587483656124106, 24419774819813602870
OFFSET
1,3
COMMENTS
A multiset is normal if it spans an initial interval of positive integers.
LINKS
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 06 2018
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Feb 05 2021
STATUS
approved