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A302545 Number of non-isomorphic multiset partitions of weight n with no singletons. 46
1, 0, 2, 3, 12, 23, 84, 204, 682, 1977, 6546, 21003, 72038, 248055, 888771, 3240578, 12152775, 46527471, 182339441, 729405164, 2979121279, 12407308136, 52670355242, 227725915268, 1002285274515, 4487915293698, 20434064295155, 94559526596293, 444527730210294, 2122005930659752 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
A multiset partition is a finite multiset of finite nonempty multisets of positive integers. A singleton is a multiset of size 1. The weight of a multiset partition is the sum of sizes of its elements. Weight is generally not the same as number of vertices.
Also non-isomorphic multiset partitions of weight n with no endpoints, where an endpoint is a vertex appearing only once (degree 1). For example, non-isomorphic representations of the a(4) = 12 multiset partitions are:
{{1,1,1,1}}
{{1,1,2,2}}
{{1},{1,1,1}}
{{1},{1,2,2}}
{{1,1},{1,1}}
{{1,1},{2,2}}
{{1,2},{1,2}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{2},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
LINKS
EXAMPLE
The a(4) = 12 multiset partitions:
{{1,1,1,1}}
{{1,1,2,2}}
{{1,2,2,2}}
{{1,2,3,3}}
{{1,2,3,4}}
{{1,1},{1,1}}
{{1,1},{2,2}}
{{1,2},{1,2}}
{{1,2},{2,2}}
{{1,2},{3,3}}
{{1,2},{3,4}}
{{1,3},{2,3}}
PROG
(PARI) \\ compare with similar program for A007716.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t)) + O(x*x^k), -k)}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 15 2023
CROSSREFS
The set-system version is A330054 (no endpoints) or A306005 (no singletons).
Non-isomorphic multiset partitions are A007716.
Set-systems with no singletons are A016031.
Sequence in context: A323126 A323127 A323111 * A320173 A107928 A130337
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 20 2018
EXTENSIONS
Extended by Gus Wiseman, Dec 09 2019
Terms a(11) and beyond from Andrew Howroyd, Jan 15 2023
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)