cp's OEIS Frontend

A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.

%I A302545 #24 Jan 16 2023 04:34:01
%S A302545 1,0,2,3,12,23,84,204,682,1977,6546,21003,72038,248055,888771,3240578,
%T A302545 12152775,46527471,182339441,729405164,2979121279,12407308136,
%U A302545 52670355242,227725915268,1002285274515,4487915293698,20434064295155,94559526596293,444527730210294,2122005930659752
%N A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.
%C A302545 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.
%C A302545 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:
%C A302545   {{1,1,1,1}}
%C A302545   {{1,1,2,2}}
%C A302545   {{1},{1,1,1}}
%C A302545   {{1},{1,2,2}}
%C A302545   {{1,1},{1,1}}
%C A302545   {{1,1},{2,2}}
%C A302545   {{1,2},{1,2}}
%C A302545   {{1},{1},{1,1}}
%C A302545   {{1},{1},{2,2}}
%C A302545   {{1},{2},{1,2}}
%C A302545   {{1},{1},{1},{1}}
%C A302545   {{1},{1},{2},{2}}
%H A302545 Andrew Howroyd, <a href="/A302545/b302545.txt">Table of n, a(n) for n = 0..50</a>
%e A302545 The a(4) = 12 multiset partitions:
%e A302545   {{1,1,1,1}}
%e A302545   {{1,1,2,2}}
%e A302545   {{1,2,2,2}}
%e A302545   {{1,2,3,3}}
%e A302545   {{1,2,3,4}}
%e A302545   {{1,1},{1,1}}
%e A302545   {{1,1},{2,2}}
%e A302545   {{1,2},{1,2}}
%e A302545   {{1,2},{2,2}}
%e A302545   {{1,2},{3,3}}
%e A302545   {{1,2},{3,4}}
%e A302545   {{1,3},{2,3}}
%o A302545 (PARI) \\ compare with similar program for A007716.
%o A302545 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A302545 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}
%o A302545 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)}
%o A302545 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
%Y A302545 The set-system version is A330054 (no endpoints) or A306005 (no singletons).
%Y A302545 Non-isomorphic multiset partitions are A007716.
%Y A302545 Set-systems with no singletons are A016031.
%Y A302545 Cf. A049311, A283877, A293606, A293607, A306008, A317533, A317794, A317795, A320665, A330053, A330055, A330058.
%K A302545 nonn
%O A302545 0,3
%A A302545 _Gus Wiseman_, Jun 20 2018
%E A302545 Extended by _Gus Wiseman_, Dec 09 2019
%E A302545 Terms a(11) and beyond from _Andrew Howroyd_, Jan 15 2023