A302545 - OEIS

%I #24 Jan 16 2023 04:34:01

%S 1,0,2,3,12,23,84,204,682,1977,6546,21003,72038,248055,888771,3240578,

%T 12152775,46527471,182339441,729405164,2979121279,12407308136,

%U 52670355242,227725915268,1002285274515,4487915293698,20434064295155,94559526596293,444527730210294,2122005930659752

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

%C 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 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   {{1,1,1,1}}

%C   {{1,1,2,2}}

%C   {{1},{1,1,1}}

%C   {{1},{1,2,2}}

%C   {{1,1},{1,1}}

%C   {{1,1},{2,2}}

%C   {{1,2},{1,2}}

%C   {{1},{1},{1,1}}

%C   {{1},{1},{2,2}}

%C   {{1},{2},{1,2}}

%C   {{1},{1},{1},{1}}

%C   {{1},{1},{2},{2}}

%H Andrew Howroyd, <a href="/A302545/b302545.txt">Table of n, a(n) for n = 0..50</a>

%e The a(4) = 12 multiset partitions:

%e   {{1,1,1,1}}

%e   {{1,1,2,2}}

%e   {{1,2,2,2}}

%e   {{1,2,3,3}}

%e   {{1,2,3,4}}

%e   {{1,1},{1,1}}

%e   {{1,1},{2,2}}

%e   {{1,2},{1,2}}

%e   {{1,2},{2,2}}

%e   {{1,2},{3,3}}

%e   {{1,2},{3,4}}

%e   {{1,3},{2,3}}

%o (PARI) \\ compare with similar program for A007716.

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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 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 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 The set-system version is A330054 (no endpoints) or A306005 (no singletons).

%Y Non-isomorphic multiset partitions are A007716.

%Y Set-systems with no singletons are A016031.

%Y Cf. A049311, A283877, A293606, A293607, A306008, A317533, A317794, A317795, A320665, A330053, A330055, A330058.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jun 20 2018

%E Extended by _Gus Wiseman_, Dec 09 2019

%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 15 2023