cp's OEIS Frontend

A321677 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with no singletons.

%I A321677 #12 Jan 16 2024 17:33:00
%S A321677 1,0,1,1,4,4,16,22,70,132,375,848,2428,6256,18333,52560,161436,500887,
%T A321677 1624969,5384625,18438815,64674095,233062429,859831186,3248411250,
%U A321677 12545820860,49508089411,199410275018,819269777688,3430680180687,14633035575435,63535672197070
%N A321677 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with no singletons.
%C A321677 The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%H A321677 Andrew Howroyd, <a href="/A321677/b321677.txt">Table of n, a(n) for n = 0..50</a>
%e A321677 Non-isomorphic representatives of the a(2) = 1 through a(6) = 16 set multipartitions:
%e A321677   {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}    {{1,2,3,4,5,6}}
%e A321677                       {{1,2},{1,2}}  {{1,2},{3,4,5}}  {{1,2,3},{1,2,3}}
%e A321677                       {{1,2},{3,4}}  {{1,4},{2,3,4}}  {{1,2},{3,4,5,6}}
%e A321677                       {{1,3},{2,3}}  {{2,3},{1,2,3}}  {{1,2,3},{4,5,6}}
%e A321677                                                       {{1,2,5},{3,4,5}}
%e A321677                                                       {{1,3,4},{2,3,4}}
%e A321677                                                       {{1,5},{2,3,4,5}}
%e A321677                                                       {{3,4},{1,2,3,4}}
%e A321677                                                       {{1,2},{1,2},{1,2}}
%e A321677                                                       {{1,2},{1,3},{2,3}}
%e A321677                                                       {{1,2},{3,4},{3,4}}
%e A321677                                                       {{1,2},{3,4},{5,6}}
%e A321677                                                       {{1,2},{3,5},{4,5}}
%e A321677                                                       {{1,3},{2,3},{2,3}}
%e A321677                                                       {{1,3},{2,4},{3,4}}
%e A321677                                                       {{1,4},{2,4},{3,4}}
%o A321677 (PARI)
%o A321677 WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o A321677 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 A321677 K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}
%o A321677 a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g), n)); s/n!)} \\ _Andrew Howroyd_, Jan 16 2024
%Y A321677 Cf. A000219, A007716, A049311, A283877, A302545, A316983, A319616.
%Y A321677 Cf. A320797, A320798, A320804, A320811, A320812, A321404, A321406.
%K A321677 nonn
%O A321677 0,5
%A A321677 _Gus Wiseman_, Nov 16 2018
%E A321677 Terms a(11) and beyond from _Andrew Howroyd_, Sep 01 2019