cp's OEIS Frontend

A320803 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic multisets.

%I A320803 #8 Jan 16 2023 14:49:39
%S A320803 1,1,3,7,21,56,174,517,1664,5383,18199,62745,223390,813425,3040181,
%T A320803 11620969,45446484,181537904,740369798,3079779662,13059203150,
%U A320803 56406416004,248027678362,1109626606188,5048119061134,23342088591797,109648937760252,523036690273237
%N A320803 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic multisets.
%C A320803 A multiset is aperiodic if its multiplicities are relatively prime.
%C A320803 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 A320803 Andrew Howroyd, <a href="/A320803/b320803.txt">Table of n, a(n) for n = 0..50</a>
%e A320803 Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions with aperiodic parts:
%e A320803   {{1}}  {{1,2}}    {{1,2,2}}      {{1,2,2,2}}
%e A320803          {{1},{1}}  {{1,2,3}}      {{1,2,3,3}}
%e A320803          {{1},{2}}  {{1},{2,3}}    {{1,2,3,4}}
%e A320803                     {{2},{1,2}}    {{1},{1,2,2}}
%e A320803                     {{1},{1},{1}}  {{1,2},{1,2}}
%e A320803                     {{1},{2},{2}}  {{1},{2,3,3}}
%e A320803                     {{1},{2},{3}}  {{1},{2,3,4}}
%e A320803                                    {{1,2},{3,4}}
%e A320803                                    {{1,3},{2,3}}
%e A320803                                    {{2},{1,2,2}}
%e A320803                                    {{3},{1,2,3}}
%e A320803                                    {{1},{1},{2,3}}
%e A320803                                    {{1},{2},{1,2}}
%e A320803                                    {{1},{2},{3,4}}
%e A320803                                    {{1},{3},{2,3}}
%e A320803                                    {{2},{2},{1,2}}
%e A320803                                    {{1},{1},{1},{1}}
%e A320803                                    {{1},{1},{2},{2}}
%e A320803                                    {{1},{2},{2},{2}}
%e A320803                                    {{1},{2},{3},{3}}
%e A320803                                    {{1},{2},{3},{4}}
%o A320803 (PARI)
%o A320803 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A320803 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 A320803 K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
%o A320803 a(n)={if(n==0, 1, my(mbt=vector(n, d, moebius(d)), s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(dirmul(mbt, sum(t=1, n, K(q, t, n)/t)))), n)); s/n!)} \\ _Andrew Howroyd_, Jan 16 2023
%Y A320803 Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A303709, A303710, A320800-A320810.
%K A320803 nonn
%O A320803 0,3
%A A320803 _Gus Wiseman_, Nov 06 2018
%E A320803 Terms a(11) and beyond from _Andrew Howroyd_, Jan 16 2023