cp's OEIS Frontend

A320804 Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.

%I A320804 #7 Jan 16 2023 14:48:58
%S A320804 1,0,1,2,6,13,41,104,326,958,3096,9958,33869,116806,417741,1526499,
%T A320804 5732931,22015642,86543717,347495480,1424832602,5959123908,
%U A320804 25407212843,110344848622,487879651220,2194697288628,10039367091586,46675057440634,220447539120814
%N A320804 Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.
%C A320804 Also the number of nonnegative integer matrices with (1) sum of elements equal to n, (2) no zero columns, (3) no rows summing to 0 or 1, and (4) no rows whose nonzero entries have a common divisor > 1, up to row and column permutations.
%C A320804 A multiset is aperiodic if its multiplicities are relatively prime.
%C A320804 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 A320804 Andrew Howroyd, <a href="/A320804/b320804.txt">Table of n, a(n) for n = 0..50</a>
%e A320804 Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 multiset partitions with aperiodic parts and no singletons:
%e A320804   {{1,2}}  {{1,2,2}}  {{1,2,2,2}}    {{1,1,2,2,2}}
%e A320804            {{1,2,3}}  {{1,2,3,3}}    {{1,2,2,2,2}}
%e A320804                       {{1,2,3,4}}    {{1,2,2,3,3}}
%e A320804                       {{1,2},{1,2}}  {{1,2,3,3,3}}
%e A320804                       {{1,2},{3,4}}  {{1,2,3,4,4}}
%e A320804                       {{1,3},{2,3}}  {{1,2,3,4,5}}
%e A320804                                      {{1,2},{1,2,2}}
%e A320804                                      {{1,2},{2,3,3}}
%e A320804                                      {{1,2},{3,4,4}}
%e A320804                                      {{1,2},{3,4,5}}
%e A320804                                      {{1,3},{2,3,3}}
%e A320804                                      {{1,4},{2,3,4}}
%e A320804                                      {{2,3},{1,2,3}}
%o A320804 (PARI)
%o A320804 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A320804 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 A320804 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 A320804 S(q, t, k)={Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t))  + O(x*x^k), -k)}
%o A320804 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)) - sum(t=1, n, S(q, t, n)/t) )), n)); s/n!)} \\ _Andrew Howroyd_, Jan 16 2023
%Y A320804 Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813.
%K A320804 nonn
%O A320804 0,4
%A A320804 _Gus Wiseman_, Nov 06 2018
%E A320804 Terms a(11) and beyond from _Andrew Howroyd_, Jan 16 2023