A321283 - OEIS

%I #19 Jan 17 2023 18:21:07

%S 1,1,2,7,21,84,214,895,2607,9591,31134,119313,400950,1574123,5706112,

%T 22572991,86933012,356058243,1427784135,6044132304,25342935667,

%U 110414556330,481712291885,2166488898387,9784077216457,45369658599779,211869746691055,1011161497851296,4871413403219085

%N Number of non-isomorphic multiset partitions of weight n in which the part sizes are relatively prime.

%C Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the row sums are relatively prime.

%C Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is aperiodic, where a multiset is aperiodic if its multiplicities are relatively prime.

%C 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 Andrew Howroyd, <a href="/A321283/b321283.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = A007716(n) - A320810(n). - _Andrew Howroyd_, Jan 17 2023

%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions with relatively prime part-sizes:

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

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

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

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

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

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

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

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

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

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

%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}}

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

%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 Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions with aperiodic multiset union:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%o (PARI) \\ See links in A339645 for combinatorial species functions.

%o seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), 1 + sum(d=1, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A,i*d)*x^(i*d)))) )))} \\ _Andrew Howroyd_, Jan 17 2023

%o (PARI) \\ faster self contained program.

%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, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}

%o a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=vector(n, t, K(q, t, n\t))); s+=permcount(q)*polcoef(sum(d=1, n, moebius(d)*exp(sum(t=1, n\d, sum(i=1, n\(t*d), u[t][i*d]*x^(i*d*t))/t, O(x*x^n)) )), n)); s/n!)} \\ _Andrew Howroyd_, Jan 17 2023

%Y Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A320800-A320810.

%K nonn

%O 0,3

%A _Gus Wiseman_, Nov 06 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 17 2023