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

1, 1, 3, 7, 21, 56, 174, 517, 1664, 5383, 18199, 62745, 223390, 813425, 3040181, 11620969, 45446484, 181537904, 740369798, 3079779662, 13059203150, 56406416004, 248027678362, 1109626606188, 5048119061134, 23342088591797, 109648937760252, 523036690273237

Offset

0,3

Comments

A multiset is aperiodic if its multiplicities are relatively prime.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Example

Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions with aperiodic parts:
  {{1}}  {{1,2}}    {{1,2,2}}      {{1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,3,3}}
         {{1},{2}}  {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{1,2,2}}
                    {{1},{1},{1}}  {{1,2},{1,2}}
                    {{1},{2},{2}}  {{1},{2,3,3}}
                    {{1},{2},{3}}  {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Prog

(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
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

Crossrefs

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A303709, A303710, A320800-A320810.

Sequence in context: A151267 A319558 A307251 * A262184 A091489 A374721

Adjacent sequences: A320800 A320801 A320802 * A320804 A320805 A320806

Keyword

nonn

Author

Gus Wiseman, Nov 06 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

Status

approved