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

1, 0, 1, 2, 6, 13, 41, 104, 326, 958, 3096, 9958, 33869, 116806, 417741, 1526499, 5732931, 22015642, 86543717, 347495480, 1424832602, 5959123908, 25407212843, 110344848622, 487879651220, 2194697288628, 10039367091586, 46675057440634, 220447539120814

Offset: 0

Author

Gus Wiseman, Nov 06 2018

Keywords

• nonn

Comments

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.

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.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 multiset partitions with aperiodic parts and no singletons:
  {{1,2}}  {{1,2,2}}  {{1,2,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,3,3}}    {{1,2,2,2,2}}
                      {{1,2,3,4}}    {{1,2,2,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,3,3}}
                      {{1,2},{3,4}}  {{1,2,3,4,4}}
                      {{1,3},{2,3}}  {{1,2,3,4,5}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}
		

Links

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

Crossrefs

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813.

Programs

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))}
S(q, t, k)={Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t))  + 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)) - sum(t=1, n, S(q, t, n)/t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2023

Extensions

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