

A320802


Number of nonisomorphic aperiodic multiset partitions of weight n whose dual is also an aperiodic multiset partition.


5



1, 1, 2, 8, 26, 89, 274, 908, 2955, 9926, 34021, 119367, 428612, 1574222, 5914324, 22699632, 88997058, 356058538, 1453059643, 6044132792, 25612530061, 110503625785, 485161109305, 2166488899640, 9835209048655, 45370059225137, 212582814591083, 1011306624492831
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

Also the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns where the multiset of rows and the multiset of columns are both aperiodic, up to row and column permutations.
A multiset is aperiodic if its multiplicities are relatively prime.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
Also the number of nonisomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods, where the period of a multiset is the GCD of its multiplicities.


LINKS

Jinyuan Wang, Table of n, a(n) for n = 0..50


FORMULA

Second Moebius transform of A007716, or Moebius transform of A303546, where the Meobius transform of a sequence b is a(n) = Sum_{dn} mu(d) * b(n/d).


EXAMPLE

Nonisomorphic representatives of the a(1) = 1 through a(4) = 26 multiset partitions:
{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1},{2}} {{1,2,2}} {{1,2,2,2}}
{{1},{1,1}} {{1,2,3,3}}
{{1},{2,2}} {{1},{1,1,1}}
{{1},{2,3}} {{1},{1,2,2}}
{{2},{1,2}} {{1,1},{2,2}}
{{1},{2},{2}} {{1},{2,2,2}}
{{1},{2},{3}} {{1,2},{2,2}}
{{1},{2,3,3}}
{{1,2},{3,3}}
{{1},{2,3,4}}
{{1,3},{2,3}}
{{2},{1,2,2}}
{{3},{1,2,3}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{1},{2,3}}
{{1},{2},{1,2}}
{{1},{2},{2,2}}
{{1},{2},{3,3}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{2},{2},{1,2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}


CROSSREFS

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320800A320810.
Sequence in context: A289595 A298189 A053956 * A052543 A026638 A307401
Adjacent sequences: A320799 A320800 A320801 * A320803 A320804 A320805


KEYWORD

nonn


AUTHOR

Gus Wiseman, Nov 06 2018


EXTENSIONS

a(26)a(27) from Jinyuan Wang, Jun 27 2020


STATUS

approved



