login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A320802 Number of non-isomorphic aperiodic multiset partitions of weight n whose dual is also an aperiodic multiset partition. 4
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 (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 non-isomorphic 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

Table of n, a(n) for n=0..25.

FORMULA

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

EXAMPLE

Non-isomorphic 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, A320800-A320810.

Sequence in context: A289595 A298189 A053956 * A052543 A026638 A307401

Adjacent sequences:  A320799 A320800 A320801 * A320803 A320804 A320805

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 06 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 28 03:49 EST 2020. Contains 331317 sequences. (Running on oeis4.)