A330058 Number of non-isomorphic multiset partitions of weight n with at least one endpoint.

0, 1, 2, 7, 21, 68, 214, 706, 2335, 7968, 27661, 98366, 357212, 1326169, 5027377, 19459252, 76850284, 309531069, 1270740646, 5314727630, 22633477157, 98096319485, 432490992805, 1938762984374, 8832924638252, 40882143931620, 192148753444380, 916747097916418

Offset

0,3

Comments

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

An endpoint is a vertex appearing only once (degree 1).

Also the number of non-isomorphic multiset partitions of weight n with at least one singleton.

Links

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

Wikipedia, Degree (graph theory)

Formula

a(n) = A007716(n) - A302545(n). - Andrew Howroyd, Jan 15 2023

Example

Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
  {1}  {12}    {122}      {1222}
       {1}{2}  {123}      {1233}
               {1}{22}    {1234}
               {1}{23}    {1}{222}
               {2}{12}    {12}{22}
               {1}{2}{2}  {1}{233}
               {1}{2}{3}  {12}{33}
                          {1}{234}
                          {12}{34}
                          {13}{23}
                          {2}{122}
                          {3}{123}
                          {1}{1}{23}
                          {1}{2}{22}
                          {1}{2}{33}
                          {1}{2}{34}
                          {1}{3}{23}
                          {2}{2}{12}
                          {1}{2}{2}{2}
                          {1}{2}{3}{3}
                          {1}{2}{3}{4}

Crossrefs

The case of set-systems is A330053 (singletons) or A330052 (endpoints).

The complement is counted by A302545.

Cf. A007716, A283877, A306005, A330054, A330055, A330059.

Sequence in context: A344500 A186240 A274203 * A220726 A347302 A127540

Adjacent sequences: A330055 A330056 A330057 * A330059 A330060 A330061

Keyword

nonn

Author

Gus Wiseman, Nov 30 2019

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 15 2023

Status

approved