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

Author

Gus Wiseman, Nov 30 2019

Keywords

• nonn

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.

Examples

			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}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• Wikipedia, Degree (graph theory)

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.

Formula

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

Extensions

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