A370646 Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.

1, 1, 2, 4, 10, 23, 62, 165, 475, 1400, 4334

Offset: 0

Author

Gus Wiseman, Mar 12 2024

Keywords

• nonn
• more

Comments

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements.

Examples

			The multiset partition {{3},{1,3},{2,3}} has unique choice (3,1,2) so is counted under a(5).
Representatives of the a(1) = 1 through a(5) = 23 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}
       {1}{2}  {1}{22}    {1}{122}      {11}{122}
               {2}{12}    {11}{22}      {1}{1222}
               {1}{2}{3}  {12}{12}      {11}{222}
                          {1}{222}      {12}{122}
                          {12}{22}      {1}{2222}
                          {2}{122}      {12}{222}
                          {1}{2}{33}    {2}{1122}
                          {1}{3}{23}    {2}{1222}
                          {1}{2}{3}{4}  {22}{122}
                                        {1}{2}{233}
                                        {1}{22}{33}
                                        {1}{23}{23}
                                        {1}{2}{333}
                                        {1}{23}{33}
                                        {1}{3}{233}
                                        {2}{12}{33}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {1}{2}{3}{44}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}
		

Crossrefs

For existence we have A368098, complement A368097.

Multisets of this type are ranked by A368101, see also A368100, A355529.

Subsets of this type are counted by A370584, see also A370582, A370583.

Maximal sets of this type are counted by A370585.

Partitions of this type are counted by A370594, see also A370592, A370593.

Subsets of this type are also counted by A370638, see also A370636, A370637.

Factorizations of this type are A370645, see also A368414, A368413.

Set-systems of this type are A370818, see also A367902, A367903.

A000110 counts set partitions, non-isomorphic A000041.

A001055 counts factorizations, strict A045778.

A007716 counts non-isomorphic multiset partitions, connected A007718.

Cf. A000612, A055621, A283877, A300913, A302545, A316983, A319616, A330223, A368095, A368412, A368422.