A321155 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with density -1 <= k < n-2.

1, 2, 1, 3, 2, 1, 6, 6, 4, 1, 10, 14, 11, 4, 1, 22, 38, 38, 20, 6, 1, 42, 94, 111, 72, 28, 6, 1, 94, 250, 348, 278, 138, 42, 8, 1, 203, 648, 1044, 992, 596, 226, 56, 8, 1, 470, 1728, 3192, 3538, 2536, 1192, 370, 76, 10, 1

Offset: 1

Author

Gus Wiseman, Oct 29 2018

Keywords

• nonn
• tabl

Comments

The density of a multiset partition of weight n with e parts and v vertices is n - e - v. The weight of a multiset partition is the sum of sizes of its parts.

Examples

			Triangle begins:
    1
    2    1
    3    2    1
    6    6    4    1
   10   14   11    4    1
   22   38   38   20    6    1
   42   94  111   72   28    6    1
   94  250  348  278  138   42    8    1
  203  648 1044  992  596  226   56    8    1
  470 1728 3192 3538 2536 1192  370   76   10    1
Non-isomorphic representatives of the connected multiset partitions counted in row 5:
{1,2,3,4,5}         {1,2,3,4,4}       {1,2,2,3,3}     {1,1,2,2,2}   {1,1,1,1,1}
{1,4},{2,3,4}       {1,2},{2,3,3}     {1,2,3,3,3}     {1,2,2,2,2}
{4},{1,2,3,4}       {1,3},{2,3,3}     {1,1},{1,2,2}   {1},{1,1,1,1}
{2},{1,3},{2,3}     {2},{1,2,3,3}     {1},{1,2,2,2}   {1,1},{1,1,1}
{2},{3},{1,2,3}     {2,3},{1,2,3}     {1,2},{1,2,2}
{3},{1,3},{2,3}     {3},{1,2,3,3}     {1,2},{2,2,2}
{3},{3},{1,2,3}     {3,3},{1,2,3}     {2},{1,1,2,2}
{1},{2},{2},{1,2}   {1},{1},{1,2,2}   {2},{1,2,2,2}
{2},{2},{2},{1,2}   {1},{1,2},{2,2}   {2,2},{1,2,2}
{1},{1},{1},{1},{1} {1},{2},{1,2,2}   {1},{1},{1,1,1}
                    {2},{1,2},{1,2}   {1},{1,1},{1,1}
                    {2},{1,2},{2,2}
                    {2},{2},{1,2,2}
                    {1},{1},{1},{1,1}
		

Crossrefs

First column is A125702. Row sums are A007718.

Cf. A007716, A030019, A052888, A056156, A134954, A147878, A317631, A319557, A320921.