# Partitions of multisets

Multisets have partitions just like normal sets. The following table shows how many of them a multiset corresponding to a particular integer partition has. Lists are linked from the table.
| | 0 | 1 2 | 2 3 | 3 2,2 | 4 4 | 5 3,2 | 6 5 | 7 2,2,2 | 8 4,2 | 9 3,3 | 10 6 | 11 3,2,2 | 12 5,2 | 13 4,3 | 14 7 | 15 2,2,2,2 | 16 4,2,2 | 17 3,3,2 | 18 6,2 | 19 5,3 | 20 4,4 | 21 8 | Σ |
| 0 | 1 | | | | | | | | | | | | | | | | | | | | | | 1 |
| 1 | 1 | | | | | | | | | | | | | | | | | | | | | | 1 |
| 2 | 2 | 2 | | | | | | | | | | | | | | | | | | | | | 4 |
| 3 | 5 | 4 | 3 | | | | | | | | | | | | | | | | | | | | 12 |
| 4 | 15 | 11 | 7 | 9 | 5 | | | | | | | | | | | | | | | | | | 47 |
| 5 | 52 | 36 | 21 | 26 | 12 | 16 | 7 | | | | | | | | | | | | | | | | 170 |
| 6 | 203 | 135 | 74 | 92 | 38 | 52 | 19 | 66 | 29 | 31 | 11 | | | | | | | | | | | | 750 |
| 7 | 877 | 566 | 296 | 371 | 141 | 198 | 64 | 249 | 98 | 109 | 30 | 137 | 47 | 57 | 15 | | | | | | | | 3255 |
| 8 | 4140 | 2610 | 1315 | 1663 | 592 | 850 | 250 | 1075 | 392 | 444 | 105 | 560 | 171 | 212 | 45 | 712 | 269 | 300 | 77 | 97 | 109 | 22 | 16010 |
triangle: A249620, columns correspond to integer partitions (A194602), row sums: A035310 col 0: A000110 (Bell), col 1: A035098 (near-Bell), col 2: A169587, col 4: A169588 end−1: A091437, end: A000041 (partition numbers)

## Right columns

The right columns give a reflection of the triangle A126442. Columns: A000041(1...) − 1 = 0, 1, 2, 4, 6, 10, 14, 21...
| | 0 | 1 2 | 2 3 | 4 4 | 6 5 | 10 6 | 14 7 | 21 8 | Σ |
| 1 | 1 | | | | | | | | 1 |
| 2 | 2 | 2 | | | | | | | 4 |
| 3 | 5 | 4 | 3 | | | | | | 12 |
| 4 | 15 | 11 | 7 | 5 | | | | | 38 |
| 5 | 52 | 36 | 21 | 12 | 7 | | | | 128 |
| 6 | 203 | 135 | 74 | 38 | 19 | 11 | | | 480 |
| 7 | 877 | 566 | 296 | 141 | 64 | 30 | 15 | | 1989 |
| 8 | 4140 | 2610 | 1315 | 592 | 250 | 105 | 45 | 22 | 9079 |
Row sums: 1, 4, 12, 38, 128, 480, 1989, 9079... Main diagonal: 1, 4, 21, 141... Diagonals on the right: A000041 = 1, 2, 3, 5, 7, 11, 15, 22... A000070 = 2, 4, 7, 12, 19, 30, 45... A082775 = 5, 11, 21, 38, 64, 105...
Another triangle is mentioned in A126442 as the second of a series. I guess that 3, 5, 8, 12... is supposed to be the sequence of integer partitions with two non-one addends, one of them being 2. That would be the columns: A248374 = 3, 5, 8, 12, 18, 25, 36, 49, 67, 90, 121, 158...
| | 3 2,2 | 5 3,2 | 8 4,2 | 12 5,2 | 18 6,2 | Σ |
| 4 | 9 | | | | | 9 |
| 5 | 26 | 16 | | | | 42 |
| 6 | 92 | 52 | 29 | | | 173 |
| 7 | 371 | 198 | 98 | 47 | | 714 |
| 8 | 1663 | 850 | 392 | 171 | 77 | 3153 |
Row sums: 9, 42, 173, 714, 3153... Diagonals on the right: A000291 = 9,16,29,47,77... A002763 = 26,52,98,171...

## Left columns

Columns: A000041(1...) = 1, 2, 3, 5, 7, 11, 12...
| | 1 2 | 2 3 | 3 2,2 | 5 3,2 | 7 2,2,2 | 11 3,2,2 | 15 2,2,2,2 | Σ |
| 2 | 2 | | | | | | | 2 |
| 3 | 4 | 3 | | | | | | 7 |
| 4 | 11 | 7 | 9 | | | | | 27 |
| 5 | 36 | 21 | 26 | 16 | | | | 99 |
| 6 | 135 | 74 | 92 | 52 | 66 | | | 419 |
| 7 | 566 | 296 | 371 | 198 | 249 | 137 | | 1817 |
| 8 | 2610 | 1315 | 1663 | 850 | 1075 | 560 | 712 | 8785 |
Row sums: 2, 7, 27, 99, 419, 1817, 8785... Main diagonal: 2, 7, 92, 850...

## Second from right columns

Columns: A000041(4...) − 2 = 3, 5, 9, 13, 20...
| | 3 2,2 | 5 3,2 | 9 3,3 | 13 4,3 | 20 4,4 | Σ |
| 4 | 9 | | | | | 9 |
| 5 | 26 | 16 | | | | 42 |
| 6 | 92 | 52 | 31 | | | 175 |
| 7 | 371 | 198 | 109 | 57 | | 735 |
| 8 | 1663 | 850 | 444 | 212 | 109 | 3278 |
Row sums: 9,42,175,735,3278... Main diagonal: 9, 52, 444... Diagonal on the right: A091437 = 9, 16, 31, 57, 109...