# 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  12 23 32,2 44 53,2 65 72,2,2 84,2 93,3 106 113,2,2 125,2 134,3 147 152,2,2,2 164,2,2 173,3,2 186,2 195,3 204,4 218 1 1 2 2 5 4 3 15 11 7 9 5 52 36 21 26 12 16 7 203 135 74 92 38 52 19 66 29 31 11 877 566 296 371 141 198 64 249 98 109 30 137 47 57 15 4140 2610 1315 1663 592 850 250 1075 392 444 105 560 171 212 45 712 269 300 77 97 109 22

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  12 23 44 65 106 147 218 1 2 2 5 4 3 15 11 7 5 52 36 21 12 7 203 135 74 38 19 11 877 566 296 141 64 30 15 4140 2610 1315 592 250 105 45 22

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...

32,2 53,2 84,2 125,2 186,2 9 26 16 92 52 29 371 198 98 47 1663 850 392 171 77

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...

12 23 32,2 53,2 72,2,2 113,2,2 152,2,2,2 2 4 3 11 7 9 36 21 26 16 135 74 92 52 66 566 296 371 198 249 137 2610 1315 1663 850 1075 560 712

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...

32,2 53,2 93,3 134,3 204,4 9 26 16 92 52 31 371 198 109 57 1663 850 444 212 109

Row sums: 9,42,175,735,3278...
Main diagonal: 9, 52, 444...
Diagonal on the right: A091437 = 9, 16, 31, 57, 109...