Partitions of multisets

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sequences from the OEIS.

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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

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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)

Multisets by integer partition

## Right columns

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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

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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

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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

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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

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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

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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

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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...