Partitions of multisets

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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
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[edit]

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[edit]

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[edit]

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...