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.

This article references sequences from the OEIS.

	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

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...