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.

triangle:,       columns correspond to integer partitions ,        row sums: col 0: (Bell),   col 1:  (near-Bell) ,   col 2: ,        col 4:         end−1: ,   end:  (partition numbers)





Right columns
The right columns give a reflection of the triangle. Columns: (1...) − 1 = 0, 1, 2, 4, 6, 10, 14, 21...

Row sums: 1, 4, 12, 38, 128, 480, 1989, 9079... Main diagonal: 1, 4, 21, 141... Diagonals on the right: = 1, 2, 3, 5, 7, 11, 15, 22... = 2, 4, 7, 12, 19, 30, 45...  = 5, 11, 21, 38, 64, 105...

Another triangle is mentioned in 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: = 3, 5, 8, 12, 18, 25, 36, 49, 67, 90, 121, 158...

Row sums: 9, 42, 173, 714, 3153... Diagonals on the right: = 9,16,29,47,77... = 26,52,98,171...

Left columns
Columns: (1...) = 1, 2, 3, 5, 7, 11, 12...

Row sums: 2, 7, 27, 99, 419, 1817, 8785... Main diagonal: 2, 7, 92, 850...

Second from right columns
Columns: (4...) − 2 = 3, 5, 9, 13, 20...

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