Algorithm to generate all multiset size-n partitions

Question

I\'ve been trying to figure out a way to generate all distinct size-n partitions of a multiset, but so far have come up empty handed. First let me show what I\'m trying to archi

Answer 1

Here's my pencil and paper algorithm:

Describe the multiset in item quantities, e.g., {(1,2),(2,2)}

f(multiset,result):
  if the multiset is empty:
    return result
  otherwise:
    call f again with each unique distribution of one element added to result and 
    removed from the multiset state


Example:
{(1,2),(2,2),(3,2)} n = 2

11       -> 11 22    -> 11 22 33
            11 2  2  -> 11 23 23
1  1     -> 12 12    -> 12 12 33
            12 1  2  -> 12 13 23


Example:
{(1,2),(2,2),(3,2)} n = 3

11      -> 112 2   -> 112 233
           11  22  -> 113 223
1   1   -> 122 1   -> 122 133
           12  12  -> 123 123

Let's solve the problem commented below by m69 of dealing with potential duplicate distribution:

{A,B,B,C,C,D,D,D,D}

We've reached {A, , }{B, , }{B, , }, have 2 C's to distribute
and we'd like to avoid `ac  bc  b` generated along with `ac  b   bc`.

Because our generation in the level just above is ordered, the series of identical 
counts will be continuous. When a series of identical counts is encountered, make 
the assignment for the whole block of identical counts (rather than each one), 
and partition that contribution in descending parts; for example,

      | identical |
ac     b      b
ac     bc     b     // descending parts [1,0]

Example of longer block:

      |    identical block     |  descending parts
ac     bcccc  b      b      b    // [4,0,0,0] 
ac     bccc   bc     b      b    // [3,1,0,0]
ac     bcc    bcc    b      b    // [2,2,0,0]
...