Re: generating submultisets with repeated elements

• To: mathgroup at smc.vnet.net
• Subject: [mg103806] Re: generating submultisets with repeated elements
• From: monochrome <bayard.webb at gmail.com>
• Date: Wed, 7 Oct 2009 07:01:40 -0400 (EDT)
• References: <ha4r9k\$d0h\$1@smc.vnet.net>

```I did a little research and found out that there are Choose(n+k-1, k)
multisets of size k from a set of size n. This made me think that
there should be a mapping from the k-subsets of n+k-1 to the k-
multisets of n. A few quick examples led me to the following function:

f[set_] := Table[set[[i]] - (i - 1), {i, Length[set]}]

This allows the following construction using the KSubsets function
from Combinatorica:

<< "Combinatorica`";
n = 6;
k = 3;
set = Range[n + k - 1];
Map[f, KSubsets[set, k]]

===OUTPUT===
{{1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {1, 1, 4}, {1, 1, 5}, {1, 1, 6}, {1,
2, 2}, {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 3, 3}, {1,
3, 4}, {1, 3, 5}, {1, 3, 6}, {1, 4, 4}, {1, 4, 5}, {1, 4, 6}, {1, 5,
5}, {1, 5, 6}, {1, 6, 6}, {2, 2, 2}, {2, 2, 3}, {2, 2, 4}, {2, 2,
5}, {2, 2, 6}, {2, 3, 3}, {2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 4,
4}, {2, 4, 5}, {2, 4, 6}, {2, 5, 5}, {2, 5, 6}, {2, 6, 6}, {3, 3,
3}, {3, 3, 4}, {3, 3, 5}, {3, 3, 6}, {3, 4, 4}, {3, 4, 5}, {3, 4,
6}, {3, 5, 5}, {3, 5, 6}, {3, 6, 6}, {4, 4, 4}, {4, 4, 5}, {4, 4,
6}, {4, 5, 5}, {4, 5, 6}, {4, 6, 6}, {5, 5, 5}, {5, 5, 6}, {5, 6,
6}, {6, 6, 6}}

On Oct 2, 5:22 am, David Bevan <david.be... at pb.com> wrote:
> I'm new to Mathematica, so if I've missed something obvious, my apologies.
>
> I want a function to generate a list of "submultisets" with up to k elements of a set s, allowing elements from s to be repeated.
>
> The following works, but is very inefficient since each multiset is generated multiple times and then sorted and then repeats deleted:
>
> coinSets[s_,k_]:=DeleteDuplicates[Sort/@Flatten[Tuples[s,#]&/@Range[k],1]]
>
> coinSets[{1,3,4},3]
>
> {{1},{3},{4},{1,1},{1,3},{1,4},{3,3},{3,4},{4,4},{1,1,1},{1,1,3},{1,1,4},{1 ,3,3},{1,3,4},{1,4,4},{3,3,3},{3,3,4},{3,4,4},{4,4,4}}
>
> I assumed there would be a suitable function in the Combinatorica package, but I can't see anything -- which would be a bit odd for a combinatorial package. What have I missed?
>
> Do I need to write my own (perhaps by looking at how KSubsets is implemented) or is there some easy way of generating these multisets?
>
> Thanks.
>
> David %^>

```

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