Re: Re: generating submultisets with repeated elements

• To: mathgroup at smc.vnet.net
• Subject: [mg103827] Re: [mg103806] Re: generating submultisets with repeated elements
• From: David Bevan <david.bevan at pb.com>
• Date: Thu, 8 Oct 2009 07:50:51 -0400 (EDT)
• References: <ha4r9k\$d0h\$1@smc.vnet.net> <200910071101.HAA00387@smc.vnet.net>

```That's an interesting bijection I wasn't aware of. Thanks.

David %^>

> -----Original Message-----
> From: monochrome [mailto:bayard.webb at gmail.com]
> Sent: 7 October 2009 12:02
> To: mathgroup at smc.vnet.net
> Subject: [mg103806] Re: generating submultisets with repeated elements
>
> 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}}
>

```

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