Re: Re: Re: generating submultisets with

• To: mathgroup at smc.vnet.net
• Subject: [mg103848] Re: [mg103827] Re: [mg103806] Re: generating submultisets with
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Fri, 9 Oct 2009 07:16:06 -0400 (EDT)
• References: <ha4r9k\$d0h\$1@smc.vnet.net> <200910071101.HAA00387@smc.vnet.net>

```g is an improvement over f, I think:

<< "Combinatorica`";

Clear[f, g, test1, test2]
f[set_] := Table[set[[i]] - (i - 1), {i, Length[set]}]
g[set_] := set - Range[0, Length@set - 1]
test1[n_, k_] := With[{set = Range[n + k - 1]},
f /@ KSubsets[set, k]]
test2[n_, k_] := With[{set = Range[n + k - 1]},
g /@ KSubsets[set, k]]

n = 15; k = 10;
Timing@Length@test1[n, k]
Timing@Length@test2[n, k]
Binomial[n + k - 1, k]

{32.9105, 1961256}

{16.3832, 1961256}

1961256

Bobby

On Thu, 08 Oct 2009 06:50:51 -0500, David Bevan <david.bevan at pb.com> wrote:

> 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}}
>>
>

--
DrMajorBob at yahoo.com

```

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