Permutations to scalars and back again.

[James A. Donald]

On 9/12/2016 8:01 PM, Georgi Guninski wrote:
> On Mon, Sep 12, 2016 at 07:50:50PM +1000, James A. Donald wrote:
>> To restate the problem:  Find a mapping between integers and injective
>> functions from N to X up to a permutation of N.
>>
>> In this case, find a mapping between integers and an injective functions
>> from 18 to 36.
>
> Sage (open source, sagemath.org) can do at least parts of what you
> are asking.
> Not sure I get the question about injective function, but AFAICT
> treating the permutation as nonnegative integer in binary will do.
>
> Example sage session:
>
> sage: l=[0]*2+[1]*2
> sage: pe=Permutations(l)
> sage: pe.cardinality()
> 6
> sage: pe[0]
> [0, 0, 1, 1]
> sage: for p in pe:  print p
> [0, 0, 1, 1]
> #...more
>
>


Found the the solution.

Combinatorial number system.

Suppose we have k cards, any one of which can be white or red, but which 
are otherwise indistinguishable and interchangeable.

Combinatorial number system gives us a one to one mapping between 
integers, and all possible subsets of an n element set.

Now I want a mapping between integers and all possible m element subsets 
of an n element set, but for m approximating n/2 the mapping is dense 
enough to be useful.