# [seqfan] Re: Energy-limited permutations?

Olivier Gerard olivier.gerard at gmail.com
Mon Jul 5 18:04:48 CEST 2010

```Hello Ron,

You might want to compare these sequences to

http://www.research.att.com/~njas/sequences/A033638

and little variations of it.

See also

http://www.research.att.com/~njas/sequences/A062870

and related ones.

You should be able to make one or several triangles.

Olivier

On Mon, Jul 5, 2010 at 16:34, Ron Hardin <rhhardin at att.net> wrote:

> Quick computation
>
> %S A000001
> 1,1,3,9,27,87,350,1675,7848,36711,196510,1177139,6908609,40106433,
> %N A000001 Number of permutations of 1..n with sum (i-p(i))^2 <= n*(n-1)/2
>
> %S A000002
> 1,2,5,13,41,151,672,3047,13796,70702,406241,2306597,13050503,81009240,
> %N A000002 Number of permutations of 1..n with sum (i-p(i))^2 <= (n+1)*n/2
>
> %S A000003
> 0,1,3,5,21,87,350,1399,6689,36711,196510,1037354,6147299,40106433,
> %N A000003 Number of permutations of 1..n with sum (i-p(i))^2 < n*(n-1)/2
>
> %S A000004
> 0,2,3,11,41,151,594,2667,13796,70702,363135,2075703,13050503,81009240,
> %N A000004 Number of permutations of 1..n with sum (i-p(i))^2 < (n+1)*n/2
>
> Take the energy of a permuation as the sum of the squares of the
> displacements i-p(i).
> What to compare this energy to, to make the count a sequence?  Half the sum
> of the elements 1..n, or elements 0..(n-1), is conveniently integer.
>
> And then you can use <= or < that energy.
>
> Any others that are not too arbitrary?
>
>  rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>

```

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