# [seqfan] Energy-limited permutations?

Ron Hardin rhhardin at att.net
Mon Jul 5 16:34:04 CEST 2010

```Quick computation

%S A000001 1,1,3,9,27,87,350,1675,7848,36711,196510,1177139,6908609,40106433,
%T A000001 256572630,1785661795
%N A000001 Number of permutations of 1..n with sum (i-p(i))^2 <= n*(n-1)/2
%O A000001 1,3

%S A000002 1,2,5,13,41,151,672,3047,13796,70702,406241,2306597,13050503,81009240,
%T A000002 546463298,3646507024
%N A000002 Number of permutations of 1..n with sum (i-p(i))^2 <= (n+1)*n/2
%O A000002 1,2

%S A000003 0,1,3,5,21,87,350,1399,6689,36711,196510,1037354,6147299,40106433,
%T A000003 256572630,1621282565
%N A000003 Number of permutations of 1..n with sum (i-p(i))^2 < n*(n-1)/2
%O A000003 1,3

%S A000004 0,2,3,11,41,151,594,2667,13796,70702,363135,2075703,13050503,81009240,
%T A000004 499992359,3350516588
%N A000004 Number of permutations of 1..n with sum (i-p(i))^2 < (n+1)*n/2
%O A000004 1,2

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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