Permutations by total distance

[franktaw at netscape.net]

I was looking at http://www.research.att.com/~njas/sequences/A072949, 
which led me to search for and find 
http://www.research.att.com/~njas/sequences/A062869.  (There should 
really be a cross reference.)

A062869 is a table with T(n,k) = the number of permutations of n where 
2k = the total distance sum_i abs(i-p(i)).  (Equivalently, k = sum_i 
max(i-p(i),0).)

A062869 contains two equivalent conjectures; these are obviously 
correct.

A072949 also contains a conjecture (in question form).  Looking at 
A062869, I see a more general conjecture: T(n,k) is even whenever k >= 
n.  I don't see any start at a proof of this; but maybe somebody else 
can.  (Or compute some more values and show it's wrong.)

Franklin T. Adams-Watters

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