[seqfan] GF or recursion requested

[Meeussen Wouter (bkarnd)]

dear all,

take a set [y1,y2,...,yn] of integers, and look at them as vertical increments on a x-y grid, defining the path
0,y1, y1+y2, y1+y2+y3,... , Sum(i); yi
then it makes sense to differentiate several paths generated by permuting the yi on the basis of 'total weight below the path';
for the example above, that would be n*y1+ (n-1)*y2 +...+  2*y_sub_(n-1) + yn.
Now I was wondering how many distinct weights there are for the set [y1,y2,...,yn] going through all permutations of  n.

I found 1, 2, 4, 11, 21, 36, 57, 85, 121, 166 known as 
A126972   Number of distinct values taken by the entropy for permutations of [1..n], where the entropy of a permutation pi is sum(k=1..n, (pi(k)-k)^2).   formula   Binomial[n+1,3] + If[  n=!=3, 1, 0]

Is it evident that these counts match?

Now, looking at the frequency with which each distinct weight occurs, we get a symmetric table
{1},
{1,1},
{1,2,2,1},
{1,3,1,4,2,2,2,4,1,3,1},
{1,4,3,6,7,6,4,10,6,10,6,10,6,10,4,6,7,6,3,4,1}
row lengths as A126972 and row sums = n! as it should be.
largest value in each row = {1, 1, 2, 4, 10, 42, 184, 1066, 6697, 50066}.

for n between 8 and 10, the above frequencies seem to avoid accidental repetitions, and each frequency then occors exactly twice.
(except for rows of odd length which have a single central element).

Put differently, the count of distinct values among the above frequencies becomes
{1, 1, 2, 4, 6, 16, 26, 43, 61, 83}
approaching Table[it=Binomial[n+1,3]+If[n=!=3,1,0];If[EvenQ[it],it/2,it/2+1/2],{n,12}]
{1, 1, 2, 6, 11, 18, 29, 43, 61, 83, 111, 144}
For low n we suffer accidental repetitions (Strong Law of Small Integers?).

Is there any ground to believe that no accidental repetitions occur for n>10?
I assume not. Any ideas?

Wouter.


===============================
This email is confidential and intended solely for the use of the individual to whom it is addressed. 
If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited.
You are explicitly requested to notify the sender of this email that the intended recipient was not reached.