Max/Min Sums From Permutations

[Emeric Deutsch]

A related question is considering the circularly extended sum 
 T = p(1)*p(2) + p(2)*p(3) + p(3)*p(4) +... + p(n-1)*p(n) + p(n)*p(1).
In this case, for a given n, the range of the values is much smaller.
For n=2 we get {4}  ("we" includes Maple);
for n=3 we get {11};
for n=4 we get {21,24,25}, each occurring 8 times;
for n=5 we get {37,38,40,42,43,45,47,48} occurring 10,20,10,20,20,10,20,10
times, respectively.
for n=6 we get {58,59,61,...,81,82}
Good luck!
Emeric 


On Fri, 28 Jan 2005, Leroy Quet wrote:

> Say we have a permutation of <1,2,3,...,n>,
> <p(1),p(2),p(3),...,p(n)>.
> 
> Next, take the sum S of products formed from the permutation,
> S = p(1)*p(2) + p(2)*p(3) + p(3)*p(4) +... + p(n-1)*p(n).
> 
> (So, every term besides p(1) and p(n) appear in 2 products, p(1) and p(n) 
> appear once each.)
> 
> For a given n, what is the maximum possible and minimum possible S?
> 
> 
> I get by hand, and not necessarily correctly, that the sequence of 
> maximums begins
> 1(or 0), 2, 9, 23, 50,...
> 
> I get the minimums (again, most likely incorrectly) as
> 1(or 0), 2, 5, 12, 22,...
> 
> Could someone compute the real sequences and submit them to the OEIS?
> 
> thanks,
> Leroy Quet
>