Max/Min Sums From Permutations

[Robert G. Wilson v]

Leroy,

	Concerning the sequences by Emeric Deutsch and using Mathematica I get:

{{1, 1}, {4, 4}, {11, 11}, {25, 21}, {48, 37}, {82, 58}, {129, 87}, {191,
     123}, {270, 169}, {368, 224}}

	I do not find either sequence in the OEIS.

	The Mathematica coding is:

f[lst_] := Block[{l = Length[lst]}, lst[[l]]*lst[[1]] +
            Sum[ lst[[i]]*lst[[i + 1]], {i, l - 1}]];

g[n_] := Block[{lst = Permutations[ Range[ n]]}, t = (f /@ lst);
          {Max[t], Min[t]}];

Table[ g[n], {n, 10}]

Bob.

Emeric Deutsch wrote:

> 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
>>
> 
>