Max/Min Sums From Permutations

[Leroy Quet]

Several people have written me off-list to send me the extended list. thanks.
And, yes, as noted by them, I did make an error for n=5. (It should be 46 instead.)

Leroy

Joshua Zucker has asked me to pass on his message to me:

>Hi Leroy,
>This email address isn't subscribed to seqfan so could you pass this
>along for me?
>
>Thanks,
>--Joshua
>
>By brute-force search on a computer, I get
>
>0 2 9 23 46 80 127 189 268
>
>which makes me wonder how you got 50!
>By hand, I think the max should be from
>1 3 5 4 2
>which gives me 46, indeed -- how do you beat it?
>
>For the minimums, I get
>0 2 5 12 22 38 59 88 124
>
>but by being intelligent I'm sure someone can get a lot more terms
>pretty easily.
>
>The obvious conjecture is that the min is found with
>n 1 n-2 3 n-4 ... 2 n-1
>and the max is found by putting n in the middle,
>n-1 and n-2 next to it,
>then n-3 next to n-1 and n-4 next to n-2,
>and so on ...
>
>If those conjectures are right, then the mins are
>0  2 5 12 22 38 59 88 124 170 225 292 370 462 567 688 824 978 1149
>1340 1550 1782 2035 2312 2612 2938 3289 3668 4074 4510 4975 5472 6000
>6562 7157 7788 8454 9158 9899 10680 11500 12362 13265 14212 15202
>16238 17319 18448 19624 20850
>(is that enough terms?)
>and the maxs are
>0 2 9 23 46 80 127 189 268 366 485 627 794 988 1211 1465 1752 2074
>2433 2831 3270 3752 4279 4853 5476 6150 6877 7659 8498 9396 10355
>11377 12464 13618 14841 16135 17502 18944 20463 22061 23740 25502
>27349 29283 31306 33420 35627 37929 40328 42826
>
>and since the conjectures seem pretty obviously true to me,
>and they match my brute-force data that came from trying every
>permutation up to 9 elements,
>I'm feeling pretty good.
>
>--Joshua Zucker
>
>On Fri, 28 Jan 05 14:06:20 -0700, Leroy Quet <qq-quet at mindspring.com> 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
>>
>>