In question 316836, I introduced the new sum

$$ S (n) = sum_ { pi in S_n} e ^ {2 pi i sum_ {k = 1} ^ nk pi (k) / n} = text {by}[e^{2pi i jk/n}]_ {1 le j, k le n} $$

and proved that $ S (2n) = 0 $ for all $ n = 1,2,3, ldots $. I also raised a conjecture about $ S (2n + 1) $ which was confirmed by Noam D. Elkies and Gjergji Zaimi in the two responses there.

I have computed $ S (2n + 1) $ for all $ n = 0.1, ldots, $ 8 and I found that

begin {gather} S (1) = 1, S (3) = – 3, S (5) = – 5, S (7) = – 105, S (9) = 81, \ S (11) = 6765, S (13) = 175747, S (15) = 30375, S (17) = 25219857. End {meet}

In view of these data, I pose the following new conjecture.

**Guess**. $ S (2n + 1)> 0 $ for all $ n = 4.5, ldots $. Further, $ S (n) ge n ^ 2 $ for all $ n = 9,11,13, ldots $.

Any ideas to solve this conjecture? Your comments are welcome!