Motivated by question 315568 mine, I am interested in the set

$$ S (n): = bigg { sum_ {k = 1} ^ n k pi (k): pi in S_n bigg }. $$

It's easy to see that

$$ S (1) = {1 }, S (2) = {4,5 } text {y} S (3) = {10,11,13,14 }. $$

For the inequality of Cauchy-Schwarz, for any $ pi in S_n $ we have

$$ bigg ( sum_ {k = 1} ^ nk pi (k) bigg) ^ 2 le bigg ( sum_ {k = 1} ^ nk ^ 2 bigg) bigg ( sum_ {k = 1} ^ n pi (k) ^ 2 bigg) $$

and therefore

$$ sum_ {k = 1} ^ nk pi (k) le sum_ {k = 1} ^ nk ^ 2 = frac {n (n + 1) (2n + 1)} 6. $$

If we leave $ sigma (k) = n + 1- pi (k) $ for all $ k = 1, ldots, n $, so $ sigma in S_n $ Y

begin {align} sum_ {k = 1} ^ nk pi (k) = & sum_ {k = 1} ^ nk (n + 1- sigma (k)) = (n + 1) sum_ { k = 1} ^ nk- sum_ {k = 1} ^ nk sigma (k)

\ ge & frac {n (n + 1) ^ 2} 2- frac {n (n + 1) (2n + 1)} 6 = frac {n (n + 1) (n + 2) } 6.

end {align}

A) Yes

$$ S (n) subseteq T (n): = left { frac {n (n + 1) (n + 2)} 6, ldots, frac {n (n + 1) (2n + 1)} 6 right }. $$

My calculation indicates that $ S (n) = T (n) $ when $ n no = 3 $. Note that

$ | T (n) | = n (n ^ 2-1) / 6 + 1 $.

Inspired by the previous analysis, here I present the following new conjecture in additive combinatorics.

**Guess**. Leave $ n $ be a positive integer and leave $ F $ be a field with $ p (F)> n + 1 $, where $ p (F) = p $ If the characteristic of $ F $ Is a cousin $ p $Y $ p (F) = + infty $ If the characteristic of $ F $ is zero Let $ A $ be any finite subset of $ F $ with $ | A | ge n + delta_ {n, 3} $, where $ delta_ {n, 3} $ is $ 1 $ or $ 0 $ according to $ n = 3 $ or not. Then, for the set.

$$ S (A): = bigg { sum_ {k = 1} ^ n ka_k: a_1, ldots, a_n text {are elements other than} A bigg }, $$

we have $$ | S (A) | ge min left {p (F), (| A | -n) frac {n (n + 1)} 2+ frac {n (n ^ 2- 1)} 6 + 1 right }. $$

QUESTION: Is my previous conjecture true?

PD: Soon I will raise another question that extends the current one to the general case.