nt.number theory – A conjectural lower limit for \$ | { sum_ {k = 1} ^ nka_k: a_1, ldots, a_n text {are elements other than} A } | \$

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.