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Statement

arXiv v5, pp. 2--3 (journal p. 415): "Theorem 1.2. Let $n\ge1$ be an integer. Then

$$(c_1+o(1))n^2\le\max_{a\in\mathrm{Sym}([n])}|S(a)|\le(c_2+o(1))n^2 \tag{6}$$

where $c_1=\frac32-\frac2{\sqrt e}=0.286\ldots$ and $c_2=\frac14+\frac\pi{16}=0.446\ldots$." The paper adds: "It would be surprising if either of the constants $c_1,c_2$ in Theorem 1.2 was optimal", and poses Question 2 (p. 3; journal p. 416): "Does there exist a constant $c>0$ such that for all $n\ge1$ we have $\max_{a\in\mathrm{Sym}([n])}|S(a)|=(c+o(1))n^2$, (7) and if so, what is the value of $c$?" Here $o(1)$ tends to $0$ as $n\to\infty$; the trivial upper bound is $|S(a)|\le\binom{n+1}2$ (p. 1).

Source. Jakub Konieczny, On consecutive sums in permutations, arXiv:1504.07156v5 (27 August 2021), pp. 2--3; J. Combinatorics 12 (2021), no. 3, 413--477, pp. 415--416. The wording is identical in both editions. Library home: konieczny_2015_consecutive_sums_permutations.

Read depth. Claims checked: the statement and Question 2 were read clause by clause in the text layer of the arXiv v5 and on the journal pages. The proofs (Sections 4 and 5) were not read; nothing here is independently reviewed.

Proof pointer

"The upper and lower bound in (6) are proved in Sections 4 and 5 respectively" (p. 3); the print names the bounds in reverse order, since Section 4 ("Lower bound", pp. 25--30) proves the lower bound and Section 5 ("Upper bound", pp. 30--42) the upper. The lower bound comes from a randomized variant of the construction of Proposition 1.1; the upper bound from an optimization argument over a functional $\Lambda$ on a class of functions, whose maximizer $f_*(x)=2x$ for $x\le1/2$, $2(1-x)$ for $x\ge1/2$ gives $\Lambda(f_*)=\pi/16$ (Corollary 5.11 and the proof of Proposition 5.4, p. 42), "perhaps the most novel contribution in this paper" (p. 2).

Dependencies

Proposition 1.1's construction for the lower bound; the analytic optimization of Section 5 for the upper bound.

Bears on

• Problem 34: the site's $f(n)=\max_{\pi\in S_n}S(\pi)$ satisfies $(0.286\cdots+o(1))n^2\le f(n)\le(0.446\cdots+o(1))n^2$, the bounds the site's commentary quotes; the constant question is Question 2. The thread (19 October 2025) notes, as the paper does (p. 25), that the lower constant is close to the random-permutation constant $(1+e^{-2})/4=0.283\ldots$ of Theorem 1.3.