---

Statement

arXiv v5, p. 3 (journal p. 416): "Theorem 1.3. Let $n\ge1$ be an integer and let $a$ be a permutation of $[n]$ chosen uniformly at random. Put $c=\frac{1+e^{-2}}4=0.283\ldots$. Then, for each $\delta>0$,

$$\mathbb P\bigl(\bigl||S(a)|-cn^2\bigr|>\delta n^2\bigr)=o(1). \tag{8}$$

"

The paper restates this as convergence of $|S(a)|/n^2$ to $c$ in probability and, since $|S(a)|/n^2$ is bounded, deduces $\mathbb E(|S(a)|)=(c+o(1))n^2$ (display (9)). Its abstract (arXiv v5, p. 1; the journal print has none) calls this the answer to "an old question of Erdős and Harzheim".

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

Read depth. Claims checked: the statement and displays (8)--(9) were read clause by clause in the text layer of the arXiv v5 and on the journal page. The proof (Sections 2--3) was not read; nothing here is independently reviewed.

Proof pointer

"The proof of Theorem 1.3 is carried out in Section 2, dealing with the expected value of $|S(a)|$, and Section 3, dealing with the second moment $\mathbb E|S(a)|^2$" (p. 3): first and second moment computations with exponential-sum notation, as the source card summarizes.

Dependencies

None named in the statement; the proof is probabilistic counting that rests on Hoeffding's inequality, mostly in its form for sampling without replacement (Theorem 2.2, p. 4, cited to Hoeffding 1963), and on Lemma 2.8(i) (p. 12), the symmetric unimodal shape of the distribution of a sum of an even number of independent uniform variables on $[n]$, whose proof cites Dharmadhikari and Joag-Dev 1988 (Thm. 1.6).

Bears on

• Problem 34: the site's "for a random permutation we have $S(\pi)\sim\frac{1+e^{-2}}4n^2$", the sense in which the site calls the $o(n^2)$ conjecture "extremely false": the typical permutation, not only an extremal one, has order $n^2$ distinct consecutive sums.