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Hegyvári proves, as Theorem 1 of his 1986 paper, that the largest number $f(n)$ of integers $a_1,\ldots,a_k$ with $1\le a_i\le n$, not required to be increasing, whose consecutive sums $\sum_{u\le i\le v}a_i$ are all distinct satisfies $(1/3+o(1))n\le f(n)\le(2/3+o(1))n$. The lower bound is a sequence of length about $n/3$ whose partial sums form a Sidon sequence, so that no two consecutive sums coincide.

The paper concerns sequences and says nothing about permutations or about $S(\pi)$. The step to Problem 34 is short and is not the paper's: a sequence of $k$ distinct integers in $[1,n]$ with all consecutive sums distinct is an initial segment of some permutation $\pi\in S_n$, and the $\binom{k+1}2$ consecutive sums inside that segment are already distinct, so $S(\pi)\ge\binom{f(n)+1}2\ge(1/18+o(1))n^2$, which is not $o(n^2)$. Konieczny makes this deduction in Section 1.5 of his paper (library home Konieczny 2015), and the site credits Hegyvári with the first counterexample on this basis; Konieczny's own explicit permutation with $S(\pi)\ge n^2/4$ has its own page.

The paper's library home is Hegyvári 1986, with a compiled page for Theorem 1; the statement was checked and the proof followed for structure only, and nothing here is independently reviewed.

Depends on. Konieczny 2015, whose Section 1.5 makes the deduction from Hegyvári's sequence to a permutation with $(1/18+o(1))n^2$ distinct consecutive sums.

Formalization. The Lean file in Boris Alexeev's lean-proofs repository that the formal-conjectures statement file points to names Hegyvári and Konieczny as its informal authors, but what it proves is Konieczny's permutation $1,n,2,n-1,\ldots$, so the link is recorded on Konieczny's page.

Acceptance. The paper is refereed: N. Hegyvári, On consecutive sums in sequences, Acta Math. Hungar. 48, no. 1--2 (March 1986), 193--200, received 2 October 1984. The site's curator, Thomas F. Bloom, marks Problem 34 disproved and credits this paper with the first counterexample. The page is dated by the first day of the issue month, since the paper's first posting carries no finer date.