Every Interior Multiple of n Is a Latin-Square Ascent Total

Madjid Mirzavaziri and Daniel Yaqubi, Latin Eulerian Numbers, arXiv:2609.25100v1, Remark 4.16:

Precisely two things remain: (i) that every multiple of n strictly between n and (n − 1)^2 − 1 is attained by some Latin square which Proposition 4.14 reduces to an explicit, checkable question about the function F, verified for n ≤ 11 but open in general; (ii) unimodality of (T_n(m)) …

Here Σ(L) is the sum over the columns of an order-n Latin square of the number of ascents read from top to bottom. Part (i) asks, for every n ≥ 5 and 2 ≤ k ≤ n − 3, for an order-n Latin square with Σ(L) = kn.

The frozen theorem D5/S3/Combinatorics/LatinEulerianMultiples.result proves part (i). Together with the paper's results it shows that the set of attained values of Σ is exactly [n − 1, (n − 1)²] with the two values n and (n − 1)² − 1 removed.

Issue 9995 records the screen made before the work. The arXiv record has only v1; the authors' companion paper arXiv:2609.28808v1 does not treat part (i); google-deepmind/formal-conjectures and conjectures.io have no entry. Nothing under Problems/, D5/ or Library/, and no entry of the three screening records, concerns this paper.

1. For a permutation p of 0, …, n − 1 let L_p(i, c) = τ((p_i + c) mod n), where τ exchanges the symbols 0 and 1. It is a Latin square.
2. For adjacent rows with cyclic difference δ, exactly n − δ columns ascend before τ is applied; τ changes only the directed pairs (0, 1) and (1, 0), which occur once when δ = 1 and once when δ = n − 1. Summing and telescoping gives Σ(L_p) = n·a(p) + p_0 − p_{n−1} − u(p) + v(p), where a counts ordinary ascents of p and u, v count cyclic differences 1 and n − 1.
3. Three explicit permutations give a = k and p_0 − p_{n−1} = u − v: one for k = 2, a four-block family for 3 ≤ k ≤ ⌊(n − 2)/2⌋, and one for the odd midpoint k = (n − 1)/2.
4. Reversing the rows exchanges ascents and descents in every adjacent pair, so Σ becomes n(n − 1) − Σ; this covers k' = n − 1 − k for the remaining targets.

The statement would fail if some interior multiple of n were not attained. It would not be the paper's statement if ascents were counted along rows or cyclically; the formal definitions count ascents in each column from top to bottom, exactly as Definition 2.3.

The construction was checked directly for every 5 ≤ n ≤ 60 and every 2 ≤ k ≤ n − 3 (1,596 cases): each square is Latin and its column-ascent total equals kn.

theorem; Tier 1 open subproblem stated in a 2026 paper, preregistered in issue 9995 before the work. The computational use is none: the delivered statement is universally quantified over every n ≥ 5 and 2 ≤ k ≤ n − 3, and no declaration is a bounded enumeration, a checker, a numeric reduction or a certified instance.

The literature screen is bounded to the sources named above; the paper was one week old when screened, so no worldwide priority claim is made. Part (ii) of Remark 4.16, the unimodality of T_n, is not addressed.