slug: mirzavaziri-yaqubi-cyclic-full-symmetry-refutation
bibkey: mirzavaziri2026cyclic
doi: 10.48550/arXiv.2609.28808
url: https://arxiv.org/abs/2609.28808v1
triage: theorem
motivation_gids:
  - D5/S3/Combinatorics/CyclicLatinEulerianRefutation.result

The Cyclic Latin Eulerian Numbers Are Not Fully Symmetric

Problem

Madjid Mirzavaziri and Daniel Yaqubi, Cyclic Latin Eulerian Numbers, arXiv:2609.28808v1, Remark 3.9:

Cyclically shifting π naturally explains an order-n cyclic symmetry for ⟨⟨n k⟩⟩_c. However, computations for n ≤ 5 (Table 1) reveal a surprising fact: ⟨⟨n k⟩⟩_c is actually invariant under the full symmetric group action on k … Identifying the hidden mechanism behind this full invariance remains an open problem.

Here ⟨⟨n k⟩⟩_c counts permutations π whose row-reordered cyclic square L_π has column-ascent vector k.

Motivation

The frozen theorem D5/S3/Combinatorics/CyclicLatinEulerianRefutation.result refutes the asserted invariance, and the module proves the mechanism that actually governs the column-ascent vectors: one cyclic shift changes the ascent number only through the value that wraps around, so the attained vectors take two adjacent values forming one cyclic interval. The failure holds for every n ≥ 4.

Gap

Issue 10060 records the screen made before the work. The arXiv record has only v1; nothing under Problems/, D5/ or Library/, and no entry of the three screening records, concerns this paper. The paper’s Table 1 lists one arrangement per multiset, which is why the asymmetry was not visible there.

Route

1. shift_step: for every n ≥ 2, the ascent number of column c + 1 differs from that of column c by [the wrapping value is in the first row] − [it is in the last row], where the wrapping value is the entry equal to n − 1 in column c.
2. no_three_changes: three consecutive changes would need three distinct wrapping values located at the two end rows, which is impossible.
3. not_fully_symmetric: for every n ≥ 4, the permutation obtained from the identity by exchanging its last two values attains a vector with two adjacent high entries; exchanging two coordinates of that vector produces three consecutive changes, so the permuted vector is attained by no permutation while the original is attained.
4. result: the asserted invariance fails, already at n = 4.

Falsifier

The statement would fail if the permuted vector were attained. It would not be the paper’s statement if rows or columns of L_π were indexed differently; the formal square is L_π(i, c) = π(i) + c over Z/n, the paper’s row-reordered cyclic square, and direct enumeration with these definitions reproduce every n = 4 entry of the paper’s Table 1.

Evidence

Direct enumeration over S_n for n = 4, …, 8: every attained vector has two adjacent values with the larger forming one cyclic interval; the sorted vector of each tested multiset is attained (2, 2, 16, 48, 384 times) while its alternating rearrangement is attained 0 times.

Triage

theorem; Tier 1 open problem stated in a 2026 paper, preregistered in issue 10060 before the work. The computational use is none: the central theorems hold for every n, and no declaration is a bounded enumeration, a checker, a numeric reduction or a certified instance.

ASSUMED-UNVERIFIED

The literature screen is bounded to the sources named above, for a paper three days old at screening; no worldwide priority claim is made.