bibkey: iurlano2026pairwise authors: Enrico Iurlano, Günther R. Raidl year: 2026 title: “Pairwise Reflection Symmetry in Generalized Latin Rectangles” doi: 10.48550/arXiv.2606.28315 url: https://arxiv.org/html/2606.28315v1 claim: “Definition 3 defines the joint row frequency and its reflection equality; Definitions 2 and 4 specify reduced arrays and the URS(n,lambda,mu) domain.” strata_touched:

• D5/S3/Combinatorics/Graph/URSComponentParity license: citation-only triage: anchor

Pairwise reflection symmetry and its source domain

Verified locator

DOI: 10.48550/arXiv.2606.28315

URL: https://arxiv.org/html/2606.28315v1

The source is arXiv:2606.28315v1. The arXiv metadata identifies Enrico Iurlano and Günther R. Raidl as the authors. This note cites the source without reproducing its text.

Definition and domain mapping

Definition 3 counts the rows simultaneously realizing prescribed symbols in two prescribed columns. Pairwise reflection symmetry requires equality of this count with the count obtained by exchanging the two symbols, for every pair of distinct columns and every pair of distinct symbols. The repository’s pairCount is this joint row frequency on the single indexed family rho : Fin (2*n) -> Equiv.Perm (Fin n).

Definition 4 defines URS(n,lambda,mu) using lambdan rows, mun columns, uniform column multiplicity lambda, uniform row multiplicity mu, and pairwise reflection symmetry. At lambda=2 and mu=1, the rows are permutations, every column-symbol fibre has size two, and the source reflection condition is precisely the joint-count hypothesis used by fibre_graph_connected, after relabeling symbols and indices with Fin.

Definition 2 calls the mu=1 array reduced when its first row is the identity tuple and its rows are in non-decreasing lexicographic order. The connectivity theorem does not require either reduction condition, so it also applies to this source-defined reduced domain.

Attribution boundary

The joint frequency and URS domain are source-defined. The graph on original row indices and its odd-order connectivity argument are repository-derived. This citation attributes the domain and count; it does not attribute a connectivity theorem to the source or establish nonzero common-kernel existence, bipartiteness, a Latin-sheet decomposition, or the source’s counting and classification conjectures.