bibkey: ernst2026italiansquares authors: A. Ernst, S. Lia, C. O’Brien, J. Sheekey, J. Zumbrägel year: 2026 title: “Generalising Latin square orthogonality and Frobenius-König with alternating sign matrices” doi: 10.48550/arXiv.2606.25884 url: https://arxiv.org/abs/2606.25884v1 claim: “Section 8, Problem 8.3 asks which signed row and column margins are realizable in W_n.” strata_touched:

• D5/S3/Combinatorics/Latin/AlternatingSignMargins license: citation-only triage: anchor

Alternating signed matrices and their margins

Verified locator

DOI: https://doi.org/10.48550/arXiv.2606.25884. Primary version: https://arxiv.org/abs/2606.25884v1, Section 4, pp. 12–13, and Section 8, p. 27, Problem 8.3.

Source statement

Section 8, p. 27: “Section 4 introduces the set W_n consisting of all (0, ±1)-matrices in which the non-zero entries of each row and column alternate in sign, and the sum of each row/column is in {0, ±1}.”

Section 8, p. 27: “Problem 8.3. For which (0, ±1)-vectors R and S of order n does there exist X ∈ W_n with row-sums R and column-sums S?”

Encoding and scope

Entries and sums are integers. Rows and columns use the natural order on Fin n. Alternation compares consecutive nonzero entries, allowing either first sign and either last sign. Zero lines are allowed. The empty order is included in the formal statement. The source asks for a characterization; equality of the two total sums is the answer proved in D5/S3/Combinatorics/Latin/AlternatingSignMargins.result, rather than a result attributed to the paper. The source’s prescribed zero-pattern and border-sign questions are outside that theorem.