Sharp asymptotic probability for prescribed parity specifications

Determine whether, for every admissible triple of row-, column-, and symbol-parity vectors in the setting of Theorem 1.2, the probability that a uniformly random order-n Latin square has exactly those prescribed parities is (1+o(1))2^{-3n}, thereby improving the established relative error factor 2^{o(n)} to 1+o(1).

Background

The paper proves that, for any vectors of row, column, and symbol parities satisfying the necessary affine parity constraint, the probability of obtaining that specification is 2{-3n+o(n)}. Equivalently, the result determines the probability only up to a multiplicative factor of 2{o(n)}.

The authors identify the most direct unresolved refinement as determining whether this multiplicative error can be reduced to 1+o(1). The question concerns all admissible parity specifications simultaneously in the setting of the paper's general parity-distribution theorem, rather than only the all-even row-parity event.

References

The most obvious open question related to this work is whether the relative error $2{o(n)}$ can be improved to $1+o(1)$: is it true that in the setting of cref{thm:main} we have $\mathbb{P}\big[(\vec\xi_{\textnormal{row}( L),\vec\xi_{\textnormal{col}( L),\vec\xi_{\textnormal{sym}( L))=(\vec r,\vec c, \vec s)\big]= (1+o(1))2{-3n}$?

— On the probability that a random Latin square is all-even  (2610.13487 - Greenhill et al., 9 Oct 2026) in Section 1, Further directions