On the Combinatorics of Pseudo-Latin Squares

We introduce a new class of combinatorial objects called consecutive pseudo-Latin squares (CPLSs), a variant of Latin squares in which at least one row or column is in consecutive or reverse-consecutive order, but every element may not appear in every row or column. We derive exact and asymptotic formulas for the number of CPLSs of order n, showing that their proportion among all pseudo-Latin squares (PLSs) rapidly approaches zero as n. We also analyze the distribution of CPLSs under uniform random sampling, and explore connections to algebraic structures, interpreting CPLSs as Cayley tables related to those of unital magmas. Finally, we supplement our theoretical results with Monte Carlo simulations for small values of n.