Latin Eulerian Number

The Latin Eulerian number is the number of order- Latin squares whose th column, read from top to bottom as a permutation, has exactly ascents. Thus Latin Eulerian numbers form a multivariate refinement of the Eulerian numbers.

For a Latin square , let be its total number of column ascents. Mirzavaziri and Yaqubi (2026) proved the sharp bounds

and proved that both bounds are attained, while the adjacent values and are unattainable for . They report using Claude 5 during exploration, proof development, editing, and revision, and state that all suggestions were substantially revised and independently verified by the authors.