Latin Eulerian Numbers

We introduce \emph{Latin Eulerian numbers} $\text{LE}\genfrac{}{}{0px}{}{n}{k_1,\dots ,k_n}$, a multivariate refinement of classical Eulerian numbers counting order-$n$ Latin squares by column ascents. We establish their fundamental symmetries, univariate marginals, and an exact multiplicity divisibility property. For the total ascent statistic $\mathrm{\Sigma }(L)={\sum }_i{k}_i(L)$, we prove the sharp, isolated bounds $n-1\le \mathrm{\Sigma }(L)\le (n-1{)}^2$, demonstrating that the adjacent values $n$ and $(n-1{)}^2-1$ are strictly unattainable. To analyze intermediate values, we show that symbol permutations typically ignored in classical enumeration directly govern $\mathrm{\Sigma }$ via an exact formula. This insight yields explicit constructions for the interior range, motivates a unimodality conjecture for the total ascent distribution.