A generalization of Eulerian numbers via rook placements

TL;DR

This paper introduces a generalized class of Eulerian numbers based on rook placements on a grid, extending the classical case and providing symmetry properties and generating functions for these new numbers.

Contribution

It generalizes Eulerian numbers through rook placements with multiple rooks per row and column, revealing symmetry and deriving generating functions for these generalized numbers.

Findings

The generalized Eulerian numbers are symmetric for any number of rooks per row and column.

Generated explicit formulas for the numbers for small values of k.

Extended the understanding of Eulerian numbers through combinatorial rook placements.

Abstract

We consider a generalization of Eulerian numbers which count the number of placements of $cn$ "rooks" on an $n\times n$ board where there are exactly $c$ rooks in each row and each column, and exactly $k$ rooks below the main diagonal. The standard Eulerian numbers correspond to the case $c=1$. We show that for any $c$ the resulting numbers are symmetric and give generating functions of these numbers for small values of $k$.