Generalized Eulerian Numbers

Let $A(n,m)$ denote the Eulerian numbers, which count the number of permutations on $[n]$ with exactly $m$ descents. It is well known that $A(n,m)$ also counts the number of permutations on $[n]$ with exactly $m$ excedances. In this report, we define numbers of the form $A(n,m,k)$, which count the number of permutations on $[n]$ with exactly $m$ descents and the last element $k$. We then show bijections between this definition and various other analogs for $r$-excedances and $r$-descents. We also prove a variation of Worpitzky’s identity on $A(n,m,k)$ using a combinatorial argument mentioned in a paper by Spivey in 2021.