2023/06/16 by Dong, David
#05A05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.11836
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.