Combinatorial proofs on the joint distribution of descents and inverse descents

Frank Z. K. Li, Xunhao Liu

Let $A_{n,i,j}$ be the number of permutations on $[n]$ with $(i-1)$ descents and $(j-1)$ inverse descents.Carlitz, Roselle and Scoville in 1966 first revealed some combinatorial an…