  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 and arithmetic properties of $A_{n,i,j}$, which
contain a recurrence of $A_{n,i,j}$. Using the idea of balls in boxes, Petersen
gave a combinatorial interpretation for the generating function of $A_{n,i,j}$,
and obtained the same recurrence of $A_{n,i,j}$ from its generating function.
Subsequently, Petersen asked whether there is a visual way to understand this
recurrence. In this paper, after observing the internal structures of
permutation grids, we present a combinatorial proof for the recurrence of
$A_{n,i,j}$. Let $I_{n,k}$ and $J_{n,k}$ be the number of involutions and
fixed-point free involutions on $[n]$ with $k$ descents, respectively. With the
help of algebraic method on generating functions, Guo and Zeng derived two
recurrences of $I_{n,k}$ and J_{2n,k}$ that play an essential role in the proof
of their unimodal properties. Surprisingly, the constructive approach to the
recurrence of $A_{n,i,j}$ is found to fuel the combinatorial interpretations of
these two recurrences of $I_{n,k}$ and $J_{2n,k}$.
