Let An,i,jβ be the number of permutations on [n] with iβ1 descents
and jβ1 inverse descents. Carlitz, Roselle and Scoville in 1966 first
revealed some combinatorial and arithmetic properties of An,i,jβ, which
contain a recurrence of An,i,jβ. Using the idea of balls in boxes, Petersen
gave a combinatorial interpretation for the generating function of An,i,jβ,
and obtained the same recurrence of An,i,jβ 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 of the recurrence of
An,i,jβ. Let In,kβ and Jn,kβ count the number of involutions and
fixed-point free involutions on [n] with k descents, respectively. With the
help of generating functions, Guo and Zeng derived two recurrences of In,kβ
and J2n,kβ that play an essential role in the proof of their unimodal
properties. Unexpectedly, the constructive approach to the recurrence of
An,i,jβ is found to fuel the combinatorial interpretations of these two
recurrences of In,kβ and J2n,kβ