Area of Catalan paths on a checkerboard

It is known that the area of all Catalan paths of length nn is equal to 4n−2n+1n, which coincides with the number of inversions of all 321-avoiding permutations of length n+1n+1. In this paper, a bijection between the two sets is established. Meanwhile, a number of interesting bijective results that pave the way to the required bijection are presented.