Area of Catalan paths on a checkerboard

It is known that the area of all Catalan paths of length n is equal to 4 n − 2 n + 1 n , which coincides with the number of inversions of all 321-avoiding permutations of length n + 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.