An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations

The symmetric q, t-Catalan polynomial Cn(q, t), which specializes to the Catalan polynomial Cn(q) when t = 1, was defined by Garsia and Haiman in 1994. In 2000, Garsia and Haglund proved the existence of statistics a(π) and b(π) on Dyck paths such that Cn(q, t) = P π qt where the sum is over all n × n Dyck paths. Specializing t = 1 gives Cn(q) = P π q and specializing q = 1 as well gives the usual Catalan number Cn. The Catalan number Cn is known to count the number of n×n Dyck paths and the number of 312-avoiding permutations in Sn, as well as at least 64 other combinatorial objects. In this paper, we define a bijection between Dyck paths and 312-avoiding permutations which takes the area statistic a(π) on Dyck paths to the inversion statistic on 312avoiding permutations. The inversion statistic can be thought of as the number of (21) patterns in a permutation σ. We give a characterization for the number of (321), (4321), . . . , (k · · · 21) patterns that occur in σ in terms of the corresponding Dyck path.