An area-to-rinv bijection between a class of generalized Dyck paths and relative words with forbidden patterns (312,212)

摘要

We provide a bijection between the lattice paths that never go below an increasing curve and some words having some forbidden patterns. This implies the following applications. Firstly, by introducing the rightmost inversion number statistics of words, we get an analogous result of Bandlow and Killpatrck's weight-preserving bijection between the Dyck paths and 312-avoiding permutations. We use a certain type of ordered trees as a bridge to equalize the rightmost inversion number of words without the patterns 312 or 212 and the area statistic of generalized Dyck paths. Secondly, we obtain a representation of matrix products for the number of words with neither of the patterns 312 and 212.