Enumeration On Catalan Words Avoiding The Pattern Of Three

Word is a very classic combinatorial structure.The study of word with specific structures is a very broad research direction in combinatorics.This article focuses on the enumerative of Catalan words.The Catalan word was proposed by J.L.Baril,Sergey Kirgizov,Vincent Vajnovszki.For any given positive integer n,we use the Dyck path to give the total number of descents and the distribution of ascents.Based on the distribution of ascents,we build an correspondence between Catalan words with k ascents to Catalan words with n-1-k ascents.We use the 2-Motzkin path to give the distribution of descents.Through the distribution of descents,we give the total number of descents of the Catalan words as well as the ascents and flat distribution.We give a combinatorial proof of the count of Catalan words and descents in avoidance pattern.