Pattern avoidance in biwords

Abstract

In this paper, we study pattern avoidance on the set of biwords with no repetitions in each block and prove that the number of those biwords avoiding π‾ is independent of the choice of π∈S3, which extends Knuth's classic result on permutations avoiding π∈S3. We present the proof in both bijective and inductive methods. As applications, we will give new combinatorial interpretations on Catalan, Riordan, and Motzkin numbers via pattern avoidance of biwords. Using the elementary theory of symmetric functions, we investigate the generating functions of biwords avoiding several specific patterns.