Pattern Avoidance of Generalized Permutations

TL;DR

This paper explores pattern avoidance in generalized permutations, revealing that the count of such permutations avoiding any pattern in S_3 is uniform, and introduces new combinatorial structures like the Catalan-Riordan path with applications to classical number sequences.

Contribution

It extends classic permutation pattern avoidance results to generalized permutations and introduces the Catalan-Riordan path, providing new combinatorial interpretations and bijections.

Findings

Number of generalized permutations avoiding any pattern in S_3 is constant.

Introduces Catalan-Riordan path as a combinatorial object.

Provides interpretations of Motzkin and Riordan numbers.

Abstract

In this paper, we study pattern avoidances of generalized permutations and show that the number of all generalized permutations avoiding $\pi$ is independent of the choice of $\pi\in S_3$, which extends the classic results on permutations avoiding $\pi\in S_3$. Extending both Dyck path and Riordan path, we introduce the Catalan-Riordan path which turns out to be a combinatorial interpretation of the difference array of Catalan numbers. As applications, we interpret both Motzkin and Riordan numbers in two ways, via semistandard Young tableaux of two rows and generalized permutations avoiding $\pi \in S_3$. Analogous to Lewis's method, we establish a bijection from generalized permutations to rectangular semistandard Young tableaux which will recover several known results in the literature.