Pattern Avoidance of Generalized Permutations

Zhousheng Mei, Suijie Wang

Published 2018-04-17Version 1

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 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.