Restricted Jacobi permutations

Abstract

Jacobi permutations, introduced by Viennot in the context of Jacobi elliptic functions, are counted by the Euler numbers $E_{n}$ appearing in the series expansion $\sec x+\tan x=\sum_{n=0}^{\infty}E_{n}x^{n}/n!$. We conduct a systematic study of pattern avoidance in Jacobi permutations, achieving a complete enumeration of Jacobi permutations avoiding a prescribed set of length 3 patterns. In the case of a single pattern restriction, we obtain refined enumerations with respect to several permutation statistics: the number of ascents (or descents), the number of left-to-right minima, and the last letter. Bijections involving certain subfamilies of binary trees and Dyck paths, as well as generating function techniques, play important roles in our proofs.