Restricted Jacobi permutations

2025/09/15 by Henke, Alyssa G., Kyle R. Hoffman, Derek H. Stephens +5
Mathematics · #05A05 #05A15 (Primary) #05A19 #05C05 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2509.11494

openalex publication_date 2025/09/15 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

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=∑_n=0^∞E_nxⁿ/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.

Related

• Parity statistics on restricted permutations and the Catalan--Schett polynomials
• Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions
• Counting permutations by alternating descents
• Pattern Avoiding Permutations Enumerated by Inversions
• Non-overlapping descents and ascents in stack-sortable permutations