Snow Leopard Permutations and Their Even and Odd Threads

TL;DR

This paper studies snow leopard permutations, focusing on their induced even and odd entries, and establishes bijections and characterizations connecting these permutations with Catalan paths, pattern avoidance, and Motzkin paths.

Contribution

It introduces a detailed analysis of even and odd threads within snow leopard permutations, including recursive bijections and pattern avoidance characterizations.

Findings

Number of snow leopard permutations of length 2n-1 equals the Catalan number C_n.

Established bijections between threads and Catalan paths.

Characterized layered threads via pattern avoidance and connected them to peakless Motzkin paths.

Abstract

Caffrey, Egge, Michel, Rubin and Ver Steegh recently introduced snow leopard permutations, which are the anti-Baxter permutations that are compatible with the doubly alternating Baxter permutations. Among other things, they showed that these permutations preserve parity, and that the number of snow leopard permutations of length $2n-1$ is the Catalan number $C_n$. In this paper we investigate the permutations that the snow leopard permutations induce on their even and odd entries; we call these the even threads and the odd threads, respectively. We give recursive bijections between these permutations and certain families of Catalan paths. We characterize the odd (resp. even) threads which form the other half of a snow leopard permutation whose even (resp. odd) thread is layered in terms of pattern avoidance, and we give a constructive bijection between the set of permutations of length $n$ which are both even threads and odd threads and the set of peakless Motzkin paths of length $n+1$.