Patterns in Inversion Sequences I

TL;DR

This paper introduces the concept of pattern avoidance in inversion sequences, establishing connections to well-known numerical sequences and extending permutation pattern studies to a new combinatorial structure.

Contribution

It defines patterns in inversion sequences and explores their enumeration, linking these patterns to classical sequences like Fibonacci, Bell, Schr"oder, and Euler numbers.

Findings

Enumeration of inversion sequences avoiding length-3 patterns

Connections to Fibonacci, Bell, Schr"oder, and Euler numbers

Foundation for further study in patterns in inversion sequences

Abstract

Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying interpretation that relates a vast array of combinatorial structures. In this paper, we introduce the notion of patterns in inversion sequences. A sequence $(e_1,e_2,\ldots,e_n)$ is an inversion sequence if $0 \leq e_i<i$ for all $i \in [n]$. Inversion sequences of length $n$ are in bijection with permutations of length $n$; an inversion sequence can be obtained from any permutation $\pi=\pi_1\pi_2\ldots \pi_n$ by setting $e_i = |\{j \ | \ j < i \ {\rm and} \ \pi_j > \pi_i \}|$. This correspondence makes it a natural extension to study patterns in inversion sequences much in the same way that patterns have been studied in permutations. This paper, the first of two on patterns in inversion sequences, focuses on the enumeration of inversion sequences that avoid words of length three. Our results connect patterns in inversion sequences to a number of well-known numerical sequences including Fibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers.