episciences.org_1323_1675649836 1675649836 episciences.org raphael.tournoy+crossrefapi@ccsd.cnrs.fr episciences.org Discrete Mathematics & Theoretical Computer Science 1365-8050 03 31 2016 Vol. 18 no. 2, Permutation... Permutation Patterns Patterns in Inversion Sequences I Sylvie Corteel Megan A. Martinez Carla D. Savage Michael Weselcouch 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. 03 31 2016 1323 French National Research Agency (ANR) ANR-08-JCJC-0011 https://arxiv.org/licenses/nonexclusive-distrib/1.0 arXiv:1510.05434 10.48550/arXiv.1510.05434 https://arxiv.org/abs/1510.05434v3 https://arxiv.org/abs/1510.05434v1 10.46298/dmtcs.1323 https://dmtcs.episciences.org/1323 https://dmtcs.episciences.org/1421/pdf https://dmtcs.episciences.org/1421/pdf