## Corteel, Sylvie and Martinez, Megan A. and Savage, Carla D. and Weselcouch, Michael - Patterns in Inversion Sequences I

dmtcs:1421 - Discrete Mathematics & Theoretical Computer Science, March 31, 2016, Vol. 18 no. 2, Permutation Patterns 2015
Patterns in Inversion Sequences I

Authors: Corteel, Sylvie and Martinez, Megan A. and Savage, Carla D. and Weselcouch, Michael

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.

Source : oai:arXiv.org:1510.05434
Volume: Vol. 18 no. 2, Permutation Patterns 2015
Section: Permutation Patterns
Published on: March 31, 2016
Submitted on: March 30, 2016
Keywords: Mathematics - Combinatorics,05A05, 05A19