Patterns in Inversion Sequences I

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 $e_i = |\{j \ | \ j \pi_i \}|$. 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

Patterns in inversion sequences I

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 (e1, e2, . . ., en) is an inversion sequence if 0 ≤ ei \u3c i for all i ∈ [n]. Inversion sequences of length n are in bijection with permutations of length n; an inversion sequence can be obtained from any permutation π = π1π2 . . . πn by setting ei = |{j | j \u3c i and πj \u3e πi}|. 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, Schröder numbers, and Euler up/down numbers

Patterns in Inversion Sequences I

Permutations that avoid given patterns have been studied in great depth fortheir connections to other fields of mathematics, computer science, andbiology. From a combinatorial perspective, permutation patterns have served asa unifying interpretation that relates a vast array of combinatorialstructures. In this paper, we introduce the notion of patterns in inversionsequences. A sequence $(e_1,e_2,\ldots,e_n)$ is an inversion sequence if $0\leq e_i \pi_i \}|$. This correspondence makes ita natural extension to study patterns in inversion sequences much in the sameway that patterns have been studied in permutations. This paper, the first oftwo on patterns in inversion sequences, focuses on the enumeration of inversionsequences that avoid words of length three. Our results connect patterns ininversion sequences to a number of well-known numerical sequences includingFibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers

Patterns in Inversion Sequences I

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 $e_i = |\{j \ | \ j \pi_i \}|$. 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