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.
Highlights
OverviewWe use the observation to define the structure of the p-avoiding inversion sequences and relate them to equinumerous combinatorial families (i) When our paper was first posted to the arXiv we were notified by Mansour that he and Shattuck had independently obtained results on |In(σ)| for the patterns σ = 012, 021, 102, 201, 210 in [16]
Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology
We show that the number of inversion sequences avoiding 012 is given by the odd-indexed Fibonacci numbers and that the inversion sequences avoiding 021 are counted by the large Schroder numbers
Summary
We use the observation to define the structure of the p-avoiding inversion sequences and relate them to equinumerous combinatorial families (i) When our paper was first posted to the arXiv we were notified by Mansour that he and Shattuck had independently obtained results on |In(σ)| for the patterns σ = 012, 021, 102, 201, 210 in [16]. We use concatenation to add an element to the beginning or end of an inversion sequence: 0 · e is the inversion sequence (0, e1, e2, . . . , en) and for 0 ≤ i ≤ n, e · i is the inversion sequence (e1, e2, . . . , en, i)
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