On 102-avoiding inversion sequences

TL;DR

This paper establishes a bijection between 102-avoiding inversion sequences and specific Schr"oder paths, introduces intermediate combinatorial objects, and provides refined enumerations based on natural statistics.

Contribution

It introduces a novel bijection linking inversion sequences avoiding 102 with Schr"oder paths and develops intermediate objects and refined enumeration methods.

Findings

Bijection between 102-avoiding inversion sequences and Schr"oder paths with restrictions.

Enumeration of these structures based on natural statistics.

Refined counts for inversion sequences avoiding 102 and another pattern of length 3.

Abstract

In this article, we provide a bijection between the set of inversion sequences avoiding the pattern 102 and the set of 2-Schr\"{o}der paths having neither peaks nor valleys and ending with a diagonal step. To achieve this, we introduce two intermediate objects, called UVD paths and labeled $F$-paths, and establish bijections among all four families. For each of these combinatorial objects, we define a natural statistic and enumerate the corresponding structures with respect to this statistic. In addition, we study inversion sequences avoiding 102 and another pattern of length 3, providing refined enumerations according to the same statistic.