# Width-$k$ Generalizations of Classical Permutation Statistics

## TL;DR

This paper introduces width-$k$ generalizations of classical permutation statistics like descents and inversions, establishing new relationships and exploring their behavior in pattern avoidance contexts.

## Contribution

It presents novel width-$k$ variants of classical permutation statistics and analyzes their properties, relationships, and behavior in pattern avoidance scenarios.

## Key findings

- Width-$k$ statistics generalize classical permutation statistics.
- New formulas and relationships are established for these statistics.
- Behavior in pattern avoidance contexts is characterized.

## Abstract

We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width-$k$ descents and width-$k$ inversions. These variations induce generalizations of the excedance and major statistics, providing a framework in which the most well-known equidistributivity results for classical statistics are paralleled. We explore additional relationships among the statistics providing specific formulas in certain special cases. Moreover, we explore the behavior of these width-$k$ statistics in the context of pattern avoidance.