Over-inversion numbers: Basic properties, log-concavity and unimodality

Abstract

In this paper, we introduce the concept of the Over-inversion number, which counts the overlined permutations of length \(n\) with \(k\) inversions, allowing the first elements associated with the inversions to be independently overlined or not. We explore its properties and combinatorial interpretations through lattice paths, overpartitions, and tilings, and provide a combinatorial proof demonstrating that these numbers form a log-concave and unimodal sequence.