# A new encoding of permutations by Laguerre histories

## TL;DR

This paper introduces a novel bijection between permutations and Laguerre histories, enabling new insights into permutation statistics and leading to a positive expansion of certain Eulerian polynomials.

## Contribution

It presents a new encoding of permutations via Laguerre histories and applies this to derive a $q$-$ ext{gamma}$-positivity expansion of $( ext{inv}, ext{exc})$-$q$-Eulerian polynomials.

## Key findings

- Established a bijection between permutations and Laguerre histories.
- Derived a $q$-$ ext{gamma}$-positivity expansion for $( ext{inv}, ext{exc})$-$q$-Eulerian polynomials.
- Provided combinatorial interpretations for permutation statistics.

## Abstract

We construct a bijection from permutations to some weighted Motzkin paths known as Laguerre histories. As one application of our bijection, a neat $q$-$\gamma$-positivity expansion of the $(\inv,\exc)$-$q$-Eulerian polynomials is obtained.