# Smirnov Trees

## TL;DR

This paper introduces Smirnov trees, a new generalization of Smirnov words involving labeled binary trees, and proves their ascent-descent generating function is e-positive using a weight-preserving bijection.

## Contribution

It extends Smirnov words to labeled binary trees and establishes e-positivity of their generating function with a novel bijective proof.

## Key findings

- Generated function for Smirnov trees is e-positive.
- Established a weight-preserving bijection for proof.
- Generalized classical Smirnov words to tree structures.

## Abstract

We introduce a generalization of Smirnov words in the context of labeled binary trees, which we call Smirnov trees. We study the generating function for ascent-descent statistics on Smirnov trees and establish that it is $e$-positive, which is akin to the classical case of Smirnov words. Our proof relies on an intricate weight-preserving bijection.