# Parity-Unimodality and a Cyclic Sieving Phenomenon for Necklaces

## TL;DR

This paper reveals parity-unimodality and a cyclic sieving phenomenon in generalized $q$-Schr"oder polynomials, connecting algebraic properties with combinatorial symmetries.

## Contribution

It demonstrates that rational $q$-Schr"oder polynomials are parity-unimodal and exhibit a $q=-1$ phenomenon, linking them to cyclic sieving via $ ext{SL}_2$-character interpretations.

## Key findings

- Rational $q$-Schr"oder polynomials are parity-unimodal.
- They exhibit a $q=-1$ phenomenon.
- A cyclic sieving phenomenon for certain $S_n$-actions is established.

## Abstract

We discuss two surprising properties of a family of polynomials that generalize the Mahonian $q$-Catalan polynomials, and more generally the $q$-Schr\"oder polynomials. By interpreting them as $\mathfrak{sl}_2$-characters, we show that the rational $q$-Schr\"oder polynomials are parity-unimodal, which means that the even- and odd-degree coefficients are separately unimodal. Second, we show that they exhibit a $q=-1$ phenomenon. This is a special case of a more general cyclic sieving phenomenon for certain transitive $S_n$-actions, deduced from Molien's formula.