Parity-Unimodality and a Cyclic Sieving Phenomenon for Necklaces

Abstract

We discuss two surprising properties of a family of polynomials that generalize the Mahonian $q$-Catalan polynomials, and more generally the $q$-Schröder polynomials. By interpreting them as ${\mathfrak{sl}}_2$-characters, we show that the rational $q$-Schröder 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.

23 pages, 6 figures