Cyclic Sieving, Necklaces, and Bracelets

Eric Stucky

Published 2018-12-11Version 1

We split the q-Schr\"oder numbers into an "even" and "odd" part. The Schr\"oder numbers are known to enumerate certain necklaces, and the even part turns out to be a $q$-analogue for the set of bracelets. The even and odd parts are symmetric and unimodal, and we seek a poset structure which explains these features. Along the way, we find a new cyclic sieving phenomenon on certain double cosets of the symmetric group which generalizes the classical enumeration of two-colored bracelets.