Cyclic sieving phenomenon on annular noncrossing permutations

TL;DR

This paper demonstrates the cyclic sieving phenomenon on annular noncrossing permutations by defining new q-analogues of classical combinatorial numbers and showing their polynomial properties and relationships.

Contribution

It introduces annular q-Kreweras, q-Narayana, and q-Catalan numbers and proves their cyclic sieving properties in the context of annular noncrossing permutations.

Findings

Polynomials exhibit cyclic sieving on annular noncrossing permutations

Sum of annular q-Kreweras numbers equals annular q-Narayana number

Sum of q-Narayana numbers equals annular q-Catalan number

Abstract

We show an instance of the cyclic sieving phenomenon on annular noncrossing permutations with given cycle types. We define annular $q$-Kreweras numbers, annular $q$-Narayana numbers, and annular $q$-Catalan number, all of which are polynomials in $q$. We then show that these polynomials exhibit the cyclic sieving phenomenon on annular noncrossing permutations. We also show that a sum of annular $q$-Kreweras numbers becomes an annular $q$-Narayana number and a sum of $q$-Narayana numbers becomes an annular $q$-Catalan number.