$q$-Kreweras numbers for coincidental Coxeter groups attached to limit symbols

TL;DR

This paper introduces a new, simplified definition of $q$-Kreweras numbers for certain Coxeter groups, exploring their combinatorial properties and connections to known $q$-analogues and cyclic sieving.

Contribution

It provides a novel, combinatorially simpler construction of $q$-Kreweras numbers for coincidental Coxeter groups, extending their properties and relations.

Findings

The $q$-Kreweras numbers are positive and well-behaved.

They relate to $q$-Narayana numbers.

They exhibit the cyclic sieving phenomenon.

Abstract

For a coincidental Coxeter group, i.e. of type $A_{n-1}, BC_n, H_3,$ or $I_2(m)$, we define the corresponding $q$-Kreweras numbers attached to limit symbols in the sense of Shoji. The construction of these numbers resembles the argument of Reiner and Sommers and these two share similar properties, but our version is slightly different from and combinatorially simpler than theirs. We also study the combinatorial properties of our $q$-Kreweras numbers, i.e. positivity, relation with $q$-Narayana numbers, and cyclic sieving phenomenon.