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

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.