Weyl group $q$-Kreweras numbers and cyclic sieving

TL;DR

This paper defines and provides explicit formulas for $q$-Kreweras numbers for finite Weyl groups, demonstrating their connection to cyclic sieving phenomena and unifying formulas across classical types.

Contribution

It introduces a new definition for $q$-Kreweras numbers for all finite Weyl groups and verifies their cyclic sieving behavior in classical types.

Findings

Explicit formulas for $q$-Kreweras numbers in all types.

Formulas depend only on the Weyl group, unifying types B and C.

Verification of cyclic sieving phenomena at roots of unity.

Abstract

The paper concerns a definition for $q$-Kreweras numbers for finite Weyl groups $W$, refining the $q$-Catalan numbers for $W$, and arising from work of the second author. We give explicit formulas in all types for the $q$-Kreweras numbers. In the classical types $A, B, C$, we also record formulas for the $q$-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types $B$ and $C$). In addition we verify that in the classical types $A,B,C,D$ that the $q$-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.