Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of 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 the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.
Bibliographical noteFunding Information:
First author supported by NSF Grant DMS-1001933, second author supported by NSA Grant H98230-11-1-0173 and by a National Science Foundation Independent Research and Development plan.
© 2018 Springer Nature Switzerland AG.
- Catalan number
- Cyclic sieving phenomenon
- Kreweras number
- Narayana number
- Nilpotent orbit
- Weyl group