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Annals of Combinatorics

, Volume 22, Issue 4, pp 819–874 | Cite as

Weyl Group \({\varvec{q}}\)-Kreweras Numbers and Cyclic Sieving

  • Victor Reiner
  • Eric Sommers
Article
  • 14 Downloads

Abstract

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 ABC, 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 ABCD the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

Keywords

Reflection Weyl group Catalan number Narayana number Kreweras number Nilpotent orbit Cyclic sieving phenomenon 

Mathematics Subject Classification

20F55 51F15 

Notes

Acknowledgements

The first author thanks Jang Soo Kim for helpful conversations.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Mathematics and StatisticsUniversity of Massachusetts-AmherstAmherstUSA

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