The cyclic sieving phenomenon

doi:10.1016/j.jcta.2004.04.009

Journal of Combinatorial Theory, Series A

Volume 108, Issue 1, October 2004, Pages 17-50

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Abstract

The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridge's q=−1 phenomenon. The phenomenon is shown to appear in various situations, involving q-binomial coefficients, Pólya–Redfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite field q-analogues.

Keywords

q-Binomial coefficient

q-Multinomial coefficient

Roots-of-unity

Principal specialization

Schur function

Kraskiewicz–Weyman

Singer cycle

Springer regular element

Polygon dissections

Noncrossing partitions

Hook formula

Ordered tree

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¹: Supported by NSF grants DMS-0245379 for Reiner, DMS-0203282 for Stanton.