Séminaire Lotharingien de Combinatoire, 95B.57 (2026), 12 pp.

Theo Douvropoulos

Extended Cyclic Sieving Phenomena

Abstract. The Cyclic Sieving Phenomenon (CSP) of Reiner-Stanton-White is a remarkable framework in which the symmetries of a cyclic action on a set X of combinatorial objects are described in terms of root-of-unity evaluations of some q-version of the cardinality of X.

We generalize the CSP, by replacing the set X with some vector space that carries a combinatorial but not necessarily permutation action of a cyclic group (for instance the action on the homology of a complex induced by some cyclic action on its vertices). In this extended Cyclic Sieving Phenomenon (eCSP), the polynomial encodes the traces of the cyclic action. We prove eCSP for a variety of examples, including the positive q-Catalan numbers, the q-Kirkman numbers resolving an open problem of Armstrong-Reiner-Rhoades since 2012, and more.

Received: November 15, 2025. Accepted: February 15, 2026. Final version: April 1, 2026.

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