Dihedral Sieving Phenomena

Abstract

Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially $q$-analogues, have useful interpretations related to actions and representations of the cyclic group. We propose a definition of sieving for an arbitrary group $G$ and study it for the dihedral group $I_2(n)$ of order $2n$. This requires understanding the generators of the representation ring of the dihedral group. For $n$ odd, we exhibit several instances of dihedral sieving which involve the generalized Fibonomial coefficients, recently studied by Amdeberhan, Chen, Moll, and Sagan. We also exhibit an instance of dihedral sieving involving Garsia and Haiman's $(q,t)$-Catalan numbers.

10 pages

Cited by in corpus (4)

• Cyclic Sieving for Plane Partitions and Symmetry
• The Cone of Cyclic Sieving Phenomena
• Symmetry of Narayana numbers and rowvacuation of root posets
• Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections