Cyclic sieving phenomena for trees and tree-rooted maps

TL;DR

This paper establishes cyclic sieving phenomena for various classes of rooted plane trees and tree-rooted planar maps under different cyclic group actions, revealing new combinatorial symmetries.

Contribution

It introduces new cyclic sieving results for rooted trees with specific properties and extends these phenomena to tree-rooted planar maps, broadening understanding of combinatorial symmetries.

Findings

Cyclic sieving phenomena proven for all trees with n nodes

Cyclic sieving established for trees with k leaves and given degree distributions

Extension of cyclic sieving to tree-rooted planar maps

Abstract

We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees with a given degree distribution of the nodes. Moreover, we consider four different cyclic group actions: (1) the root is moved to the next corner along a tour of the tree; (2) only trees in which the root is at a leaf are considered, and the action moves the root to the next leaf; (3) only trees in which the root is at a non-leaf are considered, and the action moves the root to the next non-leaf corner; (4) only trees in which the root is at a node of degree $\delta$ are considered, for a fixed $\delta$, and the action moves the root to the next corner of this type. We prove a cyclic sieving phenomenon for each meaningful combination of these sets and actions. As a bonus, we also establish corresponding cyclic sieving phenomena for tree-rooted planar maps.