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Cyclic sieving phenomena for trees and tree-rooted maps

Mireille Bousquet-Mélou, Christian Krattenthaler

TL;DR

The paper establishes cyclic sieving phenomena for corner-rooted plane trees under several root-rotation actions, including ordinary, external, internal, and degree-preserving rotations, and further extends these CSPs to tree-rooted planar maps. It provides explicit CSP polynomials in terms of $q$-Catalan, Narayana, and $q$-multinomial expressions, along with precise fixed-point counts for rotations at divisors of the group orders. The authors connect these CSPs to classical Catalan-family objects (non-crossing matchings/partitions, triangulations, dissections) and present reformulations in terms of cubic maps with Hamiltonian cycles and quadrant lattice walks, highlighting a unifying approach across diverse combinatorial structures. The work advances the CSP framework by treating a broad spectrum of enumerative families and symmetry actions, offering new intertwined perspectives among Catalan objects and map-related models. Practically, it provides a toolkit for deriving CSPs in related Catalan-like settings and clarifies how rotation symmetries interact with refined combinatorial statistics.

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 $δ$ are considered, for a fixed $δ$, 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.

Cyclic sieving phenomena for trees and tree-rooted maps

TL;DR

The paper establishes cyclic sieving phenomena for corner-rooted plane trees under several root-rotation actions, including ordinary, external, internal, and degree-preserving rotations, and further extends these CSPs to tree-rooted planar maps. It provides explicit CSP polynomials in terms of -Catalan, Narayana, and -multinomial expressions, along with precise fixed-point counts for rotations at divisors of the group orders. The authors connect these CSPs to classical Catalan-family objects (non-crossing matchings/partitions, triangulations, dissections) and present reformulations in terms of cubic maps with Hamiltonian cycles and quadrant lattice walks, highlighting a unifying approach across diverse combinatorial structures. The work advances the CSP framework by treating a broad spectrum of enumerative families and symmetry actions, offering new intertwined perspectives among Catalan objects and map-related models. Practically, it provides a toolkit for deriving CSPs in related Catalan-like settings and clarifies how rotation symmetries interact with refined combinatorial statistics.

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 nodes; (2) all trees with nodes and 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 are considered, for a fixed , 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.

Paper Structure

This paper contains 24 sections, 32 theorems, 166 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Tree enumeration
  3. The cyclic sieving phenomenon
  4. Tree rotations
  5. Ordinary tree rotation
  6. Prescribing the number of nodes
  7. Prescribing the number of nodes and leaves
  8. External tree rotation
  9. Internal tree rotation
  10. Trees with given vertex degrees under ordinary rotation
  11. Trees with given vertex degrees under $\delta$-rotation
  12. Trees with given vertex degrees under internal rotation
  13. Connections with earlier work
  14. Non-crossing perfect matchings
  15. Non-crossing partitions
  16. ...and 9 more sections

Key Result

Lemma 2

The triple $(S,G,P)$, with $G=\langle g\rangle$ of order $m$, exhibits the cyclic sieving phenomenon if and only if the number of elements of $S$ fixed by $g^e$ equals $P(\omega_0^e)$, with $\omega_0:= e^{2\pi i/m}$, for $e=0$ and for $e \in\llbracket 1, m-1\rrbracket$ a divisor of $m$.

Figures (13)

  • Figure 1: A rooted tree with $n=6$ edges, hence $7$ vertices and $12$ corners.
  • Figure 2: Rotation on rooted trees with $3$ edges.
  • Figure 3: The transformation $\Phi$, acting on an edge-centered tree $\tau$.
  • Figure 4: The transformation $\Psi_d$, with $d=3$, acting on a node-centered tree $\tau$ of central degree $\ell=6$.
  • Figure 5: A tree of $\mathop{\mathrm{T}}\nolimits_l(n,k)$ fixed by $R_l^e$, here with $k=8$, $e=2$ and $d=4$.
  • ...and 8 more figures

Theorems & Definitions (77)

  • Definition 1: Cyclic sieving phenomenon
  • Lemma 2
  • proof
  • Example 1
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • Example 2
  • ...and 67 more