Brussels Sprouts, Noncrossing Trees, and Parking Functions

TL;DR

This paper explores a variant of the Brussels Sprouts game, establishing bijections between game endstates, noncrossing trees, parking functions, and cycle factorizations, revealing deep combinatorial connections.

Contribution

It introduces a new variant of Brussels Sprouts and uncovers novel bijections linking game outcomes to well-studied combinatorial structures.

Findings

Endstates correspond to noncrossing trees

Game histories relate to parking functions

Connections to cycle factorizations in symmetric groups

Abstract

We consider a variant of the game of Brussels Sprouts that, like Conway's original version, ends in a predetermined number of moves. We show that the endstates of the game are in natural bijection with noncrossing trees and that the game histories are in natural bijection with both parking functions and factorizations of a cycle of $S_n$.