Balanced Gray Codes for Permutations and Rainbow Cycles for Associahedra

TL;DR

This paper constructs balanced Gray codes and rainbow cycles for permutations and associahedra, providing new combinatorial structures with applications in permutation enumeration and polyhedral graph theory.

Contribution

It introduces balanced Gray codes for permutations, rainbow cycles for permutations and associahedra, and extends existing results to new classes of combinatorial objects.

Findings

Constructed a balanced transposition Gray code for permutations.

Established rainbow cycles for permutations and associahedra.

Extended rainbow cycle results to permutahedra for all n.

Abstract

We settle the problem of constructing a balanced transposition Gray code for permutations of $[n] := \{1, \dots, n\}$ with $n \in \mathbb{N}\setminus\{0\}$. More generally, we obtain a~$2(m-2)!$-rainbow cycle for the permutations of $[n]$ for $m \in [n]$, a notion recently introduced by Felsner, Kleist, M\"utze, and Sering. Furthermore, we extend a result of theirs by presenting a $k$-rainbow cycle for the classical associahedron $\mathcal{A}_{n}$ for $k \in [2n + 2]$. For even $n$, we also construct a balanced Gray code for permutations of $[n]$, using only cyclically adjacent transpositions, complementing the construction for odd $n$ by Gregor, Merino, and M\"utze. Additionally, we show that the Permutahedron $P_{n}$ admits a $2$-rainbow cycle for all $n\ge5$ and a $3$-rainbow cycle for odd $n\ge3$.