Balanced Gray Codes for Permutations and Rainbow Cycles for Associahedra

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 $n\ge3$.