The Left-Regular Stabilizer of Zaks' Hamiltonian Cycle in the Pancake Graph

Let $P_n=\mathrm{C}\mathrm{a}\mathrm{y}(S_n,\{r_2,\dots ,r_n\})$ be the pancake graph, with prefix reversals acting on the right. Conjugating Zaks’ suffix-reversal permutation Gray code by the full reversal gives a distinguished Hamiltonian cycle $Z_n$ in $P_n$. We determine the stabilizer of this particular cycle under the left regular action of $S_n$. If $\rho =r_{n-1}{r}_n=[n,1,2,\dots ,n-1]$, then, for every $n\ge 3$, ${\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}_{L(S_n)}(Z_n)=⟨L_{\rho },L_{r_n}⟩\cong D_n$, where $D_n$ denotes the dihedral group of order $2n$. The inclusion $\supseteq$ follows from the recursive block decomposition $W_n=(W_{n-1}{r}_n{)}^{n-1}{W}_{n-1}$ and from the palindromy $W_n^R=W_n$. The reverse inclusion follows from a general cyclic-order rigidity lemma: if a Hamiltonian cycle on a finite group is invariant under $L_a$, with $\mathrm{o}\mathrm{r}\mathrm{d}(a)\ge 3$, then every left translation preserving the same cycle conjugates $a$ to $a$ or $a^{-1}$. For $n\ge 5$, Deng-Zhang’s automorphism theorem gives the same stabilizer inside $\mathrm{A}\mathrm{u}\mathrm{t}(P_n)$; the exceptional ranks are handled separately. We also compute the compression factor of $Z_n$: it is $n$ for $n\ge 4$ and $6$ for $n=3$.