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

Let $P_n=\mathrm{Cay}(S_n,\{r_2,\ldots,r_n\})$ 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 $\rho=r_{n-1}r_n=[n,1,2,\ldots,n-1]$, then, for every $n\ge3$, $\mathrm{Stab}_{L(S_n)}(Z_n)=\langle L_\rho,L_{r_n}\rangle\cong D_n$, 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 $\mathrm{ord}(a)\ge3$, then every left translation preserving the same cycle conjugates $a$ to $a$ or $a^{-1}$. For $n\ge5$, Deng-Zhang's automorphism theorem gives the same stabilizer inside $\mathrm{Aut}(P_n)$; the exceptional ranks are handled separately. We also compute the compression factor of $Z_n$: it is $n$ for $n\ge4$ and $6$ for $n=3$.