 Separation chain for two labels in a circular transposition shuffle

In a random transposition shuffle on a six-cycle, the shorter circular distance between two distinguished labels takes values $1,2,3$. The symmetry of the cycle and exchangeability of other labels make this a lumped Markov chain. Counting the fifteen possible transpositions gives transition matrix $P=\frac{1}{15}\left(\begin{array}{ccc}9& 4& 2\\ 4& 9& 2\\ 4& 4& 7\end{array}\right)$. Its stationary distribution is $(2/5,2/5,1/5)$ and $P=\frac{1}{3}I+\frac{2}{3}\Pi$, where each row of $\Pi$ is this stationary distribution. Consequently the probability of opposite labels after $n$ steps from adjacent labels is $(1-3^{-n})/5$.