= 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> $P=\frac1{15}\begin{pmatrix}9&4&2\\4&9&2\\4&4&7\end{pmatrix}$. Its <stationary distribution> is $(2/5,2/5,1/5)$ and $P=\frac13I+\frac23\Pi$, where each row of $\Pi$ is this <stationary distribution>. Consequently the probability of opposite labels after $n$ steps from adjacent labels is $(1-3^{-n})/5$.
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