= Solution
Let $L$ be the number of index tables, including the zero table. If distinct feasible states $A,B$ differ by an index table, then $P(A,B)=1/L=P(B,A)$, using the sign-reversal <bijection> for the reverse move. If they do not differ by an index table, both probabilities are zero. Since $\pi$ is uniform,
$$
\boxed{\pi(A)P(A,B)=\pi(B)P(B,A).}
$$
For equal states <detailed balance> is tautological. Thus the chain is reversible and its <uniform distribution> is stationary.
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