On Series of $H$-Equivalent Tuples in Markov Chains

Let $\mathbf {X}=(X_0,X_1,\ldots )$ be an irreducible Markov chain with state set $\{1,\ldots ,N\}$ and $H$ be a permutation group on the set $\{1,\ldots ,N\}$. We prove limit theorems for the number of series of $H$-equivalent $s$-tuples that start before time $n$ inclusive. These results continue the series of our works within the research direction initiated in the 1970s by A. M. Zubkov and other authors.