In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

Let O(pi) denote the number of odd parts in an integer partition p. In 2005, Stanley introduced a new statistic srank(pi) = O(pi)-O(pi'), where pi' is the conjugate of pi. Let p(r, m; n) denote the number of partitions of n with srank congruent to r modulo m. Generating function identities, congruences and inequalities for p(0, 4; n) and p(2, 4; n) were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for p(r, m; n) with m = 16 and 24. These results are refinements of some inequalities due to Swisher.