Exact Markov Probabilities from Oriented Linear Graphs

Using a theorem due to de Bruijn, van Aardenne-Ehrenfest, C. A. B. Smith and Tutte concerning the number of circuits in oriented linear graphs, an expression is found for the probability of a specified frequency count of $m$-tuples in a circular sequence where the $n$-tuple $(n < m)$ count is given. The corresponding result for linear sequences can be deduced--see [14]. The result is valid for stationary Markovity of any order up to and including the $(n - 1)$-st. A method of deriving asymptotic distributions is indicated, and a few additional observations made concerning the distribution of pairs in a circular array.