Counting Markov Types, Balanced Matrices, and Eulerian Graphs

The method of types is one of the most popular techniques in information theory and combinatorics. Two sequences of equal length have the same type if they have identical empirical distributions. In this paper, we focus on Markov types, that is, sequences generated by a Markov source (of order one). We note that sequences having the same Markov type share the same so-called balanced frequency matrix that counts the number of distinct pairs of symbols. We enumerate the number of Markov types for sequences of length over an alphabet of size . This turns out to be asymptotically equivalent to estimating the number of the balanced frequency matrices, the number of integer solutions of a system of linear Diophantine equations, and the number of connected Eulerian multigraphs. For fixed , we prove that the number of Markov types is asymptotically equal to d(m) n
^m
2
-m
/(m
²
-m)! where we give an integral representation for d(m). For m →∞, we conclude that asymptotically the number of types is equivalent to √2m
^3m/2
e
^m
2
/m
^2m2
2
^m
π
^m/2
n
^m2
-m provided that m = o(n
^1/4
). These findings are derived by analytical techniques ranging from analytic combinatorics, to multidimensional generating functions, to the saddle point method.