Markov Types Again Revisited

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 n over an alphabet of size m. This turns out to coincide with the number of the balanced frequency matrices as well as with the number of special linear diophantine equations, and also balanced directed multigraphs. For fixed m we prove that the number of Markov types is asymptotically equal to d(m) n 2 −m (m −m)! , where d(m) is a constant for which we give an integral representation. For m → ∞ we conclude that asymptotically the number of types is equivalent to √ 2me 2 m2m22mπm/2 n 2 −m provided that m = o(n) (however, our techniques work for m = o( √ n)). We also extend these results to r order Markov sources. These findings are derived by analytical techniques ranging from multidimensional generating functions to the saddle point method. Index Terms – Markov types, integer matrices, linear diophantine equations, multidimensional generating functions, saddle point method. This work was partly supported by the Esprit Basic Research Action No. 7141 (Alcom II). This author was supported by NSF Grant DMS-0800568 and NSA Grant H98230-08-1-0092 The work of this author was supported by the NSF Grants DMS-05-03745, CCF-0513636, DMS-0800568, and CCF-0830140, and NSA Grant H98230-08-1-0092, and the AFOSR Grant FA8655-08-1-3018.