Counting occurrences of some subword patterns

Counting the number of words which contain a set of given strings as substrings a certain number of times is a classical problem in combinatorics. This problem can, for example, be attacked using the transfer matrix method (see [20, Section 4.7]). In particular, it is a well-known fact that the generating function of such words is always rational. For example, in [20, Example 4.7.5] it is shown that the generating function for the number of words in [3]n where neither 11 nor 23 appear as two consecutive digits is given by 3 + x− x2 1−2x− x2 + x3 . In this paper, we present, in several cases, a complete solution for the problem of the enumeration of words containing a subword pattern (see below for the precise definition) of length l exactly r times. For example, we find the number of words in [3]n containing the subword pattern 111 exactly r times, that is, the number of words which contain 111, 222, and 333 as substrings a total of r times. Régnier and Szpankowski [18] used a combinatorial approach to study the frequency of occurrences of strings (which they also called a “pattern”) from a given set in a random word, when overlapping copies of the “patterns” are counted separately (see [18, Theorem 2.1]). We note that the term “pattern” in [18] is used to denote an exact string rather than its type with respect to order isomorphism. For example, the “pattern” 112 in [18] is the actual string 112, whereas in our setting an occurrence of the subword pattern 112 is any substring aab of the ambient string with a < b. Although, in principle, it is possible to deduce our results from the result by Régnier and Szpankowski, our direct derivations are much simpler.