The Mu Pattern In Words

In this paper, we study the distribution of the number of occurrences of the simplest frame pattern, called the mu pattern, in words. Given a word w = w(1)...w(n). {1,..., k}(n), we say that a pair < w(i), w(j)> matches the ae pattern if i < j, w(i) < w(j), and there is no i < k < j such that w(i) <= w(k) <= w(j). We say that < w(i), w(j)> is a trivial mu-match if w(i) + 1 = w(j) and is a nontrivial mu-match if w(i) + 1 < w(j). The main goal of this paper is to study the joint distribution of the number of trivial and nontrivial mu-matches in {1,..., k}*.