### Abstract:

This thesis is concerned with the probability that a word of length n avoid adjacent patterns. Three topics are investigated. First, the formulae are constructed for the probability that a word of length n avoids given adjacent patterns. In particular, where possible, formulae are constructed for the probability that a word of length n avoids one, two and three letter adjacent patterns. The probability generating functions (if possible) for words avoiding adjacent patterns, is also derived. Since the counting of permutations satisfying various constraints is usually achieved by exponential generating functions (EGF), therefore derive g-analogues of the EGFare derived. A g-analogue of the EGF for the number of words of length n avoiding an adjacent 123... k pattern, k &qe; 2, will also be derived. Next, permutations of length n avoiding adjacent patternsare considered. This is a consequence of the probability that a word of length n avoid adjacent pattern, deriveed earlier. For the sake of completeness, the results regarding permutations in S<sub>n</sub> avoiding four or five adjacent patterns are included. Finally, permutations satisfying various constraints are considered. These will eventually give rise to Olivier functions. Therefore some q-analogues analogues of Olivier functions will be derived.