Abstract. For words of length n, generated by independent geometric random variables, we consider the mean and variance of the number of inversions and of a parameter of Knuth from permutation in situ. In this way, q–analogues for these parameters from the usual permutation model are obtained. 1.

For words of length n, generated by independent geometric random variables, we consider the average and variance of the number of distinct values (= letters) that occur in the word. We then generalise this to the number of values which occur at least b times in the word. 1.

For words of length n, generated by independent geometric random variables, we consider the mean and variance, and thereafter the distribution of the number of runs of equal letters in the words. In addition, we consider the mean length of a run as well as the length of the longest run over all words of length n. 1.

Abstract. We study the asymptotic probability that a random composition of an integer n is gap-free, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distributed random variables with a geometric distribution is likewise gap-free. 1. introduction A composition of a natural number n is said to be gap-free if the part sizes occuring in it form an interval. In addition if the interval starts at 1, the composition is said to be complete. Example Of the 32 compositions of n = 6, there are 21 gap-free compositions arising from permuting the order of the parts of the partitions

For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.

Abstract. For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth left–to–right maximum counted from the right, for fixed r and n → ∞. This complements previous research [5] where the analogous questions were considered for the rth left–to–right maximum counted from the left. 1.