Combinatorial Problems Related to Geometrically Distributed Random Variables

Motivated from Computer Science problems we consider the following situation (compare [2] and [3]). In these papers the reader will find a more complete description as well as additional references. Let X denote a geometrically distributed random variable, i.e. P{X = k} = pqk−1 for k ∈ N and q = 1 − p. Assume that we have n independent random variables X1, . . . , Xn according to this distribution. The first parameter of interest is the number of left-to-right maxima, where we say that Xi is a left-to-right maximum (in the strict sense) if it is strictly larger than the elements to left. A left-to-right maximum in the loose sense is defined analogously but “larger” is replaced by “larger or equal”. The second parameter of interest is the (horizontal) path length, i.e. the sum of the left-to-right maxima in the loose sense of all the sequences Xi, . . . , Xn, where i is running from 1 to n. Example. Consider the sequence 4, 5, 2, 3, 5. It has 2 left-to-right maxima in the strict sense (4–5) and 3 left-to-right maxima in the loose sense (4–5–5). For the path length we must consider the subsequences