Combinatorial problems related to geometrically distributed random variables (Q1382881)

Motivated from computer science problems we consider the following situation (compare [\textit{P. Kirschenhofer} and \textit{H. Prodinger}, The path length of random skip lists, Acta Inf. 31, No. 8, 775-792 (1994)] and [\textit{H. Prodinger}, Discrete Math. 153, No. 1-3, 253-270 (1996; Zbl 0853.60006)]). 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. \(\mathbb{P}\{X=k\} =pq^{k-1}\) for \(k\in\mathbb{N}\) and \(q=1 -p\). Assume that we have \(n\) independent random variables \(X_1, \dots, X_n\) according to this distribution. The first parameter of interest is the number of left-to-right maxima, where we say that \(X_i\) 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 \(X_i, \dots, X_n\), where \(i\) is running from 1 to \(n\).