The visibility parameter for words and permutations

We investigate the visibility parameter, i.e., the number of visible pairs, first for words over a finite alphabet, then for permutations of the finite set {1, 2, ... , n}, and finally for words over an infinite alphabet whose letters occur with geometric probabilities. The results obtained for permutations correct the formula for the expectation obtained in a recent paper by Gutin et al. [Gutin G., Mansour T., Severini S., A characterization of horizontal visibility graphs and combinatorics on words, Phys. A, 2011, 390(12), 2421-2428], and for words over a finite alphabet the formula obtained in the present paper for the expectation is more precise than that obtained in the cited paper. More importantly, we also compute the variance for each case.