Ascents of size less than d in compositions

A composition of a positive integer n is a finite sequence π₁π₂...π_m of positive integers such that π₁+...+π_m = n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if π_i+1 ≥ π_i+d (respectively, π_i < π_i+1 < π_i + d). Recently, Brennan and Knopfmacher determined the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. In this paper, we find an explicit formula for the multi-variable generating function for the number of compositions of n according to the number of parts, ascents of size d or more, ascents of size less than d, descents and levels. Also, we extend the results of Brennan and Knopfmacher to the case of ascents of size less than d. More precisely, we determine the mean, variance and limiting distribution of the number of ascents of size less than d in the set of compositions of n.