Parameter augmentation for two formulas.

Zhang, Caihuan

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A composition of a positive integer n is a finite sequence π1π2...πm of positive integers such that π1+...+πm = n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if πi+1 ≥ πi+d (respectively, πi < πi+1 < πi + 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...