A probabilistic approach to the descent statistic (Q1601427)

Given a permutation \(\sigma= \sigma_1\cdots \sigma_{n+1}\) of \(\{1,\dots, n+1\}\), the decent word \(u= u_1\cdots u_n\) is the word in variables \(a\) and \(b\) with \(u_i= a\) if \(\sigma_i> \sigma_{i+1}\) and \(b\) otherwise. For an \(ab\)-word \(u\), the decent statistic is the number of permutations having \(u\) as decent word. The authors present a probabilistic approach to studying the decent based upon a two-variable probability density deriving quadratic inequalities for the descent statistic. Using Fourier series, they give exact expressions for the Euler numbers and the alternating \(r\)-signed permutations. They also obtain a probabilistic interpretation of the sin function.