Enumeration of Restricted Permutation Triples

Let [n] stand for the first n natural numbers {1, 2, · · · , n} and Sn for the permutations of [n]. Given a permutation π = (a1, a2, · · · , an) ∈ Sn, a rise (shortly as “R”) at the kth position refers to ak < ak+1, while a fall (shortly as “F”) at the same position refers to ak > ak+1, where the position index k runs from 1 to n − 1. It is classically well–known (cf. Comtet [2, §6.5]) that the number of the permutations of [n] with exactly k − 1 rises is equal to the Eulerian number A(n, k), which admits the following bivariate generating function 1 + ∑