New permutation statistics: Variation and a variant (Q1026169)

The authors introduce the variation of a permutation \(\pi=\pi_1\ldots\pi_n\) as the total number of elements \(\pi_j\) to the right of some \(\pi_i\pi_{i+1}\) such that \(\pi_i<\pi_j<\pi_{i+1}\) or \(\pi_i>\pi_j>\pi_{i+1}\) and give an expression for the generating function of the number of permutations \(\pi\in S_n\) regarding their variation. (In terms of generalized permutation patterns, the variation counts the number of all occurrences of \(13-2\) and \(31-2\) in \(\pi\).)