Permutations with exactly one copy of a monotone pattern of length \(k\), and a generalization (Q2155548)

The authors provide an injection from the set of permutations of length \(n+2\) that contain exactly one copy of the decreasing pattern of length \(k\) to the set of permutations of length \(n+2\) that avoid that pattern. They prove that the generating function counting the former is not rational, and in the case when \(k\) is even and \(k\geq 4\), it is not even algebraic. They extend this injection and the nonrationality result to a larger class of patterns.