Quadrant marked mesh patterns in alternating permutations

This paper is a continuation of the systematic study of the distribution of quadrant marked mesh patterns initiated in [J. Integer Sequences, 12 (2012), Article 12.4.7]. We study quadrant marked mesh patterns on up-down and down-up permutations, also known as alternating and reverse alternating permutations, respectively. In particular, we refine classical enumeration results of André [C. R. Acad. Sci. Paris 88 (1879), 965-967; J. Math. Pur. Appl. 7 (1881), 167-184] on alternating permutations by showing that the distribution with respect to the quadrant marked mesh pattern of interest is given by (sec(xt))1/x on up-down permutations of even length and by $ \int_0^t (\sec(xz))^{1+\frac{1}{x}}dz$ on down-up permutations of odd length.