Alternating permutations containing the pattern 123 or 321 exactly once

Abstract: Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the affirmative; in particular, we show that for m > 1, a_(2m)(123) = 10 (2m)!/((m - 2)! (m + 3)!), a_(2m)(321) = 4(m - 2) (2m + 3)!/((m + 1)! (m + 4)!), and a_(2m + 1)(123) = a_(2m + 1)(321) = 3(3m + 4)(m - 1) (2m + 2)!/((m + 1)! (m + 4)!) where a_n(p) is the number of alternating permutations of length n containing the pattern p exactly once.