  A classical result of Euler states that the tangent numbers are an
alternating sum of Eulerian numbers. A dual result of Roselle states that the
secant numbers can be obtained by a signed enumeration of derangements. We show
that both identities can be refined with the following statistics: the number
of crossings in permutations and derangements, and the number of patterns 31-2
in alternating permutations.
  Using previous results of Corteel, Rubey, Prellberg, and the author, we
derive closed formulas for both q-tangent and q-secant numbers. There are two
different methods to obtain these formulas: one with permutation tableaux and
one with weighted Motzkin paths (Laguerre histories).
