  We define a tournament to be alternation acyclic if it does not contain a
cycle in which descents and ascents alternate. Using a result by Athanasiadis
on hyperplane arrangements, we show that these tournaments are counted by the
median Genocchi numbers. By establishing a bijection with objects defined by
Dumont, we show that alternation acyclic tournaments in which at least one
ascent begins at each vertex, except for the largest one, are counted by the
Genocchi numbers of the first kind. Unexpected consequences of our results
include a pair of ordinary generating function formulas for the Genocchi
numbers of both kinds and a new very simple model for the normalized median
Genocchi numbers.
