  The Eulerian numbers count permutations according to the number of descents.
The two-sided Eulerian numbers count permutations according to number of
descents and the number of descents in the inverse permutation. Here we derive
some results for Eulerian and two-sided Eulerian numbers using an elementary
"balls-in-boxes" approach. We also discuss an open conjecture of Ira Gessel
about the two-sided Eulerian numbers.
