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

Two-sided Eulerian numbers via balls in boxes