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Two-Sided Eulerian Numbers via Balls in Boxes

T. Kyle Petersen
Mathematics Magazine
Vol. 86, No. 3 (June 2013), pp. 159-176
DOI: 10.4169/math.mag.86.3.159
Stable URL: http://www.jstor.org/stable/10.4169/math.mag.86.3.159
Page Count: 18
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Two-Sided Eulerian Numbers via Balls in Boxes
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Abstract

Summary 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.

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