Two-sided Eulerian numbers via balls in boxes
T. Kyle Petersen
TL;DR
This paper introduces a simple balls-in-boxes method to analyze two-sided Eulerian numbers, providing new insights and discussing an open conjecture related to these combinatorial objects.
Contribution
It presents an elementary approach to study two-sided Eulerian numbers and explores an open conjecture in the field.
Findings
Derived new formulas for two-sided Eulerian numbers
Provided elementary combinatorial proofs
Discussed an open conjecture by Ira Gessel
Abstract
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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