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{\bf Hoda Bidkhori and Seth Sullivant}
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{\bf Eulerian-Catalan Numbers}
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We show that the Eulerian-Catalan numbers enumerate Dyck permutations.   We provide two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analogue of the Chung-Feller theorem.

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