Eulerian-Catalan Numbers

TL;DR

This paper demonstrates that Eulerian-Catalan numbers count Dyck permutations, providing two proofs: one geometric involving alcoved polytopes and one combinatorial via an Eulerian-Catalan Chung-Feller theorem analogue.

Contribution

It introduces a novel combinatorial interpretation of Eulerian-Catalan numbers and offers two distinct proofs, enriching the understanding of their combinatorial significance.

Findings

Eulerian-Catalan numbers enumerate Dyck permutations

Two proofs provided: geometric and combinatorial

Establishment of an Eulerian-Catalan Chung-Feller theorem analogue

Abstract

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.