The Eulerian numbers can D.I.E

TL;DR

The paper develops two complementary D.I.E. sign-reversing involution proofs of alternating sign identities for Eulerian numbers. The first uses anchored barred permutations and extends to $P$-Eulerian numbers via $P$-partitions, connecting to lattice-point counts in order polytopes. The second develops a topological perspective with decorated set compositions and abstract simplicial complexes, relating Eulerian numbers to the $h$-vector of partitionable complexes and to Stirling numbers through a barycentric subdivision framework. Together, these results illustrate the versatility of the D.I.E. method in combinatorics and topology, yielding multiple equivalent expressions for Eulerian numbers and their generalizations. The work reinforces Eulerian numbers as a nexus between permutation statistics, posets, and topological combinatorics.

Abstract

The Eulerian numbers form a triangular array with many interesting properties. The numbers arise from various combinatorial and probabilistic interpretations, and have been studied in a variety of mathematical contexts. In this article we examine two distinct alternating sign formulas for the Eulerian numbers and show how they can be proved using a sign-reversing involution technique described by Benjamin and Quinn known as the ``D.I.E.'' method. Each of these arguments lends itself to a broad generalization, shedding light on different parts of mathematics.

Figures (7)

• Figure 1: The Hasse diagram of a poset $P$ and its linear extensions.
• Figure 2: The Hasse diagram of a poset $P$.
• Figure 3: Two visual representations of a nonpure simplicial complex.
• Figure 4: A pure partitionable simplicial complex.
• Figure 5: A $2$-dimensional simplex $\Sigma_3$ and its barycentric subdivision $\Delta_3$.