A geometric proof of the Brenti--Welker identity

TL;DR

This paper provides a geometric proof of the Brenti--Welker identity by constructing a hypersimplicial subdivision of dilated hypersimplices, offering new geometric insights into the combinatorial identity.

Contribution

It introduces a novel hypersimplicial subdivision approach to geometrically prove the Brenti--Welker identity, connecting combinatorics and geometry.

Findings

Constructed a hypersimplicial subdivision of dilated hypersimplices.

Provided a geometric proof of the Brenti--Welker identity.

Bridged combinatorial identities with geometric constructions.

Abstract

We construct a hypersimplicial subdivision of the $r$-dilation of the $i$-th hypersimplex of dimension $d$ that provides a geometric proof of the Brenti--Welker identity.