Combinatorial Proof of Selberg's Integral Formula

TL;DR

This paper provides a combinatorial proof of Selberg's integral formula, connecting graph theory, induction, and generalizations of the Vandermonde determinant, thereby solving a problem posed by R. Stanley.

Contribution

It introduces a novel combinatorial approach to prove Selberg's integral formula, expanding the understanding of its underlying structures.

Findings

Established a bijective proof for a related topological order theorem

Derived Selberg's integral formula through induction based on combinatorial arguments

Generalized the Vandermonde determinant in the context of the proof

Abstract

In this paper we present a combinatorial proof of Selberg's integral formula. We start by giving a bijective proof of a Theorem about the number of topological orders of a certain related directed graph. Selberg's Integral Formula then follows by induction. This solves a problem posed by R. Stanley in 2008. Our proof is based on Andersons analytic proof of the formula. As part of the proof we show a further generalisation of the generalised Vandermonde determinant.