In this paper we present a combinatorial proof of Selberg's integral formula. We prove a theorem about the number of topological orderings of a certain related directed graph bijectively. Selberg's integral formula then follows by induction. This solves a problem posed by R. Stanley in 2009. Our proof is based on Anderson's analytic proof of the formula. As part of the proof we show a further generalisation of the generalised Vandermonde determinant.

中文翻译：

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塞尔伯格积分公式的组合证明

在本文中，我们提出了塞尔伯格积分公式的组合证明。我们双射地证明了一个关于某个相关有向图的拓扑排序数的定理。塞尔伯格的积分公式然后通过归纳遵循。这解决了 R. Stanley 在 2009 年提出的一个问题。我们的证明基于 Anderson 对公式的解析证明。作为证明的一部分，我们展示了广义 Vandermonde 行列式的进一步推广。