We give a graph theoretic interpretation of r-Lah numbers, namely, we show that the r-Lah number \({\left\lfloor {\matrix{n \cr k \cr } } \right\rfloor _r}\) counting the number of r-partitions of an (n + r)-element set into k + r ordered blocks is just equal to the number of matchings consisting of n − k edges in the complete bipartite graph with partite sets of cardinality n and n + 2r − 1 (0 ⩽ k ⩽ n, r ⩾ 1). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for r-Stirling numbers of the second kind.

中文翻译：

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完全二部图中的匹配和 r-Lah 数

我们给出了r -Lah 数的图论解释，即我们证明了r -Lah 数\({\left\lfloor {\matrix{n \cr k \cr } } \right\rfloor _r}\)计数的数目- [R的（的-partitions ñ + - [R ）-元素组成ķ + ř有序块刚好等于由匹配数的数目ñ - ķ与部集的基数的完全二部图的边缘ñ和ñ + 2 r − 1 (0 ⩽ k ⩽ n, r⩾ 1). 我们提出了五个独立的证明，包括一个直接的、双射的证明。最后，我们对第二类r -Stirling 数以类似的结果结束我们的工作。