Matchings in complete bipartite graphs and the r-Lah numbers

We give a graph theoretic interpretation of r-Lah numbers, namely, we show that the r-Lah number ⌊nk⌋r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left\lfloor {\matrix{n \cr k \cr } } \right\rfloor _r}$$\end{document} 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.