Matchings in complete bipartite graphs and the r-Lah numbers

Abstract

We give a graph theoretic interpretation of r-Lah numbers, namely, we show that the r-Lah number ⌊nk⌋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.