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Matchings in complete bipartite graphs and the r -Lah numbers

Gábor NyulGabriella Rácz — 2021

Czechoslovak Mathematical Journal

We give a graph theoretic interpretation of r -Lah numbers, namely, we show that the r -Lah number n k 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 + 2 r - 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.

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