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We give a graph theoretic interpretation of -Lah numbers, namely, we show that the -Lah number counting the number of -partitions of an -element set into ordered blocks is just equal to the number of matchings consisting of edges in the complete bipartite graph with partite sets of cardinality and (, ). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for -Stirling numbers of the second kind.
We introduce the notion of a matroid over a commutative ring , assigning to every subset of the ground set an -module according to some axioms. When is a field, we recover matroids. When , and when is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, i.e. tropical linear spaces, respectively. More generally, whenever is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and...
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