Laguerre polynomials, weighted derangements, and positivity
SIAM Journal on Discrete Mathematics
On the cover polynomial of a digraph
Journal of Combinatorial Theory Series B
Variations sur le thème E+E=XY
Advances in Applied Mathematics
Matching Polynomials And Duality
Combinatorica
The algebra of set functions I: The product theorem and duality
European Journal of Combinatorics
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Triesch (1997) [25] conjectured that Hall's classical theorem on matchings in bipartite graphs is a special case of a phenomenon of monotonicity for the number of matchings in such graphs. We prove this conjecture for all graphs with sufficiently many edges by deriving an explicit monotonic formula counting matchings in bipartite graphs. This formula follows from a general duality theory which we develop for counting matchings. Moreover, we make use of generating functions for set functions as introduced by Lass [20], and we show how they are useful for counting matchings in bipartite graphs in many different ways.