##### This material has been published in
J. Combin. Theory Ser. A
**74** (1996), 351-354,
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## Christian Krattenthaler

# Combinatorial proof of the log-concavity of the
sequence of matching numbers

### (4 pages)

**Abstract.**
For *k>=l* we construct an injection from the set of
pairs of matchings in a given graph *G* of sizes *l*-1 and *k*+1 into
the set of pairs of matchings in *G* of sizes *l* and *k*. This
provides a combinatorial proof of the log-concavity of the sequence
of matching numbers of a graph. Besides, this injection implies that
a certain weighted version of the matching numbers is strongly
**x**-log-concave in the sense of Sagan (Discrete Math. **99** (1992),
289-306).

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