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A Combinatorial Proof of the Log-Concavity of the Numbers of Permutations with k Runs

✍ Scribed by Miklós Bóna; Richard Ehrenborg


Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
178 KB
Volume
90
Category
Article
ISSN
0097-3165

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Combinatorial Proof of the Log-Concavity
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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 impl

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If a graph G with cycle rank p contains both spanning trees with rn and with n end-vertices, rn < n, then G has at least 2p spanning trees with k end-vertices for each integer k, rn < k < n. Moreover, the lower bound of 2p is best possible. [ l ] and Schuster [4] independently proved that such span