Gábor Hetyei
Published 2017-04-24Version 1
We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. By establishing a bijection with objects defined by Dumont, we show that alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind. Unexpected consequences of our results include a pair of ordinary generating function formulas for the Genocchi numbers of both kinds and a new very simple model for the normalized median Genocchi numbers.
Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind
On an alternating sum of factorials and Stirling numbers of the first kind: trees, lattices, and games
Two new identities involving the Bernoulli numbers, the Euler numbers, and the Stirling numbers of the first kind
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