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Article Dans Une Revue Journal of Combinatorial Theory, Series A Année : 2011

Bijections for Baxter Families and Related Objects

Stefan Felsner
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Marc Noy
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David Orden
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Résumé

The Baxter number can be written as $B_n = \sum_0^n \Theta_{k,n-k-1}$. These numbers have first appeared in the enumeration of so-called Baxter permutations; $B_n$ is the number of Baxter permutations of size $n$, and $\Theta_{k,l}$ is the number of Baxter permutations with $k$ descents and $l$ rises. With a series of bijections we identify several families of combinatorial objects counted by the numbers $\Theta_{k,l}$. Apart from Baxter permutations, these include plane bipolar orientations with $k+2$ vertices and $l+2$ faces, 2-orientations of planar quadrangulations with $k+2$ white and $l+2$ black vertices, certain pairs of binary trees with $k+1$ left and $l+1$ right leaves, and a family of triples of non-intersecting lattice paths. This last family allows us to determine the value of $\Theta_{k,l}$ as an application of the lemma of Gessel and Viennot. The approach also allows us to count certain other subfamilies, e.g., alternating Baxter permutations, objects with symmetries and, via a bijection with a class of plan bipolar orientations also Schnyder woods of triangulations, which are known to be in bijection with 3-orientations
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Dates et versions

hal-00714699 , version 1 (05-07-2012)

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Stefan Felsner, Eric Fusy, Marc Noy, David Orden. Bijections for Baxter Families and Related Objects. Journal of Combinatorial Theory, Series A, 2011, 118 (3), pp.993-1020. ⟨10.1016/j.jcta.2010.03.017⟩. ⟨hal-00714699⟩
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