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Bijections for Baxter Families and Related Objects · Around

2008

Bijections for Baxter Families and Related Objects

Felsner, Stefan, Fusy, Éric, Noy, Marc et al.

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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.

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