Bijections for Baxter families and related objects

The Baxter number Bn can be written as Bn = n k=0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1)2 (k + 2) k + ℓ k k + ℓ + 1 k k + ℓ + 2 k . These numbers have first appeared in the enumeration of so-called Baxter permutations; Bn is the number of Baxter permutations of size n, and Θk,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 Θk,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 Θk,ℓ as an application of the Lemma of Lindstr¨om Gessel-Viennot. The approach also allows us to count certain other subfamilies, e.g., alter...

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: Stefan Felsner, Éric Fusy, Marc Noy, David Orden
• Keywords: Baxter Permutations, Bipolar Orientations, JCT 2011, So-called Baxter Permutations