Bijections for Baxter families and related objects

Abstract

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.Postprint (published version

Rectangle and Square Representations of Planar Graphs

“…If v is black, the outgoing edge is the last ome in its color in clockwise order (see Figure 9). Separating decompositions have been studied in [dFOdM01], [FHKO10], and [FFNO11]. To us they are of interest because of the following lemma:…”

“…I have decided to include it because it nicely and unexpectedly combines some combinatorial structures. Details can be found in [FFNO11]. To begin with we need some facts.…”

“…The study of the oriented counterpart of these classical bijections goes at least back to [RT86] and was continued in [Ros89], [dFOdMR95], [dFOdM01], and [FFNO11]. Let B be a plane bipolar orientation on black vertices with the root edge (s, t) on the outer face.…”

On the Characterization of Plane Bus Graphs

“…If v is black, the outgoing edge is the last one in its color in clockwise order (see Figure 5). Separating decompositions have been studied in [2], [6], and [5]. In particular it is known that every plane quadrangulation admits a separating decomposition.…”

Uniformly Random Generation of Floorplans

“…An algorithm can be designed to generate a uniformly random mosaic floorplan with f faces in linear time; this can be done by generating uniformly random "watermelon structures" [4], which have a bijection into mosaic floorplans [6]. However, by slightly modifying our algorithm, we can also generate mosaic floorplans with various additional properties that are not possible with the linear-time method.…”