  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.
