1. Introduction
This article is concerned with the enumerative properties of two fascinating families of discrete geometric structures, corner polyhedra and rigid orthogonal surfaces. Corner polyhedra, see Figure 2(a), were introduced by Eppstein and Mumford [eppstein2010steinitz] who were interested in the possibility to give an elegant characterization à la Steinitz of the graphs that can be realized as 1skeleton for certain classes of orthogonal polyhedra. On the other hand, rigid orthogonal surfaces, see Figure 3(a), were considered by Felsner [felsner2003geodesic] in relation with the order dimension of 3polytopes.
It turns out that these geometric structures can be described in a very similar way by certain underlying combinatorial structures, polyhedral orientations for corner polyhedra, see Figure 2(b), and (connected) Schnyder labelings for rigid orthogonal surfaces, see Figure 3(b). As illustrated by Figure 1, and as already partially observed by several authors, e.g. [eppstein2010], these combinatorial counterparts are similar to those that were already observed for horizontal/vertical contact segments and rectangular tilings.
After recalling in Section 2 the definition of polyhedral orientations and Schnyder labelings, and how to recast them in terms of bipolar orientations, we move on in Section 3 to set up exact correspondences with certain weighted bimodal models of socalled tandem quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson [kenyon2019bipolar]. The resulting bijections, stated as Proposition 1 and Proposition 2, allow us to describe in Section 4 polynomial time algorithms to count these structures, and moreover to determine in Theorem 2 their exact asymptotic growth constants, see also Figure 1. An exact enumeration formula is known for the case of Schnyder labelings on triangulations [bernardi2009intervals]; we show in Section 3.3 that it can be recovered from our results.
As can be observed in Figure 1, our results on the enumeration of corner polyhedra and 3connected Schnyder labelings parallel those for the number of plane bipolar orientations, and for the number of transversal structures. However the analysis is made more difficult by the fact that the tandem walks that we have to deal with have a bimodal behavior: the step set available at a current point depends on the parity of the ordinate of this point. This puts the complete analysis of the asymptotic behavior out of reach of our current understanding of these models, based on DenisovWachtel approach [denisov2015random]. Resorting to a plausible but conjectural version of their argument, we are able to state Conjecture 1 on the polynomial corrections, which would imply that the associated series are not Dfinite.
A nice final touch on the emerging global picture is the possibility to recast these results in terms of colored pseudoline contact systems (before last line in Figure 1), as explained in Section 5.
horizontal/vertical  rectangular tilings  corner polyhedra  rigid orthogonal 
contact segments ([Tamassia])  ([kant1997regular])  ([eppstein2010steinitz], Sec 1)  surfaces ([felsner2003geodesic], Sec 1) 
separating decompositions  transversal structures  polyhedral orientations  Schnyder labelings 
([de1995bipolar])  ([kant1997regular, fusy2009transversal])  ([eppstein2010steinitz], Sec 2.1)  ([felsner2000convex], Sec 2.1) 
plane bipolar  transverse bipolar  admissible bipolar  transverse bipolar 
orientations ([de1995bipolar])  orientations ([Na20])  orientations (Sec 2.3)  orientations (Sec 2.3) 
tandem walks  admissible tandem  admissible tandem  admissible tandem 
([Na20])  walks ([Na20])  walks (Sec 3.2)  walks (Sec 3.2) 
free bicolored  rigid bicolored  free tricolored  rigid tricolored 
contactsystems (Sec 5)  contactsystems (Sec 5)  contactsystems (Sec 5)  contactsystems (Sec 5) 
([baxter2001dichromatic])  ([Na20])  (Sec 4.2)  (Sec 4.2) 
2. Presentation of the two models
2.1. Definitions
A planar map is a connected multigraph embedded on the oriented sphere up to orientationpreserving homeomorphism. It is rooted by marking a corner, whose incident face is taken as the outer face in planar representations. Vertices and edges are called inner or outer depending on whether they are incident to the outer face or not. A map is called Eulerian if its vertices have even degree, in which case the faces can be uniquely bicolored in light and dark faces so that the outer face is light (any edge has a dark face on one side and a light face on the other side). Dually, a map is bipartite iff all faces have even degree, in which case the vertex bicoloration is unique, up to choosing the color of a given vertex.
A triangulation is a planar map where all faces have degree . It is known that a triangulation is 3colorable iff it is Eulerian. In that case, the coloration of vertices (say in blue, green, red) is unique once the colors around a given triangle are fixed. If the triangulation is rooted, we take the convention that the rootvertex is red, and the outer vertices are colored red, green, blue in clockwise order around the outer face (i.e., walking along the outer contour with the outer face on the left). Note that every edge is also canonically colored red, green, or blue: it receives the color it misses (e.g. an edge connecting a green vertex and a blue vertex is colored red). In an orientation of a planar map, a corner is called lateral if exactly one of is ingoing at (the other one being outgoing), it is called extremal otherwise (either are both ingoing or both outgoing at ). For a rooted Eulerian triangulation, a polyhedral orientation of is an orientation of such that (see Figure 2(c) for an example):

There is no extremal corner at the outer vertices, and the outer contour is a cw cycle.

Every inner vertex is incident to exactly two extremal corners, and all the extremal corners are incident to light faces (hence dark face contours are either cw or ccw).
Remark 1.
Based on a counting argument, it can be checked that there must be exactly two extremal corners in every inner light face (indeed, every inner light face has either zero or two extremal corners; but by the Euler relation the number of inner light faces equals the number of inner vertices, so that the number of extremal corners is twice the number of inner light faces).
Remark 2.
Not every Eulerian triangulation admits a polyhedral orientation: in fact it is the case if and only if all its red/blue/green ccw triangles are facial, as first shown in [eppstein2010steinitz]. These socalled corner triangulations are enumerated in [dervieux].
From now on, we call polyhedral orientation a (corner) triangulation endowed with a polyhedral orientation. We let be the set of polyhedral orientations, and let (resp. ) be the set of polyhedral orientations with inner vertices (resp. with red inner vertices, blue inner vertices, and green inner vertices).
A dissection is a planar map whose outer face is a simple cycle of length , and whose inner faces have degree . Such a map is bipartite, and if rooted (which is assumed here) the vertexbicoloration (in black and white vertices) is the unique one such that the rootvertex is white. The outer vertices are labeled in ccw order around the outer face, starting with the rootvertex. An outer vertex is called isolated if it has degree (i.e., is not incident to an inner edge). A Schnyder labeling of is a coloration of the inner edges of in blue, green, red, such that (see Figure 3(c)):

The two outer vertices labeled (resp. ) have their incident inner edges red (resp. blue, green).

The edges at each inner vertex form, in cw order, 3 nonempty groups: of red, green and blue edges.
Remark 3.
It is known [felsner2000convex, fusy2008dissections] that a dissection admits a Schnyder labeling iff it has no multiple edge and every cycle delimits a face. These dissections are counted in [mullin1968enumeration, fusy2008dissections, bouttier2014irreducible].
Remark 4.
One can classically associate to a planar (essentially connected) map , which is obtained from by adding in each inner face an edge that connects the two opposite white vertices, and then erasing all edges and black vertices of . Via this mapping, our definition of Schnyder labelings matches the one of Felsner [felsner2000convex] (precisely, he considers those corresponding to the case where the outer white vertices are nonisolated), see Figure 4.
From now on, we call Schnyder labeling a dissection endowed with a Schnyder labeling. We let be the set of Schnyder labelings, and let be the subset of those where the outer white vertices are nonisolated. We let (resp. ) be the set of elements in (resp. ) with inner faces, and let (resp. ) be the set of elements in (resp. ) with white vertices and black vertices. The Euler relation ensures that a dissection with inner faces has vertices, hence (resp. ).
Remark 5.
Based on a counting argument [felsner2000convex, Lem.1], it can be checked that, for , if the outer edges incident to the black vertex labeled (resp. , ) are colored red (resp. green, blue), then the contour of every inner face has at least one edge in every color, with one of the three colors represented by two edges that are consecutive along the face contour.
2.2. Link to 3D representations
A simple orthogonal polyhedron [eppstein2010steinitz] is a 3D shape with the topology of a ball, and whose boundary is made of flat portions (called flats), each flat being orthogonal to one of the coordinate axes. It is also required that if two flats share a boundary then their orthogonal directions are not the same, and that at most flats can intersect at a point on the boundary. The boundary yields a trivalent map (embedded on a topological 2Dsphere), whose vertices correspond to the boundary points where flats meet, and whose faces correspond to the flats. A corner polyhedron (see Figure 2(a) for an example) is a simple orthogonal polyhedron where exactly
flats have a negative coordinate in their orthogonal vector. These
flats have to intersect, and the intersection is taken as the origin. The dual of the associated trivalent map is an Eulerian triangulation (a corner triangulation), whose outer face is taken as the one dual to the origin, with red (resp. blue, green) vertices associated to the flats whose orthogonal direction is the one of the axis (resp. axis, axis). Moreover, is naturally endowed with a polyhedral orientation, upon orienting every inner blue (resp. green, red) edge in the direction of increasing (resp. , ), see Figure 2(b). It is also shown in [eppstein2010steinitz] that every polyhedral orientation can be obtained in this way. Polyhedral orientations can thus be considered as the combinatorial types of corner polyhedra. Thus, (resp. ) gives the number of combinatorial types of corner polyhedra with facets (resp. with facets, among which are red, are blue, and are green), including the nonvisible facets.The union of the boundary of a corner polyhedron and of the three 2Dquadrants corresponds to a (nondegenerate, axial) orthogonal surface in the sense of [felsner2003geodesic]. There is another natural way to associate a decorated map to such a surface [felsner2003geodesic]. Let (resp. ) be the extremal vertex of on the axis (resp. axis, axis). Let , and . The white points of are and the points of (excluding the origin), called inward points of , where flats meet and the border rays leave the point in the coordinateincreasing way. The black points of are and the points of , called outward points of , where flats meet and the border rays leave the point in the coordinatedecreasing way. A black point and a white point are adjacent if coordinatewise.
This adjacency relation yields a bipartite graph formed by black and white points that is drawn on (with edges as segments). The surface is called rigid if the drawing is crossingfree (the edges can meet only at common extremities); the example of Figure 2(a) is nonrigid (there would be a crossing in the blue flat on the right) whereas the one in Figure 3(a) is rigid. In that case, every edge is such that share exactly one coordinate. Then is considered blue (resp. green, red) if and have same coordinate (resp. coordinate, coordinate). Upon adding the outer hexagon (labeled ), this embedded graph exactly gives a Schnyder labeling in , see Figure 3(b). It is shown in [felsner2003geodesic, felsner2008schnyder] that every Schnyder labeling in can be obtained in this way. Schnyder labelings in can thus be considered as the combinatorial types of rigid orthogonal surfaces. The coefficient gives the number of combinatorial types of rigid orthogonal surfaces that arise from corner polyhedra with flats. The coefficient gives the number of combinatorial types of rigid orthogonal surfaces that arise from corner polyhedra with inward points and outward points.
Remark 6.
It would also be possible to consider the bivariate refinement (where parameters would have the same meaning as in ) and the trivariate refinement (where parameters would have the same meaning as in ). However, we feel that the trivariate refinement is more natural for polyhedral orientations (the parameters are intrinsic to the underlying corner triangulation, they do not depend on which polyhedral orientation the triangulation is endowed with). Similarly, the bivariate refinement seems more natural for Schnyder labelings (the parameters are intrinsic to the underlying dissection).
2.3. Encoding by (constrained, decorated) plane bipolar orientations
A plane bipolar orientation is a planar map endowed with an acyclic orientation with a unique source and a unique sink that are both incident to the outer face. It is known [de1995bipolar] that a plane bipolar orientation is characterized by the following local properties, illustrated in Figure 5:

Apart from , each vertex has two lateral corners (so the incident edges form two groups: ingoing and outgoing edges).

Each face (including the outer one) has two extremal corners, so that the contour is partitioned into a left lateral path and a right lateral path that share their origins and ends, which are called the bottom vertex and top vertex of the face.
The type of a face is the integer pair such that the left (resp. right) lateral path of the face has length (resp. ). The outer type of the orientation is the type of the outer face.
If the underlying map of the orientation is bipartite, i.e., the type of every inner face is such that is even, then the vertex bicoloration is chosen such that is white. An inner face is called a blacktip face (resp. whitetip face) if its top vertex is black (resp. white).
A bipartite plane bipolar orientation is called admissible iff

it has outer type for some even ,

the type of every blacktip (resp. whitetip) inner face is such that (resp. ).
Proposition 1.
Polyhedral orientations with inner vertices, among which are red, are blue, and are green, are in bijection with admissible bipolar orientations with edges, white vertices, black vertices, and inner faces.
Proof.
From a polyhedral orientation , its image is obtained by removing all the green vertices, and recoloring the blue vertices as black, and the red vertices as white, see Figure 6. Clearly, is bipartite, and it forms a plane bipolar orientation [eppstein2010steinitz, Lem 18] whose source (resp. sink) corresponds to the outer blue (resp. red) vertex of . Moreover, has outer type , with the degree of the green outer vertex of . Hence, (PA1) holds for . Note that every corner in correspond to exactly two corners in , respectively in a light and in a dark triangle. The light one is called the attached corner of . (This gives a 1to1 correspondence between the corners of and the light corners at blue or red vertices in .) Moreover, since the dark corners are always lateral, a corner in is extremal iff its attached corner is lateral. For a green inner vertex of , and with the corresponding inner face of , the corner attached to the top corner (resp. bottom corner) of has thus to be lateral. Hence, by Remark 1, the corner at in the same light triangle has to be extremal. Thus, the two extremal corners of are those that are in the light triangle (resp. ) touching the topvertex (resp. the bottomvertex) of . The fact that these light triangles are different easily ensures that the left contour of contains an edge from a black to a white vertex, and the right contour of contains an edge from a white to a black vertex (see Figure 7 showing the cases, depending on the colors of the topvertex and bottomvertex of ). This is equivalent to Condition (PA2) being satisfied by . Hence, is admissible.
Now, from a admissible bipolar orientation , its image is constructed as follows:

[label=()]

Add a socalled facevertex in every face of , and connect to all corners around .

Recolor all black (resp. white) vertices in blue (resp. in red); and all facevertices in green.

Let be the facevertex of ’s outer face: we orient its incident edges toward blue (former black) vertices and away from red (former white) vertices. In particular, the edge between and the sink (resp. source) vertex of goes toward (resp. away from) and edge directions alternate around .

For every inner face of , we mark every corner attached to a lateral corner in and we mark the two corners of incident to the two light triangles incident to the bottom and top corners of (these two light triangles are distinct, due to (PA2)). Then there is exactly one way to orient blue and red edges so that the extremal corners within are the marked ones, see Figure 7.
The bipartite nature of and 2 ensures that must be 3colorable, and (PO1) follows from 3. By construction, all dark corners are lateral, and (PO2) is satisfied at all inner green vertices. As for red and blue inner vertices, the construction is again such that a corner of is extremal iff the attached corner in is lateral. Since each nonpole vertex of has two lateral corners, the corresponding red or blue inner vertex in must have two extremal corners. Hence, (PO2) is satisfied at red and blue inner vertices.
The two mappings are clearly inverse to each other, hence give a bijection. ∎
An transverse bipolar orientation is a dissection , where the edges are partitioned into plain edges that are directed, and transversal edges that are undirected, so that the following conditions are satisfied:

Plain edges span all vertices of , and form a (bipartite) plane bipolar orientation of outer type , with at least one inner face.

Each transversal edge is within an inner face of , and it connects a black vertex in the interior of the left lateral path of and a white vertex in the interior of the right lateral path of . Moreover, in every inner face , every black (resp. white) vertex in the interior of the left (resp. right) lateral of is incident to at least one transversal edge.
Proposition 2.
Schnyder labelings with inner faces, white vertices and black vertices, and whose two outer vertices are nonisolated, are in bijection with transverse bipolar orientations with vertices, among which are white and are black.
Proof.
The bijection is defined as follows. Given a Schnyder labeling whose outer vertices are nonisolated, let (resp. ) be the outer black (resp. white) vertex labeled . The left (resp. right) lateral path of is the path of outer edges from to with the outer face on its left (resp. right). We color (red, blue, red) the edges on the left (resp. right) lateral path. We orient the red edges from black to white and the blue edges from white to black (in particular, the left lateral path and right lateral path are directed from to ), and leave the green edges undirected. We claim that we obtain an transverse bipolar orientation, with source and sink . The main point is to check that the oriented map formed by the oriented edges (which spans all the vertices of ) is acyclic.
Assume that has a directed cycle , and consider a minimal one, i.e., whose interior does not contain the interior of another directed cycle of . The local conditions (SL1)(SL2) imply that if is clockwise (resp. counterclockwise), then any incidence of a transversal edge in with a vertex on must be such that is black (resp. white), and moreover every black (resp. white) vertex on must be incident to at least one transversal edge in . Hence, if is clockwise (resp. counterclockwise), then there must be a transversal edge leaving a black vertex on and whose other extremity is strictly in . Note that are the only extremal vertices of , and they are exterior to . Hence, from starts a path of outgoing edges (the next edge at each step is an outgoing edge of the current vertex), which can not loop by minimality of , hence has to reach at some vertex . Similarly, from starts a path of ingoing edges (the next edge at each step is an ingoing edge of the current vertex), which can not loop by minimality of , hence has to reach at some vertex . These two paths, together with the path on connecting to , form a directed cycle whose interior is in , a contradiction.
Hence, is acyclic. Since it has a single source and a single sink (both incident to the outer face), it is a plane bipolar orientation, and clearly its left and right outer paths are the left and right outer paths of defined above. Hence, has outer type , and every transversal edge is within an inner face of . Moreover, the local conditions (SL1)(SL2) easily ensure that every transversal edge within an inner face of has to connect a black vertex in the interior of the left lateral path of to a white vertex in the interior of the right lateral path of , and that every black (resp. white) vertex in the interior of the left (resp. right) lateral path of is incident to at least one transversal edge (the fact that in the two outer are nonisolated is necessary to have this property satisfied at these two vertices).
The inverse mapping is defined as follows. Starting from an transverse bipolar orientation, with the part made by the plane bipolar orientation, we color green the transversal edges, and color red (resp. blue) the edges of that are directed from black to white (resp. from white to black), and we forget the edge directions and the colors of the outer edges. The outer vertices of the obtained edgecolored map are labeled in counterclockwise order around the outer face, starting from the sink of . Condition (B) of plane bipolar orientations and condition (ST2) imply that condition (SL2) is satisfied, and moreover that the two outer vertices are incident to at least one transversal edge, and every incidence of a transversal edge with an outer vertex must be with one of the two outer vertices labeled . Condition (ST2) also implies that every inner face of has at least one black vertex in the interior of its left lateral path and at least one white vertex in the interior of its right lateral path. This easily implies that the left outer has indegree and outdegree in , so that in all inner edges incident to the left outer (and similarly, the right outer ) are green. Similarly, the left outer has indegree in , so that in all inner edges incident to this vertex (and similarly, to the outer on the right side) are blue. And all inner edges incident to one of the two outer vertices labeled (which are the source and sink of the bipolar orientation) are red. Hence, Condition (SL1) is satisfied, so that is a Schnyder labeling (whose two outer are nonisolated).
Finally, the two mappings are clearly inverse to each other, hence give a bijection. ∎
Remark 7.
For enumerative purposes, the constraint that the two outer vertices labeled are nonisolated is mild. Indeed, if denotes the number of Schnyder labelings with inner faces (with ), and denotes the number of those where the two outer vertices labeled are nonisolated, then for (the three terms correspond to having , , or isolated vertices among the two outer Green ). Similarly, if denotes the number of Schnyder labelings with white vertices and black vertices, and denotes the number of those where the two outer vertices labeled are nonisolated, then for and .
Remark 8.
Let be an transverse bipolar orientation and let be an inner face of , with the quadrangular faces within , ordered from bottom to top. Let be the path from the first to the last black vertex on the strict left boundary of (i.e., excluding the extremal vertices of ), and let be its length. Let be the path from the first to the last white vertex on the strict right boundary of , and let be its length. It is easy to see that for , either has two edges on and none on , or the opposite. We can thus attach to a word in giving the types of ( if the face has two edges on , otherwise). It completely encodes the configuration of transversal edges within , and any such word is a valid encoding. Hence the configuration can be encoded by an integer in . Note also that the degree of is in all cases (in particular, all inner faces of have degree at least ).
3. Bijections with walks in the quadrant
Similarly as in [Na20], once our models have been set in bijection to certain models of plane bipolar orientations, they can be set in bijection to specific quadrant walks by specializing a bijection due to Kenyon, Miller, Sheffield and Wilson (shortly called the KMSW bijection), which we use as a bijective black box. An example of the KMSW bijction is given on Figure 9.
3.1. KMSW bijection
A tandem walk is a walk on the lattice , with steps in . A step that is not a SE step (i.e., a step of the form ) is called a facestep.
Theorem 1 ([kenyon2019bipolar]).
There is a bijection between plane bipolar orientations of outer type and tandem walks from to staying in the quadrant . For a plane bipolar orientations and the corresponding tandem walk, the number of edges of corresponds to one plus the length of , each inner face of type in corresponds to a facestep in , and each nonpole vertex corresponds to a SE step of .
Remark 9.
The bijection is easy to specialize to the bipartite setting (we will use the bijection in this setting only). A plane bipolar orientation is bipartite iff in the corresponding walk, each facestep is such that is even; such a tandem walk is called even. Moreover, the nonpole white and black vertices of correspond to the SE steps that start at even
and odd
, respectively (this is due to the property that the where the step starts indicates a pathlength in between and the vertex corresponding to the step). Similarly, whitetip inner faces and blacktip inner faces correspond to facesteps that start at even and at odd , respectively, see Figure 9 for an example.3.2. Application to the two models
We first specialize the KMSW bijection (in the bipartite setting) to the admissible bipolar orientations. A admissible tandem walk is an even tandem walk where every facestep starting at even (resp. odd) has (resp. ). Via Proposition 1 we obtain:
Proposition 3.
Polyhedral orientations are in bijection with admissible quadrant tandem walks starting at the origin and ending on the axis. If the polyhedral orientation has inner vertices, among which are red, are blue, and are green, then the corresponding admissible tandem walk has length , with SE steps starting at even , SE steps starting at odd , and facesteps.
We then specialize the KMSW bijection to the Stransverse bipolar orientations. For this (given Remark 8), we need a weighted terminology: a step in a tandem walk is said to be weighted by if comes with an integer in (for the enumeration, the weights of the steps composing the walk have to be multiplied, those where no weight is indicated are implicitly assumed to have weight ). An admissible tandem walk is defined as an even tandem walk such that every facestep with even entries is of the form and is weighted by , every facestep with odd entries and starting at even is of the form and is weighted by , and every facestep with odd entries and starting at odd is of the form and is weighted by . Via Proposition 2 and Remark 8, we obtain:
Proposition 4.
Schnyder labelings whose two outer vertices are nonisolated are in bijection with admissible tandem walks in the quadrant that start at and end at , and have length at least (i.e., have at least one facestep). If the Schnyder labeling has inner faces, with white vertices and black vertices, then the corresponding admissible tandem walk has SE steps, among which start at even , and start at odd .
3.3. Specialization of Proposition 4 to triangulations
It is well known that Schnyder labelings on triangulations with vertices are in bijection to noncrossing pairs of Dyck walks of length . We explain here how this bijective result can be recovered from Proposition 4. Let be the set of Schnyder labelings with white vertices and black vertices, and let be the subset of those with no outer isolated. We define a 1aligned tandem walk as a tandem walk such that all facesteps have . Let be the set of 1aligned quadrant tandem walks starting at the origin and ending on the axis, with SE steps (note that the number of facesteps must also be , and the length is ).
Lemma 1.
The family is empty for . For , it is in bijection with .
Proof.
By Proposition 4, is in bijection with the set of admissible quadrant tandem walks from to having (resp. ) SE steps starting at even (resp. odd) . Let be such a walk. An easy case inspection ensures that the first two steps of have to be SE steps. Let be deprived from the first two steps (hence, starts at ). The projection of to the axis yields a 1D nonnegative walk from to with rising steps (corresponding to facesteps of ) and downsteps (corresponding to SE steps of ). Seen as a directed 2D walk, is thus a (maximal non negative prefix of) Lukasiewicz walk. A step of is called even or odd whether it starts at even or odd . By the conditions on , has even downsteps and odd downsteps, and moreover each rising step has increment if is even or has increment if is odd.
Note that each downstep of going ordinate to is classically associated with the closest preceding rising step covering these ordinates. For a rising step , we let and be respectively the numbers of even and odd downsteps associated with (so that ), and we let . Given the condition on rising steps of , it is easy to see that , and that iff is an even step. Moreover, an easy case inspection ensures that has to end with two SE steps, which have to be preceded by a step of the form or for some . The corresponding rising step in has (and ) in the first case, and (and ) in the second case. Let be the sum of over the rising steps of . Note that . Moreover, the preceding discussion ensures that (i.e., must be at most , as claimed), and (extremal case ) iff all rising steps are even steps, except for the last rising step that is an even step starting from the horizontal axis. We conclude that if is counted by then it starts with two SE steps, all its facesteps start at even , and they have except for the last one that has to reach the point (before the final two SE steps to reach ). Note also that all facesteps in such a walk must have weight .
From , it is easy to produce such a walk as follows (we specify it by the sequence of steps, and the starting point, that has to be ): turn each SE step into two successive SE steps (combined, these have the effect of a step ), and turn each step into two consecutive steps: a step followed by a SE step (combined, these have the effect of a step ). Finally, if the walk ends at , append the three steps , prepend two SE steps, and choose the starting point at . (The mapping is such that every facestep starting from in becomes a facestep starting from in the corresponding walk .) It is also easy to check that the mapping is bijective (every walk with can be obtained in that way). ∎
Remark 10.
Another way to see that is empty for is to argue via vertexdegrees. For every inner black vertex and the outer black vertex labeled have degree at least , while the two other outer black vertices have degree at least . Moreover, by the Euler relation, the number of edges is . Hence, , giving . In the extremal case , all the inequalities have to be tight. Similarly, for the inequalities are the same, except that the black outer is allowed to have degree . Hence, the extremal case is , in which case all inner black vertices have degree , and the outer black vertices have degree . There is an easy bijection from to that consists in removing the outer black vertex and its two incident edges. Moreover, via the mapping mentioned in Remark 4, corresponds to the family of Schnyder labelings on triangulations with vertices.
The bijective link between and can also be established via the KMSW bijection. Indeed, the plane bipolar orientations corresponding to 1aligned tandem walks ending on the axis are those having a left outer boundary of length , and such that every inner face has a right boundary of length ; and there is an easy bijection between and such orientations with vertices [fusy2009bijective, Sec.5]. The resulting link between and via maps is more direct, but the interest of Lemma 1 is that it establishes the result by sole inspection of the properties of admissible tandem walks.
A Dyck walk of length is a walk from the origin to with steps in , staying in the region . A pair of Dyck walks of length is called noncrossing if for every the unique step starting at height in is weakly left of the unique step starting at height in . We now recover the following wellknown bijective correspondence [bernardi2009intervals]:
Proposition 5.
Schnyder labelings on triangulations with vertices are in bijection with noncrossing pairs of Dyck walks of length .
Proof.
As we have seen in Remark 10, identifies to . And this set is in bijection with (by Lemma 1, or alternatively Remark 10). It remains to give a bijection between walks in and noncrossing pairs of Dyck walks of length . For a noncrossing pair of Dyck walks of length , and for each point on after the origin, let be the horizontal distance between and the step of arriving at height , and let be the vertical distance between and the diagonal . With the sequence of points of after the origin, we define as the quadrant walk starting from the origin and visiting successively the points for from to . Clearly, ends on the axis (since ). It is also easy to see that each east step in yields a SE step in , while each north step in yields a facestep that increases by . Hence, is in . The mapping is easy to invert, hence gives a bijection. ∎
4. Enumerative results
4.1. Exact enumeration
A system of two equations with two catalytic variables , can easily be written for the series and of admissible tandem walks staying in the quadrant, with even or odd final positions, along the lines for instance of [Beat20, Thm 3], and the same can be done for admissible tandem walks. The resulting equations are however somewhat cumbersome to manipulate and it turns out to be more efficient to reduce the problem to small step walk problems, in the spirit of [Na20, Prop. 4], but taking into account the final parity.
We start with Schnyder labelings, which lead to simpler recurrences due to the weights on facesteps:
Proposition 6.
Let denote the number of Schnyder labelings with inner faces. Let moreover , and be given by the following recurrences:
(1)  
(2)  
(3)  
with null boundary conditions for all coefficients with or or or except .
Then, for , where .
Proof.
According to Proposition 4, is the number of (weighted) admissible tandem walk from to with SE steps.
Observe now that a weighted admissible tandem walk identifies with an unweighted tandem walk with step set such that steps (resp. steps) always start from an even (resp. odd) position, and steps never follow steps. Indeed, the weight on a facestep in an admissible tandem walk exactly corresponds to the number of ways to convert such a step into a sequence of steps of starting with a step in and followed with a sequence of steps in (with the constraint that the starting step is allowed only if starting at odd , and the starting step is allowed only if starting at even ).
For , let (resp. ) denote the number of admissible quadrant tandem walks starting at position , ending at position and with steps of type , whose associated unweighted tandem walk with step set ends with a step (respectively with a step of ). A last step removal decomposition of these unweighted paths then directly yields Equations (1)–(2), upon setting except for , to ensure propre initialization.
Finally, observe that an admissible quadrant tandem walk ending on the horizontal axis has to finish with a SE step, hence . ∎
The recurrence allows us to compute the first terms in polynomial time (using additions on integers of size ). The first terms are
Proposition 6 can be refined to take into account the number of black and white vertices, since these quantities respectively correspond to the numbers of iterations through Eq (1) at even or odd values of . Thus, letting and , we have (for and ) , and (for ) , where is obtained from the same recurrence as in Proposition 6, except that the righthand side in Eq (1) is to be multiplied by (resp. ) if is odd (resp. even). The first terms are
Pushing further the expansion, we recognize that the coefficient of matches the number of noncrossing pairs of Dyck walks of length , as expected from Section 3.3. This sequence starts as [OEIS, A005700].
Remark 11.
We are also interested in counting Schnyder labelings whose outer white vertices are nonisolated, due to their link to rigid orthogonal surfaces, as discussed in Section 2.2. The counting sequence can easily be obtained from the counting sequence , by a similar argument as in Remark 7. The counting series is given by
For bivariate enumeration, with , we have
Similarly, a recurrence can be obtained for polyhedral orientations, although 3 sequences are necessary, due to a further restriction on the set of admissible small steps walks to consider:
Proposition 7.
Let denote the number of polyhedral orientations with inner vertices, which is also the number of admissible tandem walk of length starting from the origin and ending on the axis. Let moreover , , and be given by the following recurrences:
(4)  
(5)  
(6)  
with boundary conditions for all or or or , except .
Then .
Proof.
Our strategy is again to identify admissible tandem walks with certain marked tandem walks on a well chosen small step set, namely . In order to do that in absence of weighting, we break the symmetry arbitrarily and we decompose face steps as follows:

each face step of the form is mapped to a sequence starting with a marked step , followed with steps and ended by steps ;

each face step of the form that starts at even ordinate is mapped to a sequence starting with a marked step , followed by steps , and ending with steps ;

each face step of the form that starts at odd ordinate is mapped to a sequence starting with a marked step , followed by steps , and ending with steps ;
Observe that the decomposition of a face step always starts with its unique marked step. Therefore, a admissible can be recovered from the concatenation of the image of its steps, and its length corresponds to the number of marked or steps in its image.
Now we observe that the subset of marked tandem walks on that correspond to admissible walks is characterized by the following local rules:

a step can follow any kind of step,

a marked step of type can follow any kind of step,

a marked step of type (resp. ) can follow any kind of step provided it starts at even (resp. odd) ordinate,

an unmarked step can only follow a marked step or an unmarked step,

an unmarked step can only follow a marked step or a unmarked step or an unmarked step .
For , let then (resp. , resp. ) denote the number of admissible quadrant tandem walks starting at position , ending at position , whose associated marked tandem walk on has marked or steps and ends with a step (resp. with a marked step or unmarked step, resp. with a step). The local rules then clearly imply the announced last step removal decomposition equations (4)–(6), upon assuming except for , to ensure that the counted paths start with a marked step as expected.
Finally, observe that a admissible quadrant tandem walk ending on the horizontal axis has to finish with a SE step, so that . ∎
The first terms, computed from the recurrence, are
Remark 12.
It is easy to see that if two rigid corner polyhedra yield the same Schnyder labeling, then they yield the same polyhedral orientation. This gives a mapping from to , which is surjective (the surjectivity will appear clearly in Section 5, via a formulation in terms of tricolored contactsystems of curves). Hence, , which we observe on the initial terms (the coefficients start to differ from ).
Again the proposition can be refined, to take into account the number of red, blue, and green vertices: blue inner vertices and red inner vertices correspond respectively to the numbers of iterations in Equation (4) at even or odd respectively. Thus, letting be the number of polyhedral orientations with inner red vertices, inner blue vertices, and inner green vertices, we have , where is obtained from the same recurrence as in Proposition 6, except that the righthand side in Eq (4) is to be multiplied by (resp. ) if is odd (resp. even). The first terms are