The combinatorics of planar maps (i.e., planar multigraphs endowed with an embedding on the sphere) has been a very active research topic ever since the early works of W.T. Tutte . In the last few years, after tremendous progresses on the enumerative and probabilistic theory of maps [7, 2, 14, 23], the focus has started to shift to planar maps endowed with constrained orientations. Indeed constrained orientations capture a rich variety of models [15, 13] with connections to (among other) graph drawing [26, 4], pattern-avoiding permutations [3, 24, 5], Liouville quantum gravity , or theoretical physics . From an enumerative perspective, these new families of maps are expected to depart (e.g. [16, 12]) from the usual algebraic generating function pattern followed by many families of planar maps with local constraints . From a probabilistic point of view, they lead to new models of random graphs and surfaces, as opposed to the universal Brownian map limit capturing earlier models. Both phenomena are first witnessed by the appearance of new critical exponents in the generic asymptotic formulas for the number of maps of size .
A fruitful approach to oriented planar maps is through bijections (e.g. ) with walks with a specific step-set in the quadrant, or in a cone, up to shear transformations. We rely here on a recent such bijection  that encodes plane bipolar orientations by certain quadrant walks called tandem walks, and that was recently used in the article  to obtain counting formulas for plane bipolar orientations with control on the face-degrees: we show in Section 2 that it can be furthermore adapted to other models by introducing properly chosen weights. Building on these specializations, in Section 3 we obtain exact enumeration results for plane bipolar posets and transversal structures. In particular we show that the number of plane bipolar posets on vertices is equal to the number of plane permutations of size introduced in  and recently further studied in , and that a reduction to small-steps quadrant walks models (which makes coefficient computation faster) can be performed for the number of plane bipolar posets with edges and the number of transversal structures on vertices. In Section 4 we then obtain asymptotic formulas for the coefficients all of the form with and with explicit. Using the approach of  we then deduce from these estimates that the generating functions for and are not D-finite. Finally in Section 5 we provide a direct bijection between plane permutations of size and plane bipolar posets with vertices, which is similar to the one  between Baxter permutations and plane bipolar orientations.
2. Oriented planar maps and tandem walks in the quadrant
A plane bipolar orientation is a planar map (with a marked face taken as the outer face) endowed with an acyclic orientation having a single source and a single sink , which both lie in the outer face, see Figure 1(a). It is known that the contour of each face of (including the outer one) splits into a left lateral path and a right lateral path (which share the same origin and end); the type of is the pair where (resp. ) is the length of (resp. ). The outer type of is the type of the outer face. The pole-type of is the pair such that is the degree of and is the degree of .
On the other hand, a tandem walk (see Figure 1(c)) is defined as a walk on with steps in ; it is a quadrant walk if it stays in all along. Every step in such a walk is called a face-step, and the pair is called its type. We will crucially rely on the following bijective result:
Theorem 1 (KMSW bijection ).
Plane bipolar orientations of outer type with edges are in bijection with quadrant tandem walks of length from to . Every non-pole vertex corresponds to a SE-step, and every inner face corresponds to a face-step, of the same type.
An edge is called transitive if there is a path from to avoiding . If has no transitive edge it is called a plane bipolar poset. On the other hand, in a tandem walk, a face-step is called transitive if its type has at least one zero entry.
Let be a plane bipolar orientation. Then is a plane bipolar poset iff it has no inner face whose type has a zero entry. Hence the KMSW bijection specializes into a bijection (with same parameter-correspondence) between plane bipolar posets of outer type and quadrant tandem walks from to and with no transitive face-step.
Assume has such an inner face , e.g. has length . Then is an edge , and it is transitive since avoids and goes from to . Conversely, assume has a transitive edge , and let be a path from to such that the region enclosed by is minimal. Then it is easy to see that the interior of is a face, which has as one of its lateral paths. ∎
Note that in Claim 2, the primary parameter of the poset (the one corresponding to the walk length) is the number of edges (minus ). We will see below another way to relate plane bipolar posets to (weighted) quadrant tandem walks, this time with the number of vertices as the primary parameter (associated to the walk length)111Though we will focus on univariate enumeration, both approaches make it possible to compute the number of plane bipolar posets having vertices and edges..
Another kind of oriented maps to be related below to weighted quadrant tandem walks are transversal structures [19, 17] (these also naturally encode the combinatorial types of generic rectangulations). A 4-triangulation is a map whose outer face contour is a (simple) 4-cycle and whose inner faces are triangles, such that all 3-cycles delimit a face; the outer vertices are denoted in clockwise order, and denotes the set of inner vertices. A transversal structure on such a map (see Figure 2) is an orientation and bicoloration of its inner edges (in blue or red) so that the following local conditions are satisfied: the edges incident to are respectively outgoing blue, ingoing red, ingoing blue, and outgoing red; and for each inner vertex the incident edges in clockwise order around form four (non-empty) groups: outgoing red, outgoing blue, ingoing red, ingoing blue. It is known that the map formed by the red (resp. blue) edges is a plane bipolar poset (spanning all vertices except for the red poset, and all vertices except for the blue poset).
For a function from to , a -weighted plane bipolar orientation is a bipolar orientation where every inner face carries an integer in , with the type of . Similarly a -weighted tandem walk is a tandem walk where every face-step carries an integer in , with the type of .
For , plane bipolar posets of pole-type , with vertices and inner faces, are in bijection with -weighted plane bipolar orientations of outer type , with edges and vertices. These correspond (via KMSW) to -weighted quadrant tandem walks of length from to with SE-steps.
For (with if or ), transversal structures having inner vertices and blue edges are in bijection with -weighted plane bipolar posets of outer type having vertices and edges. These correspond (via KMSW) to -weighted quadrant tandem walks from to of length with SE-steps.
The first correspondence (see Figure 3) is adapted from . Starting from a plane bipolar orientation , insert a square vertex in the middle of each edge (these are to be the non-pole vertices of the bipolar poset). Then in each inner face , with its type, insert non-crossing edges from the square vertices on the left lateral path to the square vertices on the right lateral path; there are precisely ways to do so (so the chosen way can be encoded by an integer ). Finally create a square vertex (resp. ) in the left (resp. right) outer face and connect it to all square vertices on the left (resp. right) lateral path of . Then the bipolar poset is obtained by erasing the vertices and edges of in the obtained figure.
The second correspondence relies on the fact that a transversal structure is completely encoded by its red bipolar poset (augmented by the 4 outer edges oriented from to ) and the knowledge of how each inner face is transversally triangulated by blue edges: if the face has type then there are precisely ways to do so. ∎
3. Exact counting results
Let denote the generating series of -weighted quadrant tandem walks starting in position , with respect to the number of steps (variable ), end positions (variables and ) and number of SE steps (variable ). A last step decomposition immediately yields the following master equation in the ring of formal power series in and with coefficients that are Laurent series in :
3.1. Plane bipolar posets by edges
The case of bipolar posets counted by edges corresponds to having (cf Claim 2). The master equation then becomes
and the coefficient gives the number of plane bipolar posets of outer type with edges.
By some algebraic manipulations similar to those performed in [10, Sect.5.2], we can relate the enumeration of plane bipolar posets by edges to a simpler model of quadrant walks:
For let be the number of plane bipolar posets with edges. Then is equal to the number of quadrant excursions of length with steps in .
Note that, for , is also (by adding a path of length from source to sink on the left side) the number of plane bipolar posets with edges and left boundary of length , thus . Let under the change of variable relation (note that ). Via the change of variable, the functional equation for becomes
This resembles the functional equation for the series of quadrant walks (starting at the origin) with steps in , whose functional equation is
By coefficient extraction in each of these two functional equations, we recognize that and appear to be related as . This relation can then be easily checked. Indeed, if we substitute by in the first functional equation, we recover the second functional equation (multiplied by ). As a consequence for . Thus for , (and for one manually checks that ).
While the series are non D-finite (as discussed in the next section) the reduction to a quadrant walk model with small steps ensures that the sequence can be computed with time complexity and using bit space. The sequence starts as .
3.2. Plane bipolar posets by vertices
In the case of plane bipolar posets enumerated by vertices, we have (cf Proposition 3) for , so that in . The master equation then rewrites
and the coefficient gives the number of plane bipolar posets of pole-type with vertices.
From the functional equation, we now prove that plane bipolar posets counted by vertices are equinumerous to so-called plane permutations introduced in  and that have been recently studied in . These are the permutations avoiding the vincular pattern (and also the permutations such that the dominance poset of the point diagram is planar).
Let denote the number of plane bipolar posets with vertices. Then is equal to the number of plane permutations on elements.
Note that is also, by adding a new source of degree (connected to the former source), the number of plane bipolar posets of pole-type with vertices and arbitrary , so that . Let , so that, under the change of variable , we have , and . In these terms, the main equation for rewrites
This equation for is exactly [10, Eq. (2)] (they use for our ), where gives the number of plane permutations of size . This concludes the proof, since and . ∎
3.3. Transversal structures
Finally in the case of transversal structures, for , otherwise, so that in . The master equation then rewrites
and gives the number of transversal structures with vertices and red edges (or, upon removing the vertex and adjoining the path to the red bipolar poset, a properly weighted bipolar poset with outer-type for some , vertices and red edges). As shown in the next section this series (in the variable , with ) is non D-finite. However, similarly as for plane bipolar posets counted by edges, we can make the coefficient computation faster by reduction to a model of quadrant walks with small steps (however this time with some forbidden two-step sequences).
Let be the number of transversal structures on vertices. Let be the coefficients given by the recurrence, valid for :
with boundary conditions for any with or or , with the exception (initial condition) of . Then is equal to the coefficient .
Upon adjoining the paths and to the red poset, is the number of -weighted quadrant tandem walks —for — from to with SE steps.
Note that a -weighted quadrant tandem walk identifies with an unweighted quadrant walks with small steps (and with the same endpoints and same number of SE steps) upon turning each step into a step followed by a portion in (there are indeed ways to do that). The resulting walks have steps in , with the condition that every step in comes after a step in . Letting (resp. ) be the number of those walks starting at and ending at with a final SE step (resp. non SE step) such that their length plus number of NW steps is , a last step decomposition directly translates into the recurrence (1). Finally it is easy to see that in such a walk (from to ) the length plus number of NW steps is of the form , with the number of SE steps (this can be checked by tracing how evolves along the path). ∎
Similarly as for plane bipolar posets counted by edges, the recurrence makes it possible to compute the sequence with bit operations using bit space, giving an alternative (with same complexity order) to the recurrence in . Another recurrence yielding a polynomial computation time is described in  (where a bijection to a certain family of pattern-avoiding permutations family is also introduced). The sequence starts as .
4. Asymptotic counting results
We adopt here the method by Bostan, Raschel and Salvy  (itself relying on results by Denisov and Wachtel ) to obtain asymptotic estimates for the counting coefficients of plane bipolar posets (by vertices and by edges) and transversal structures (by vertices). Let be the tandem step-set. Let satisfying the symmetry property . The induced weight-assignment on is for and for . Let be the weighted number (i.e., each walk is counted with weight ) of quadrant tandem walks of length , for some fixed starting and ending points. Let
let , and let be the radius of convergence (assumed here to be strictly positive) of . Let be the modified weight-distribution where are adjusted so that
is a probability distribution (i.e.) and the drift is zero, which is here equivalent to having solution of (one solves first for and then takes ; note also that is increasing on so that is unique if it exists). Then as shown in [11, 6] we have, for some ,
As shown in , the dependance of on the starting point and ending point is of the form for some absolute constant and a function from to , (which is a discrete harmonic function for the walk model). In the special case where the starting point is and ending point is , the dependence of on and is thus of the form for some function from to , in which case we say that has separate dependence on and .
Proposition 8 (Plane bipolar posets counted by edges).
For fixed , let be the number of plane bipolar posets of outer type with edges. Let be the unique positive root of . Let , , and . Then there exists a positive constant (with separate dependence on and ) such that
Moreover, the associated generating function is not D-finite. These results also apply to the coefficient (number of plane bipolar posets with edges) via the relation (valid for ).
This case corresponds to taking . By Claim 2, is the number of -weighted tandem walks of length from to . We have
Hence , of radius of convergence . Next we have ; and thus has a positive root in which is the unique positive root of the polynomial . Since is the minimal polynomial of , any rational expression (with coefficients in ) in reduces to a polynomial expression in of degree smaller than . Clearly and have rational expressions in (since is rational in ), and the expressions we obtain by reduction are and .
For a generating function with , a known sufficient condition for being not D-finite is that with . As shown in , if is an algebraic number with minimal polynomial , then is rational iff (the numerator of) has a cyclotomic polynomial among its prime factors. Here the minimal polynomial of is (since ), and the numerator of is the prime polynomial which is not cyclotomic (as recalled in , all cyclotomic polynomials of degree at most have their coefficients in ). Hence for any fixed , the generating function is not D-finite.
Finally, the claimed relation follows from the fact that for a plane bipolar poset with edges identifies to a plane bipolar poset of outer type with edges, upon adding a left outer path of length and a right outer path of length . ∎
Proposition 9 (Plane bipolar posets counted by vertices).
For fixed , let be the number of plane bipolar posets of pole-type with vertices. Then there exists a positive constant (with separate dependence on and ) such that
These results also apply to the coefficient (number of plane bipolar posets with vertices) via the relation .
This case correspond to . By Proposition 3, is the number of -weighted tandem walks of length from to . We have
Hence of radius of convergence . Next we have thus has a positive root in which is the unique positive root of the polynomial . Any rational expression in thus reduces to an expression with in . We find , and (we actually have , as for plane bipolar posets counted by edges), and .
Finally, the claimed relation follows from the fact that for a plane bipolar poset with vertices identifies to a plane bipolar poset of pole-type with vertices, upon creating a new sink (resp. a new source) connected to the former sink (resp. source) by a single edge. ∎
We recover, as expected in view of the previous section, the asymptotic constants and for plane permutations, which were obtained in  (where was also explicitly computed).
Proposition 11 (Transversal structures).
Let be the number of transversal structures on vertices. Then there exists a positive constant such that
Moreover, the associated generating function is not D-finite.
This case corresponds to and the associated series is . By Proposition 3, for is the number of -weighted tandem walks from to with SE steps. Consider such a walk . Since there is no step of the form (as such a step has weight here) it starts with a SE step, and since there is no step of the form it also ends with a SE step. Let be the walk obtained from by deleting the last SE step, and aggregating the other steps into groups formed by a SE step followed by a (possibly empty) sequence of non-SE steps; precisely each such group yields the aggregated step with weight . The obtained weighted walk starts at , ends at (since the last SE step of has been deleted), and the condition that stays in the quadrant translates to the condition that stays in the shifted quadrant (indeed it is easily checked that stays in the quadrant iff the starting point of every SE step is in the shifted quadrant). Note also that the series corresponding to one aggregated step is
Hence , of radius of convergence . Next we have , which cancels at . Since is rational, so are and ; we find and .
To show that the associated series is not D-finite, we consider the minimal polynomial of , which is . We have , whose numerator is not cyclotomic as it has coefficients of absolute value larger than . Hence is not D-finite according to . ∎
The case of plane bipolar posets by vertices, this time with corresponding to the outer type rather than the pole-type, can be treated from the weight-assignment (used for plane bipolar posets counted by edges) and aggregating the steps into groups formed by a SE step followed by a (possibly empty) sequence of non-SE steps. In that case the series for one aggregated step is
5. Bijection between plane permutations and plane bipolar posets
5.1. Presentation of the construction and a first proof of bijectivity
We give here a bijection between the set of plane permutations of size and the set of plane bipolar posets on vertices. It resembles (both the construction, in fact more straighforward here, and the proof strategy based on generating trees) the bijection introduced in  between Baxter permutations and plane bipolar orientations. A permutation is called a plane permutation if it avoids the vincular pattern . We adopt the standard diagrammatic representation of , i.e., we identify to the set , whose elements are called the points of . The dominance order on the points of is the poset where iff and . The dominance diagram of is obtained by drawing a segment for each pair of the poset. It is known  that a permutation is plane iff its dominance diagram is crossing-free. The completion of is the permutation on where , and for . Clearly if is plane then so is . We let be the dominance diagram of and let be the underlying plane bipolar poset on vertices, see Figure 4 for an example.
Note that the neighbors of the source of correspond to the left-to-right minima of , the neighbors of the sink of correspond to the right-to-left maxima of , the non-pole vertices on the right boundary of correspond to the right-to-left minima of , and the non-pole vertices on the left boundary of correspond to the left-to-right maxima of .
Let be a plane bipolar poset , with its set of non-pole vertices, its source and its sink. The left tree (resp. right tree ) of is the spanning tree of where for every its parent-edge is its leftmost (resp. rightmost) ingoing edge. We order the non-pole vertices by first visit in clockwise order around . For we let be the rank (in ) of for this order. And similarly we let be the rank (in ) of first visit of (among vertices in ) during a counterlockwise tour around . Let be the permutation whose point-diagram is . Then the following property holds (see Figure 5):
For every , we have .
Let . For let be the point , and let be the vertex of associated to . We want to show that and , which amounts to show that and for (we only detail the argument for , the argument for is completely symmetric). If is an ascent, i.e. , then clearly so that there is an edge from to and moreover is the leftmost outgoing edge of and the rightmost ingoing edge of . Hence and is traversed just after the discovery of . Hence the discovery of occurs just after the discovery of , so that . If is a descent, let be the path from to in . In the path is a broken line (from to ), and it follows from the definition of that there can be no point under (i.e., in the area delimited by , the horizontal segment between the origin and , and the vertical segment between