Bijections for generalized Tamari intervals via orientations
Éric Fusy Abel Humbert






Bijections for generalized Tamari intervals via orientations]Bijections for generalized Tamari intervals via orientations











ÉF: Laboratoire Informatique Gaspard Monge, 
Université Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France

eric.fusy@u-pem.fr







AH: IRIF, Université Paris-Diderot, France
ahumbert@irif.fr






















Generalized Tamari intervals have been recently introduced by Préville-Ratelle and Viennot, and have been proved to be in bijection with (rooted planar) non-separable maps by Fang and Préville-Ratelle. We present two new bijections between generalized Tamari intervals and non-separable maps.  Our first construction 
proceeds via separating decompositions on simple bipartite quadrangulations (which are known to be in bijection with non-separable maps). It can be seen as an extension of the Bernardi-Bonichon bijection between Tamari intervals and minimal Schnyder woods.  
On the other hand, our second construction relies on a specialization of the Bernardi-Bonichon bijection to so-called 
synchronized Tamari intervals, which are known to be in one-to-one correspondence with generalized Tamari intervals. 
It yields a trivariate generating function expression that interpolates between the bivariate generating function for generalized Tamari intervals, 
and the univariate generating function for Tamari intervals. 


[
    Abel Humbert
    December 30, 2023
=====================












§ INTRODUCTION

The ν-Tamari lattice (ν) (for ν an arbitrary directed walk with steps in {N,E}) has been recently 
introduced by Préville-Ratelle and Viennot <cit.>, and further studied in <cit.>, with connections to geometric combinatorics.  
It is a lattice on the set of directed walks weakly above ν and with same endpoints as ν, and it 
generalizes the Tamari lattice <cit.> (in size n, case where ν=(NE)^n) 
and the m-Tamari lattices <cit.> 
(in size n, case where ν=(NE^m)^n).   
 
The enumeration of intervals (i.e., pairs formed by two elements x,x' with x≤ x') in Tamari lattices has attracted a lot of attention <cit.>, due in particular to their (conjectural) connections to dimensions of diagonal coinvariant spaces <cit.>, and to their bijective connections to planar maps <cit.>, as well as intriguing symmetry properties <cit.>. A planar map (shortly, a map) is a connected multigraph embedded in the plane, up to continuous deformation. A rooted map is a map with a marked corner incident to the outer face (all maps in this article are assumed to be rooted if not specified otherwise). A map is called non-separable (or 2-connected) if it is either the loop-map, or is loopless and M\ v is connected for every vertex v∈ M. Chapoton <cit.> proved that the number of Tamari intervals of size n is 2/n(n+1)4n+1n-1, which coincides with the number of simple triangulations with n+3 vertices <cit.>. Bernardi and Bonichon <cit.> subsequently gave a bijective proof of this formula, relying on so-called Schnyder woods (orientations and colorations of the inner edges in 3 colors, with specific local constraints).     
Regarding ν-Tamari lattices, if we let _ν be the set of intervals in (ν), then it has recently been shown by Fang and Préville-Ratelle <cit.> that 
_n:=∪_ν∈{N,E}^n_ν (generalized Tamari intervals of size n) is in bijection with the set _n of non-separable maps with n+2 edges, and more precisely that _i,j:=∑_ν∈𝔖(E^iN^j)_ν is in bijection with the set _i,j of non-separable maps with i+2 vertices and j+2 faces (it is known <cit.> that |_n|=2(3n+3)!/(n+2)!(2n+3)! and |_i,j|=(2i+j+1)!(2j+i+1)!/(i+1)!(j+1)!(2i+1)!(2j+1)!). 
They have a first recursive bijection based on parallel decompositions with a catalytic variable, and then make the bijection more explicit via certain auxiliary labeled trees. 
As shown in <cit.>, their bijection also has interesting symmetry properties, as it commutes with natural involutions on the two classes (duality on maps, and a mirror duality for Tamari intervals).    

A quadrangulation is a map with all faces of degree 4; by a bipartite quadrangulation we mean a quadrangulation endowed with its unique coloration of 
vertices in black or white such that adjacent vertices have different colors,   
and the root-vertex (vertex at the root-corner) is black. By a classical correspondence <cit.>, _i,j
 is in bijection with the set _i,j of bipartite simple quadrangulations with i+2
black vertices and j+2 white vertices.

In this article, we give two new bijections between _i,j and _i,j. Each one relies on seeing _i,j as included in a certain superfamily, and specializing a bijection
involving oriented maps. In our first bijection (Section <ref>) 
we see _i,j as a subfamily of non-intersecting
triples of lattice walks (a so-called Baxter family) and specialize a bijection (closely related to the one in <cit.>
and also to a recent bijection by Kenyon et al. <cit.>) 
with so-called separating decompositions on simple quadrangulations. We also show that this construction gives an extension of the Bernardi-Bonichon bijection (which is recovered as the case ν=(NE)^n).   
In our second bijection (Section <ref>) we see _i,j as a subfamily (synchronized intervals) 
of classical Tamari intervals of size i+j+1, to which we specialize the Bernardi-Bonichon bijection <cit.>, which we compose with a bijection <cit.> to certain tree-structures  on which we can characterize the property of being synchronized. 

Several parameters can be tracked by the first construction, which 
 gives a model of maps for intervals in the m-Tamari lattices, and reveals certain symmetry properties on _i,j. The second construction yields a trivariate generating function expression (Corollary <ref>) 
that interpolates between the bivariate generating function of generalized Tamari intervals and the univariate 
generating function of classical Tamari intervals. 



§ THE Ν-TAMARI LATTICE, AND GENERALIZED TAMARI INTERVALS

We recall from <cit.> the definition of ν-Tamari lattices, and how they are related to the classical Tamari lattice. 
We consider walks in ℕ^2 starting at the origin and having steps North or East 
(these are equivalent to words on the alphabet {N,E}). For two such walks γ,γ', we say that γ' 
is weakly above γ if γ and γ' have the same endpoint, and no East step of γ is strictly
above the East step of γ' in the same vertical column. A Dyck walk of length 2n thus corresponds to a walk γ that is weakly above (NE)^n. More generally, for ν a walk, we let _ν be the set of walks weakly above ν. 
For γ∈_ν and for p=(x,y) a point on γ, we let x'≥ x be the abscissa of the North step of ν from ordinate y to y+1 (with the convention that x'=i if y=j), and we let ℓ(p):=x'-x. If p is preceded by E and followed by N we let p' be the next point after p along γ such that ℓ(p')=ℓ(p), and we let _p(γ) be the walk γ' obtained from γ by moving the E preceding p to be just after p' (see Figure <ref> for an example); we say that γ' covers γ. The Tamari lattice for ν is defined as  (ν)=(_ν,≤) where ≤ is the 
transitive closure of the covering relation.  
The classical Tamari lattice _n corresponds to the special case _n=((NE)^n), and more generally for m≥ 1, the m-Tamari lattice _n^(m) corresponds 
to the special case _n^(m)=((NE^m)^n) . 



Interestingly, for ν of length n, (ν) can also be obtained
 as a sublattice of _n+1, the classical Tamari lattice on Dyck walks of length 2n+2. 
For γ=E^α_0NE^α_1⋯ NE^α_k a Dyck walk of length 2k, 
the canopy-word of γ is the word (γ)=(w_0,…,w_k)∈{E,N}^k+1 such that for 
r∈ 0,k, w_r=E if α_r=0 and w_r=N if α_r≥ 1  
 (note that we always have w_0=E and w_k=N). 
 Then (ν) is isomorphic to the sublattice of _n+1 induced by the Dyck walks
whose canopy-word is equal to Eν N.


 











Let _i,j (resp. _n) be the set of triples (ν,γ,γ') such 
that γ≤γ' in (ν), and ν ends at (i,j) (resp. ν has length n).  
Elements of _i,j (resp. _n) are called generalized Tamari intervals
 with endpoint (i,j) (resp. of size n).  
We now make two remarks based on properties shown in <cit.> (each remark is associated with a bijection for _i,j described later, respectively in Section <ref> and Section <ref>, the second remark is also used for the bijection in <cit.>). 


 Since γ≤γ' in (ν) implies that γ' is weakly above γ, _i,j
is a subfamily of the family _i,j of triples of walks (ν,γ,γ'), each starting at the origin and ending at (i,j), such that
 γ' is weakly above γ, itself weakly above ν. 



On the other hand, let _n be the set of intervals in _n (classical Tamari intervals, on Dyck words of length 2n). 
An interval (γ,γ')∈_n is called synchronized if (γ)=(γ'). Let _n⊂_n 
be the set of synchronized Tamari intervals of size n. 
Then the above sublattice characterization of (ν) implies that 
_n is in bijection with _n+1. More generally, if we let _i,j be the set of synchronized intervals such that the common canopy-word is in S(E^i+1N^j+1), then _i,j is in bijection with _i,j.  




§ BIJECTION USING SEPARATING DECOMPOSITIONS

Several bijections are known between _i,j and other combinatorial families (which are called Baxter families, a survey is given in <cit.>). 



Our aim here is to pick one such bijection and show that it specializes 
nicely to the subfamily _i,j⊂_i,j. We pick the bijection (called here Φ) from <cit.> for separating decompositions,  but have to slightly 
modify it (the modified bijection is called Φ') so that it specializes well. As we will see in Section <ref>, our construction Φ' is also closely related to a recent bijection by Kenyon et al. <cit.>.   



 §.§ Separating decompositions

For Q∈_i,j, let s,s',t,t' be the outer vertices of Q 
in clockwise order around the outer face, with s the one
at the root.  
A separating decomposition of Q is given by an orientation and coloration (blue or red) of each edge of Q such that:

 
  * All edges incident to s (resp. t) are incoming blue (resp. incoming red).
 
  * Every vertex v∉{s,t} has one outgoing edge in each color. Moreover, if v is white (resp. black), then every incoming edge at v
   has the color of the next outgoing edge in clockwise (resp. counterclockwise) order around v, see Figure <ref>(a). 
  
  
  An example of separating
decomposition is given in Figure <ref>(b).  
It can be shown <cit.> that the blue edges form a spanning tree of Q\ t and the red 
edges form a spanning tree of Q\ s. By a slight abuse of notation, we also call separating decomposition a pair S=(Q,X),
where Q is a simple quadrangulation, and X is a separating decomposition on Q. 
We let _i,j be the set of separating decompositions with i+2 black vertices and j+2 white vertices. 
A separating decomposition is called minimal  if it has no clockwise cycle. 






A general property of outdegree-constrained orientations of planar maps <cit.>  
ensures that each simple quadrangulation
 has a unique minimal  separating decomposition. Hence, _i,j is in bijective correspondence to the set of separating decompositions in _i,j that are minimal.   



 §.§ Presentation and statement of the bijection

We first recall the bijection Φ introduced in <cit.> 
 between _i,j and _i,j. For  S∈_i,j, we let 
 be the blue tree of S, and let v_0,…,v_j+1 be the  
white vertices, ordered according to first visit in a clockwise walk around  starting
at the root. For r∈ 0,j, let β_r be the number of incoming red edges at v_r+1. 
Then Φ(S) is the triple of walks (γ_,γ_|,γ_) (written here as binary words) obtained as follows: 


  * Let w_1 be the word obtained from a clockwise walk around , where we write an E each time we traverse an edge from white to black while getting farther from the root,  and write an N each time we traverse an edge from white to black while getting closer to the root. 
Since the rightmost child t' of s is a leaf of , w_1 ends with two occurences of N.
Let γ_ be w_1 without its two last N letters,

  * Let w_2 be the word obtained from a clockwise walk around , where we write an N each time we traverse an edge from black to white while getting farther from the root,  and write an E each time we traverse an edge from black to white while getting closer to the root. Then w_2 starts with N, and (again due to the rightmost child of s being a leaf) ends with N. 
Let γ_| be w_2 without its first and last N letters.

  * The walk γ_ is E^β_0NE^β_1… NE^β_j.



We now introduce a mapping Φ' that is a modified version of Φ (see Figure <ref> for an example), better suited in view
of the specialization to generalized Tamari intervals.   
 For S a separating decomposition, 
 Φ'(S) is the triple  (γ_,γ_|,γ_) of walks 
where γ_| and γ_ are obtained as above, and γ_ is modified to be 
E^α_0NE^α_1… NE^α_j, with α_r the number of incoming blue edges at v_r for r∈ 0,j. 
For instance, for the separating decomposition of Figure <ref>, γ_ is EEENEENENNNE when applying Φ, and is EEENNEENENNE when applying Φ'. 
 

For i,j≥ 0, the mapping Φ' is a bijection between _i,j and _i,j. In addition,
for S∈_i,j, S is minimal if and only if Φ'(S)∈_i,j. Hence, Φ' yields a bijection
between _i,j and _i,j.  




The proof is delayed to Section <ref>. 


 





 For (,,)∈_i,j,  
a value r∈ 1,j is called a level-value of type (p,q)  
if β_r-1=p and α_r=q. 
On the other hand, for Q∈_i,j, an inner white vertex is said to be of type (p,q) 
if it has p incoming red edges and q incoming blue edges.  
Then clearly in the bijection 
Φ', α_0 is mapped to the degree of s' minus 2, β_j is mapped to the degree of t' minus 2, and 
each level-value in 1,j corresponds to an inner white vertex of the same type.

 



From the parameter-correspondence in Remark <ref>, we can see that our bijection for _i,j differs  (under the classical
correspondence of _i,j with _i,j) from the one in <cit.>. 
Indeed, in their bijection, the 
parameter α_0 corresponds to the length (minus 1)
of the leftmost branch in their labelled DFS trees. But that parameter does not correspond to a face-degree (e.g. for one of the two faces adjacent to the root-edge) nor to a vertex-degree (e.g. for one of the two extremities of the root-edge)  in the associated  non-separable map. 


Remark <ref> also yields a model of maps for m-Tamari intervals. 
 For m≥ 1, we let _n^(m) be the subfamily of _mn,n where each inner white vertex has m incoming blue edges in the minimal separating decomposition, and s' has no incoming blue edge.  
Then Φ' yields a bijection between _n^(m) and intervals of _n^(m). 
It is known <cit.> (extension of the formula for m=1 discovered in <cit.>) that the number I_n^(m) of intervals in _n^(m) is given by the beautiful formula

    I_n^(m)=m+1/n(mn+1)(m+1)^2n+mn-1.

The family _n^(1) is in bijection (via contraction of the blue edges directed toward 
a white vertex <cit.>) with simple triangulations with n+3 vertices, endowed with their minimal Schnyder wood. 
Under this correspondence, one can check that our bijection coincides with the one by Bernardi and Bonichon <cit.>  
(recalled and exploited in Section <ref>) between _n and simple triangulations with n+3 vertices.
Simple triangulations with n+3 vertices can then be bijectively enumerated (we will recall a correspondence
to certain mobiles in Section <ref>), giving a bijective proof of (<ref>) for the case m=1.  
It would be interesting to provide a bijective proof of (<ref>) working for all m≥ 1, based on such an approach (edge-contractions or similar operations applied to maps in _n^(m), so as to obtain maps or hypermaps amenable to bijective enumeration). 








 §.§ A more symmetric formulation of the bijection Φ'
   
We reformulate here the bijection Φ' in a more symmetric way (in terms of the roles played by blue and red edges).  
We recall from <cit.> the notions of equatorial line and 2-book embedding of a separating decomposition S=(Q,X).  
A separating decomposition has the property that each inner face f has two bicolored corners. We may then draw a green curve inside f to join these two corners.   
It is shown in Lemma 3.1 of <cit.> that the union of these green curves forms a simple curve L   
from s' to t' that visits all inner faces and all vertices except s and t; then L is called the equatorial line of S,  see Figure <ref>(c). 

One can then stretch L into a horizontal line where the vertices of Q are equally spaced (with t as the left
extremity and s as the right extremity), and planarly draw the blue (resp. 
red) edges of S as half-circles in the lower (resp upper) half-plane, see Figure <ref>. 
This canonical drawing is called the 
2-book embedding of S. It is shown in Theorem 2.14 of <cit.> (see also Proposition 3.3 in <cit.>) 
that the 2-book embedding of S satisfies the following so-called alternating condition: 



(A): for all blue (resp. red) edges the right (resp. left) extremity is black. 



This property, and planarity of the set of arcs (see the discussion on fingerprints in Section 3 of <cit.>), implies that when applying the bijection Φ', the word corresponding to the middle walk is exactly the word read by traversing the line from left to right (excluding the outer vertices {s,t,s',t'}), writing E (resp. N) every time we meet a 
black (resp. white) vertex, see Figure <ref>. It also implies that the white vertices v_0,…,v_j+1 (ordered according to first visits in a clockwise walk around the blue tree starting at the root-corner)
 are ordered left-to-right  along the line. 

For S∈_i,j we let τ(S) be the separating decomposition obtained as the half-turn rotation of  S, i.e.,
the roles of s and t are exchanged and the colors are swapped. On the other hand, for 
R=(,,)∈_i,j we let τ(R) 
be the half-turn rotation of R, i.e., 
if mir(c_1,…,c_n):=(c_n…,c_1) denotes the mirror of a word on {N,E},  
then τ(R):=(mir(),mir(),mir()). 
Clearly, the symmetric reformulation of Φ' 
ensures that if S∈_i,j is mapped by Φ' to R∈_i,j, 
 then τ(S) is mapped to τ(R). Since τ is an involution on _i,j that preserves the property
of being minimal, we obtain:




For R∈_i,j we have τ(R)∈_i,j if and only if R∈_i,j.

To be precise, in our proof of Theorem <ref> given next,    
we will use a characterization of elements of _i,j
in terms of a certain arc diagram representation (Lemma <ref>), such that 
 Corollary <ref> already follows from this representation.  




 §.§ Proof of Theorem <ref>





  §.§.§ Proof that Φ' is a bijection from _i,j to _i,j


We define a blue-red arc diagram as obtained by concatenating (for some j≥ 0) 
 j+1 horizontal segments S_0,…,S_j, where for r∈ 0,j  
the segment S_r is made of three successive (possibly empty) groups of dots: blue dots, then black dots, then red dots, such 
that the total numbers of blue dots, black dots, and red dots are the same;  
 in addition, in the upper (resp. lower) half-plane, there is a red (resp. blue) planar matching of the black dots with the red dots (resp. of the blue dots with the black dots), 
see the right-part of Figure <ref> for an example. 
We denote by α_r (resp. μ_r,β_r) the number of blue (resp. black, red) dots in S_r, for r∈ 0,j. 
We let _i,j be the set of blue-red arc diagrams made of j+1 segments
and having i black dots.  
There is a straightforward bijection ξ from _i,j to _i,j:  the 
triple of walks (,,)∈_i,j associated to A∈_i,j is the one where  (resp. , )
has α_r (resp. μ_r, β_r) East steps at height r for r∈ 0,j. 
The property that  is weakly above  is equivalent to ∑_k=0^rα_k≥∑_k=0^rμ_k
for all r∈ 0,j, which is equivalent to the fact that the blue dots can be matched to the black
dots. Similarly, the property that  is weakly above  is equivalent to the fact that the black dots 
can be matched to the red dots. 

We will now describe a bijection χ from _i,j to _i,j. Before giving it, we state the following property that refines (A) and follows from <cit.> (see
the discussion about uniqueness of alternating layouts of rooted plane trees before Proposition 3.3): 

Every separating decomposition has a unique 2-book embedding satisfying (A), the vertices being equally spaced on the equatorial line, with t as the leftmost vertex and s
as the rightmost vertex. In this representation, for each vertex v∉{s,t}, 
the outgoing edge (edge going to the parent) of v in the upper (resp. lower) half-plane is the outermost arc incident to v.   




The bijection χ is done in two steps. Let _i,j' be the set
of arc-diagrams specified as follows (see the left-part of Figure <ref> for an example): 


  * There are i+j+2 vertices aligned along the horizontal axis, among which i are black and j+2 are white, the left-most point s' and rightmost point t' being white.

  * In the upper half-plane, there is a planar arc-system, such that the left (resp. right) extremity of every arc is black (resp. white), and every black vertex is incident to 
exactly one arc,

  * In the lower half-plane, there is a planar arc-system, such that the left (resp. right) extremity of every arc is white (resp. black), and every black vertex is incident to 
exactly one arc.
 



For S∈_i,j, let S' be obtained as follows, see the left-part of Figure <ref>:


  * take the 2-book embedding of S with property (A),

  * delete the vertices s,t and their incident edges,

  * erase all edges with a white origin.

Let χ_1 be the mapping that associates S' to S.


The mapping χ_1 is a bijection from _i,j to _i,j'.


For S∈_i,j, the fact that χ_1(S)∈_i,j' is a direct consequence of the property (A) of the 2-book embedding. 

The inverse construction is given as follows.    
For S'∈_i,j', insert two black vertices at the left and right extremity on the line, called respectively t and s. 
For w a white vertex on the horizontal line, the upper parent of w is defined as follows: if 
w is “covered" by a red arc a (a is the first arc crossed by an upward vertical line starting from w), then the upper-parent of w 
is the (black) vertex b at the left extremity of a; otherwise the upper-parent of w is t. Similarly, the lower parent of w is defined as follows: 
if w is “covered" by a blue arc a (a is the first arc crossed by a downward vertical line starting from w), then the lower parent of w 
is the (black) vertex b at the right extremity of a; otherwise the lower parent of w is s. 
We let S be obtained from S' by orienting its arcs from black to white, then for each white vertex w, by inserting a red (resp. blue) arc from w 
to its upper parent (resp. to its lower parent) in the upper (resp. lower) half-plane.  We claim that S is a separating decomposition in _i,j, 
endowed with its unique 2-book embedding as characterized in Lemma <ref>. First, the arc insertions yield no crossing (e.g. 
for each upper arc a of S', letting Σ_a be the face of the arc-system on the interior-side of a, all inserted arcs connected to the left extremity of a occur within Σ_a).   
The alternation property (A) is clearly satisfied, as well as the fact that the outgoing edge of every vertex in the upper (resp. lower) half-plane is the outermost one. 
These two properties (and planarity of the arc-system) also easily imply that the graph in the upper (resp. lower) half-plane is a tree rooted at t (resp. s), indeed
the directed path in the upper (resp. lower) half-plane starting from a given vertex gives a sequence of edges whose arcs must have increasing width, hence the path can not cycle, it thus has to end at the unique sink, which is t (resp. s). Hence, the structure we obtain is indeed a separating decomposition S endowed with its unique 2-book embbedding satisfying
(A). Let δ_1 be the mapping that sends S' to S. Clearly, we have χ_1∘δ_1(S')=S' for every S'∈_i,j'. 
To check that δ_1∘χ_1(S)=S for every S∈_i,j, we observe that, for every white vertex w∈ S'=χ_1(S), the unique black vertex allowed
to receive the outgoing arc of w in the upper (resp. lower) half-plane is the upper parent (resp. lower parent) of w. Indeed, this is the only choice so as to satisfy planarity and 
the property that the outgoing edge at every black vertex b is the outermost arc incident to b.  



We now describe the second part (from _i,j' to _i,j), which is illustrated in Figure <ref>.     
Let S'∈_i,j', with v_0,…,v_j+1 the white vertices from left to right along the horizontal axis. 
For r∈ 0,j, let α_r be the blue degree of v_r, and let β_r be the red degree of v_r+1.  
We turn v_0 into a group of α_0 blue dots, and v_j+1 into a group of β_j red dots. 
Then, for r∈ 1,j, we turn v_r into a group of β_r-1 red dots, followed by a segment-separator, followed by a group of α_r blue dots. 
We let A be the obtained arc-diagram, and let χ_2 be the mapping that associates A to S'. 


The mapping χ_2 is a bijection from _i,j' to _i,j. 


Clearly, with the notation above, A=χ_2(S') is a blue-red arc-diagram, which has j+1 segments (initially, there is one segment, then each white vertex in v_1,…,v_j creates
a separator), with α_r (resp. β_r) blue (resp. red) dots in the rth segment for r∈ 0,j. 
The number μ_r of black dots in the rth segment is the number of black vertices between v_r and 
v_r+1 on the horizontal line of S' for r∈ 0,j. Since the number i of black vertices becomes the number of black dots, A is in _i,j. The 
inverse mapping δ_2 is defined so as to reverse the construction. For A∈_i,j, we contract the group of blue dots in the first segment (with their incident blue arcs) into a white vertex s' having blue degree α_0 (and no incident red arc), we  contract the group of red dots in the last segment (with their incident red arcs) into a white vertex t' having red degree β_j (and no incident blue arc), and for each r∈ 1,j, we contract the group of β_r-1 red dots in the (r-1)th segment (with their
incident red arcs) and the group of α_r blue dots in the rth segment (with their incident blue arcs) into a white vertex v_r, which has blue degree α_r and
red degree β_r-1. We obtain an arc-diagram S'∈_i,j' (the number of white vertices is j+2, and the number of black vertices is i, as it is equal to the number of 
black dots in A). Let δ_2 be the mapping that associates S' to A. By construction, the two mappings χ_2,δ_2 are inverse of each other. 


To conclude, χ:=χ_2∘χ_1 is a bijection from _i,j to _i,j, and by construction, ξ∘χ coincides with the symmetric reformulation of Φ' as 
given in Section <ref>. Hence,  Φ' is a bijection from _i,j to _i,j.  









  §.§.§ Proof that S∈_i,j is minimal if and only if Φ'(S)∈_i,j
 
In a blue-red arc diagram, a Z-pattern is a pair made of a blue arc a and a red arc a'
such that the blue extremity of a is enclosed within a' and the red extremity of a' is enclosed within a,
see Figure <ref>(a). 



Our proof that S∈_i,j is minimal if and only if Φ'(S)∈_i,j relies on the two following lemmas.


Let S∈_i,j and let A∈_i,j be the corresponding blue-red arc diagram (i.e., A=χ(S)). 
Then S is minimal
if and only if A has no Z-pattern. 
 

It is known (see e.g. <cit.>) that, if S is not minimal, then it contains a clockwise 4-cycle C (not necessarily the contour of a face),
and any edge in the interior of C and incident to a vertex v∈ C is incoming at v. 
The local conditions (Figure <ref>(a)) imply that the colors are as shown in Figure <ref>(b). Then, in the 2-book embedding, the alternating
property (A) ensures that the 4 edges of C are as shown in Figure <ref>(c). Hence, the two arcs of A resulting from the two edges of C going out of a black vertex form a Z-pattern. 

Conversely, assume A has a Z-pattern. A Z-pattern is called minimum if the distance along the line
between its blue dot and its red dot is smallest possible. 
Let a,a' be a pair forming a minimum Z-pattern.   
Let e={b,w} and e'={b',w'} be the corresponding edges in the 2-book embedding of S. 
Let e_1' be the outgoing red edge of w and let b̃ be its black extremity. 
Assume b̃≠ b'. Then, by Lemma <ref>, the outgoing red edge e_2' of b̃ is in the area between e_1' and e', and 
it ends between w and w' (w excluded, since the quadrangulation is simple). Let a_2' be the arc of A that corresponds to e_2. 
Then clearly the pair of arcs a,a_2' forms a Z-pattern 
in A, contradicting the fact that the pair a,a' is minimum. Hence e_1' ends at b'. Similarly,  
the outgoing blue edge of w' has to end at b. Hence the vertices b,w,b',w' form a clockwise 4-cycle, which implies that 
S is not minimal.  





Let R=(,,)∈_i,j and let A∈_i,j be the corresponding blue-red arc diagram (i.e.,
 R=ξ(A)). Then R∈_i,j 
if and only if A has no Z-pattern. 



The modified arc diagram Ã of A is the same as A except that the arcs in the upper part 
match planarly the blue dots to the red dots instead of matching planarly the black dots to the red dots, see Figure <ref>.
 (the fact that the planar matching is doable in the upper half-plane follows from the fact that  is weakly above ).   
Let b_1,…,b_i be the blue dots of Ã ordered from left to right, and let ν:=. 
For k∈ 1,i, let U_k∈ 0,j be the index of the segment containing
the black dot matched with b_k, and let V_k∈ 0,j be the index of the segment containing
the red dot matched with b_k, see the left part of Figure <ref>. The vectors   
U⃗=(U_1,…,U_i) and  V⃗=(V_1,…,V_i) are the bracket vectors <cit.> 
of  and 
with respect to ν (compared to <cit.> we omit the fixed underlined entries).    
It is shown in Section 4 of <cit.> that
 ≤ in (ν) iff U⃗≤V⃗ (component by component). 
  
Clearly, this is equivalent to the fact
that the modified arc diagram Ã avoids the pattern shown in the right part of Figure <ref>, i.e., for each blue point, its 
matched red point (via the incident arc in the upper half-plane) is on the right of its matched black point (via the incident arc in the lower half-plane).  
Indeed, the index of the segment to which the red dot belongs has to be greater or equal to the index of the segment to which the black dots belongs; 
since the group of red dots comes after the group of black dots in each segment, we conclude that the red dot has to be on the right of the black dot. 

Assume R∉_i,j. Then Ã contains a pattern as in the right part of Figure <ref>. In this pattern, let d_1,d_2,d_3 be the blue dot, red dot and black dot, 
let a”=(d_1,d_2) be the gray arc above and a=(d_1,d_3) the blue arc below. Let E be the 
part of the line strictly between d_1 and d_2, and   
let ,, be respectively the numbers of blue dots, black dots,
and red dots in E. We have = since d_1 and d_2 are matched 
(for the upper diagram). We have ≥ since in the lower diagram d_1 is matched to a black dot 
that is on the right of d_2. Hence ≥. In other word, in E∪ d_2 
 we have more red dots than black dots. This implies that, in the upper diagram of A,   
there is a red dot in E∪ d_2 that is matched to a black dot on the left of d_1. 
Hence, if we let a' be the arc formed by this matched pair, then the pair a,a' is a Z-pattern in A. 

Assume now that A contains a Z-pattern, and let a,a' be a pair of arcs forming a minimum Z-pattern.  
We denote by d_1 the blue extremity of a, by d_2 the red extremity of a', and by E the 
part of the line strictly between d_1 and d_2. 
Let ,, be respectively the numbers of blue dots, black dots,
and red dots in E.  
The fact that the pattern is minimum easily implies that there is no arc (neither in the upper nor in the lower
diagram) starting from E and ending outside of E. Hence ==. In the upper diagram of Ã, d_1 has thus to be matched with a red dot belonging to E∪ d_2. Clearly, this arc a” together with a form a pattern
as in the right part of Figure <ref>, hence R∉_i,j.    



 







§ LINK TO A BIJECTION BY KENYON ET AL. <CIT.>
 


A plane bipolar orientation is a map endowed
with an acyclic orientation having a unique source s and a unique sink t both incident to the outer face, with s the root-vertex, 
see Figure <ref> left.
It is known <cit.> that a plane bipolar orientation is characterized by the following local conditions:


  * Every non-pole vertex (vertex ∉{s,t})  has its incident edges partitioned into a non-empty interval of incoming edges
and a non-empty interval of outgoing edges. 

  * Every inner face has its incident edges partitioned
into a non-empty interval of clockwise edges and a non-empty interval of counterclockwise edges. 

An inner face consisting of p+1 clockwise edges and q+1 counterclockwise edges is said to have type (p,q).  
Let _i,j be the set of plane bipolar orientations with i non-pole vertices and j inner face,
and let _i,j[a,b] be the subset of those
 where the left (resp. right) outer boundary has length a+1 (resp. b+1). 
Let _i,j[a,b] be the subset of _i,j where s' has degree a+2 and t' has degree b+2. 
There is a direct bijection ι (illustrated in Figure <ref>) from _i,j[a,b] to 
 _i,j[a,b] where each vertex corresponds to a black vertex, and each inner face corresponds 
to an inner white vertex of the same type <cit.>.  





On the other hand, a tandem walk is a 2d walk where each step is either (1,-1) (SE step), or a step of the form (-p,q) for some p,q≥ 0. 
Let _i,j[a,b] be the set of tandem walks of length i+j with i SE steps, starting
at (0,a), ending at (b,0) and staying in the quadrant ℕ^2. 
Let _i,j[a,b] be the subset 
of elements in _i,j where α_0=a and β_j=b. We now recall a bijection σ from _i,j[a,b]
to _i,j[a,b] recently described  in <cit.> (where we slightly change the convention: 
the lower and upper walks are constructed line by line here, and column by column in <cit.>). For R=(,,)∈_i,j[a,b], we let n=i+j and let u_1,…,u_n be the successive steps in . 
For m∈ 1,n, we let s_m be a SE step if u_m
is a horizontal step, and let s_m=(-p,q) if u_m is the rth vertical step of  (with r∈ 1,j), where p=β_r-1 and q=α_r. Then σ(R) is defined as the walk (in _i,j[a,b]) of length n starting at (0,a) and with successive steps s_1,…,s_n, see Figure <ref> for an example. Combining the bijection Φ' with ι and σ, and with the parameter-correspondence of Φ' stated in Remark <ref>, we obtain:


The mapping Λ:=σ∘Φ'∘ι is a bijection from _i,j[a,b] to _i,j[a,b]. Each non-pole 
vertex corresponds to a SE step, and each inner face of type (p,q) corresponds to a step (-p,q). 




A bijection from _i,j[a,b] to _i,j[a,b] with the same parameter correspondence 
has been recently introduced by Kenyon et al. <cit.>. 
We recall their consruction, and then prove that it coincides with Λ. 

For a bipolar orientation B∈_i,j[a,b], the rightmost tree
 T(B) of B is the spanning tree of B\{s} obtained by selecting every edge
of B that is the rightmost outgoing edge at its origin, with the exception that the rightmost outgoing edge ẽ of s is not selected 
(see the left-part of Figure <ref>). 
An internal edge is an edge in T(B). An external edge is an edge not in T(B), and different from ẽ.  
Clearly, there is a one-to-one correspondence between internal edges and non-pole vertices: every internal edge e is the rightmost outgoing edge of exactly 
one non-pole vertex, which is denoted v(e). 
Similarly, there is a one-to-one correspondence between external edges and inner faces: every external edge e is the bottom-left edge of 
exactly one inner face, which is denoted f(e).  

A counterclockwise 
walk around T(B) yields an ordered list[More generally, such an ordering of the edges can be considered 
for any map endowed with a spanning tree, see <cit.> where
it is exploited to get bijective insights on the Tutte polynomial.] γ=(e_1,…,e_i+j) of the edges of B\ẽ.  
Starting with γ=∅, each time we walk along an internal edge e away from the root s, we 
append e to γ, and each time we cross the incoming half of an external edge e, we append e to γ.
Let W be the walk starting at (0,a), with successive steps s_1,…,s_i+j, 
obtained as follows. For every k∈ 1,i+j, if e_k is internal, then s_k is a SE step, while if 
e_k is external, then s_k=(-p,q), where (p,q) is the type of the inner face f(e_k), see Figure <ref> for an example. 
It is shown in <cit.> that W∈_i,j[a,b], and that the mapping that associates W to B is a bijection from _i,j[a,b] to _i,j[a,b]. 


We have presented here the KMSW with mirror conventions compared to <cit.> (which relies on the leftmost tree of B). 







The bijection Λ coincides with the KMSW bijection (with mirror conventions).


Let B be a plane bipolar orientation with n+1 edges, and left boundary of length a+1. Let S=ι(B) be the corresponding separating decomposition.  
The bijection Λ amounts to visit the inner vertices of S along the equatorial line H, producing a SE step when visiting a black vertex, and
producing a step (-p,q) when visiting a white vertex having p incoming red edges and q incoming blue edges (the produced walk starting at (0,a)).  
Let ẽ be the bottom edge on the right outer boundary of B, and let L=(e_1,…,e_n) be the ordered list of edges of B\ẽ used in the KMSW bijection.
For every edge e of B\ẽ that is internal (resp. external), we let o(e) be the black (resp. white) inner vertex of S corresponding to v(e) (resp. to f(e)).  
Via the mapping ι, the KMSW bijection amounts to produce (with starting point (0,a)) the step sequence s_1,…,s_n, where, for k∈ 1,n, s_k is a SE step if o(e_k)
is black, and is a step (-p,q) if o(e_k) is a white vertex having p incoming red edges and q incoming blue edges. 
Thus, we just have to check that o(e_1),…,o(e_n)
gives the list of inner vertices of S ordered along H.  
This property (which can be visualised in Figure <ref>(b)) amounts to check that, for two consecutive edges e',e” along L, the vertices o(e') and o(e”) are
adjacent on H. This can be checked by a case-by-case analysis. Consider the case where e' is external, and let f(e') 
be the corresponding inner face of B, i.e., e' is the bottom edge on the left boundary of f(e'). Note that, apart from e',  all the edges on the left boundary of f(e') are internal.    
 This easily implies that e” (whether external or internal) has to be the top edge on the right boundary of f(e').  An easy inspection ensures that, if e” is internal (resp. external), then the white vertex o(e') is adjacent to the black (resp. white) vertex o(e”) on H. On the other hand, if e' is internal, let v(e') be the corresponding non-pole vertex, i.e., v(e') is the origin of e'. Then e” (whether external or internal) has to be the leftmost incoming edge at v(e'). Again, by inspection, if e” is internal (resp. external),  
 then the black vertex o(e')=v(e') is adjacent to the black (resp. white) vertex o(e”) on H.













 
















Remark. Another bijection from _i,j to _i,j is presented in <cit.>. 
It relies on the rightmost incoming tree of the (dual) bipolar orientation (i.e., the tree formed by the rightmost incoming edges of non-pole vertices),   
and it  is closely related to the bijection σ^-1∘Λ. 
However, similarly as when taking Φ instead of Φ', one of the three walks (the upper one) differs. 




§ BIJECTION USING SCHNYDER WOODS

We first recall the definitions of Schnyder woods and Schnyder labelings <cit.>. 
For T a simple triangulation, the outer 
vertices are called u_B,u_G,u_R in clockwise order, with u_B the one incident to the root-corner. 
A Schnyder wood of T is  an orientation and coloration
(in blue, green or red) of every inner edge  of T such that all edges incident to u_B,u_G,u_R
are incoming of color blue (resp. green, red), and every inner vertex has outdegree 3 and satisfies the local condition 
shown in Figure <ref>(a).  
A Schnyder wood induces a coloring of the corners. For each corner c at an inner vertex v, the edge opposite to c is the second outgoing edge encountered after c
in clockwise order around v. Then each corner at an inner vertex inherits the color of its opposite edge, and each corner at an outer vertex v receives the color of v. 
It can be checked that,  
around each inner face, there is one corner in each color and these occur as blue, green, red in clockwise order. 
It is known that the local conditions of Schnyder woods imply that the graph in every color is a tree spanning all the internal vertices (plus the outer vertex of the same color, which is the root-vertex of the tree).    
A Schnyder wood is called minimal if it has no clockwise cycle. Any simple triangulation has a unique minimal Schnyder wood <cit.>. 



Let _n be the set of pairs (γ,γ') of Dyck walks of length 2n such that γ' is weakly above γ. 
The Bernardi-Bonichon construction <cit.> 
starts from a simple triangulation with n+3 vertices endowed with a Schnyder wood, 
and outputs a pair (γ,γ')∈_n. Precisely (see Figure <ref> for an example), we let  
be the blue tree of the Schnyder wood plus the outer edge e'={u_B,u_R}, and let v_0,…,v_n=u_R be the vertices
of \{u_B} ordered according to the first visit in a clockwise walk around  starting at u_B. 
Then γ is obtained as the contour walk of \ e' and γ' 
is NE^β_1NE^β_2⋯ NE^β_n, with β_r the number of incoming red edges at v_r
 for r∈ 1,n. Bernardi and Bonichon 
show <cit.> that this construction gives a bijection between Schnyder woods on simple triangulations with n+3 vertices and 
_n; and they show that it specializes into a bijection between minimal Schnyder woods with n+3 vertices 
 and _n⊂_n. Theorem <ref> can be seen as an extension of this statement to  separating decompositions, using
the fact <cit.> that Schnyder woods correspond bijectively to separating decompositions where s' has blue indegree 0, 
and all inner white vertices have blue indegree 1, and the bijection preserves the property of having no clockwise cycle.   

On the other hand, minimal Schnyder woods are themselves known to be in bijection to certain 
tree structures <cit.>. We will use here the bijection from <cit.>.  
A 3-mobile is a (non-rooted) plane tree T where vertices have degree in {1,3},  
respectively called leaves and nodes, and with an additional color structure given by the following conditions:


  * The nodes are colored black or white, so that adjacent nodes have different colors, and there is at least one white node. 

  * All leaves are adjacent to black nodes.

  * The edges are colored blue, green or red, such that around each node, the incident edges in clockwise order are blue, green and red.


In a 3-mobile, an edge is called a leg if it is incident to a leaf and is called a plain edge  otherwise. 
For n≥ 1, let _n be the set of 3-mobiles with n white nodes. 
From a simple triangulation M on n+3 vertices, endowed with its minimal Schnyder wood,    
one builds a 3-mobile T∈_n as follows (see Figure <ref>): 


  * Orient the outer cycle clockwise.

  * Insert a black vertex b_f in each inner face f of M.

  * For each edge e=u→ v, with (f, f') the faces
on the left and on the right of e, create a plain edge {u,b_f} (if f is an inner face), and create  a leg at b_f' pointing (but not reaching) to v.
Give to the plain edge (resp. to the leg) the color of the corresponding corner of M. 

  * Erase the outer vertices and the edges of M. 



Composing both constructions, we get a bijection between _n and _n.  
Let (γ,γ')∈_n, with n≥ 1. For dc∈{EE,NN,EN}, we say that a position 
r∈ 0,n is of type dc if there is c (resp. d) at position r in (γ) (resp. in (γ')). 
On the other hand, let S be the minimal Schnyder wood with n+3 vertices associated to (γ,γ') via the Bernardi-Bonichon construction, and let T 
be the 3-mobile associated to S. As described above,  
we let v_0,…,v_n=u_R be the vertices
of \{u_B} ordered according to the first visit in a clockwise walk around  starting at u_B. 
For r∈ 0,n, let e_r be the outgoing blue edge of v_r (with the convention that e_n is the outer edge {u_R,u_B}).  
Let f_r be the face on the right of e_r, and let b_r be the corresponding black node in  T. 
This gives a 1-to-1 correspondence between 0,n and black nodes of T whose blue edge is a leg.  
Such a node is said to be of type EE if its green edge is a leg, of type NN if its red edge is a leg, 
and of type EN otherwise (only the blue edge is a leg), see the last column in Figure <ref>.  
We claim that the type is preserved under this correspondence, i.e., a 
position r∈ 0,n is of type EE (resp. NN,EN) iff b_r is of the same type.  
Indeed, by the Bernardi-Bonichon construction, (γ') has E at position r if and only if there is no incoming red edge at v_r, 
and (γ) has N at position r if and only if e comes just after a `valley' in a clockwise walk around . 
From the local conditions of Schnyder woods, it is easy to see
that each of the 3 possible types for r correspond to f=f_r being in each of the configurations shown in the left column
of Figure <ref>. By the local rules of the mobile constructions, these configurations correspond to b=b_r being respectively of type 
 EE, NN, and EN. To summarize, we obtain the following bijective result: 




 


Let n≥ 1. The composition of the Bernardi-Bonichon construction and of the 3-mobile construction gives a bijection between _n and _n such that each position r∈ 0,n corresponds to a black node b_r (whose blue edge is a leg) of the same type. 


















Let i,j,k≥ 0, and n=i+j+k+1. We denote by a[i,j,k] the number of intervals in _n having i+1 positions of type EE, 
j+1 positions of type NN and k positions of type EN, and we let F(x,y,z):=∑_i,j,ka[i,j,k]x^i+1y^j+1z^k
be the associated generating function. Note that F(x,y,0)=∑_i,j|_i,j|x^i+1y^j+1=∑_i,j|_i,j|x^i+1y^j+1.  


The generating function F≡ F(x,y,z) is given by





    F=xR+yG+zRG-RG/(1+R)(1+G),
 



where R,G are the trivariate series (in x,y,z) specified by the system 

    {[                    R =(y+zR)(1+R)(1+G)^2,;                    G =(x+zG)(1+G)(1+R)^2. ].


Before proving the corollary, we note that   
F(x,y,0) coincides (upon setting G=u/(1-u) and R=v/(1-v)) with the known expression <cit.>
 of the bivariate series ∑_i,j|_i,j|x^i+1y^j+1, and we recover 
|_i,j|=|_i,j| (we will also give a bijective argument at the end of the section); 
and t+F(t,t,t) coincides (upon setting G=R=θ/(1-θ)) with the known expression <cit.> of the series counting  simple triangulations by the number of vertices minus 2. 

A planted 3-mobile T is defined similarly as a 3-mobile
 except that (exactly) one of the leaves is adjacent to a white node. 
This leaf is called the root of T, and its incident edge is called the root-edge. A planted 3-mobile
is called blue-rooted (resp. red-rooted, green-rooted) if its root-edge is blue (resp. red, green). 
 We keep the same definition of 
black nodes of types EE, NN, EN as for 3-mobiles.  We let B,R,G be the trivariate (variables x,y,z)  generating functions of blue-rooted, red-rooted, and green-rooted planted 3-mobiles, where x (resp. y, z) is conjugate 
to the number of black nodes of type EE (resp. type NN, type EN). A 2-levels decomposition at the root 
translates into the following equation-system:

    {[                            B =(B(1+G)+x+zG)(B(1+R)+y+zR),;                            R    =(B(1+R)+y+zR)(1+R)(1+G),;                            G    =(B(1+G)+x+zG)(1+R)(1+G). ].

In a 3-mobile T, the number of blue legs is one more
than the number of blue plain edges. Indeed, letting S be the associated minimal Schnyder wood,  there is a blue plain edge in T associated to every red edge (outgoing part) of S, and there is a blue leg associated to every blue edge (incoming part) of S, plus an extra blue leg associated to the outer edge (u_B,u_R). 
Hence, by Theorem <ref>, 
F=F_1-F_2 where F_1 is the trivariate series of 3-mobiles with a marked blue leg, 
and F_2 is the trivariate series of 3-mobiles with a marked blue plain edge. A decomposition
at the marked blue leg gives F_1=xR+yG+zRG, and a decomposition at the marked blue plain edge gives F_2=B(1+R)(1+G). We now simplify the equation-system 
by eliminating B. We look at the quantity B(1+R)/G, where we substitute B and G by their 
respective expressions in the equation-system. After simplification, this gives B(1+R)/G=B(1+R)+y+zR/1+G, 
so that B(1+R)=yG+zRG. Similarly, looking at the quantity B(1+G)/R, we obtain B(1+G)=xR+zRG. Substituting each occurence of B(1+R) (resp. B(1+G))
by yG+zRG (resp. by xR+zRG) into the three-line system above, we obtain 

    {[                        B =(1+R)(x+zG)(1+G)(y+zR),;                        R     =(y+zR)(1+R)(1+G)^2,;                        G     =(x+zG)(1+G)(1+R)^2. ].

Then, substituting B by its expression (given in the first line) into F_2=B(1+R)(1+G), we obtain F_2=(x+zG)(y+zR)(1+R)^2(1+G)^2=RG/(1+R)(1+G).







A bijection between _i,j and _i,j via mobiles. A 3-mobile (with at least
one white node) is called synchronized if it has no black node of type EN, i.e., every black node  b 
having a blue leg is incident to (exactly) one other leg. If this other leg is
 red (resp. green) then b has type NN
(resp. type EE). We let _i,j^syn be the set of synchronized 3-mobiles
with i+1 black nodes of type EE and j+1 black nodes of type NN. It follows
from Theorem <ref> 
 that _i,j^syn is in bijection with  _i,j (synchronized intervals such that the common canopy-word is in S(E^i+1N^j+1)), 
 itself in bijection with _i,j. 

 

An unrooted ternary tree is a plane tree where all vertices have degree in {1,4},
called respectively leaves and nodes. 
An unrooted ternary tree T is said to be bicolored if its edges are 
 colored green or red and are oriented such that, around each node, the incident edges in clockwise order are incoming red,
outgoing green, outgoing red, and incoming green. A leaf ℓ is called outgoing (resp. incoming) if 
its incident edge e is outgoing (resp. incoming) at ℓ, and is called red (resp. green) if e is red (resp. green). 
It is easy to see that T has as many red leaves that are outgoing as incoming, and as many green leaves that 
are outgoing as incoming. We let _i,j 
be the set of bicolored unrooted ternary trees having i+1 outgoing red leaves and j+1 outgoing green leaves. 
A bijection between _i,j and _i,j has been 
introduced in Section 2.3.3 of <cit.> and recovered in <cit.> (to obtain a ternary tree from a minimal separating decomposition, it actually uses the same local rules as those to obtain a 3-mobile from a minimal Schnyder wood, the inverse construction relies on the repeated use of so-called “local closure" operations
 that yield the dual map of a simple quadrangulation).   

Hence, to derive another bijection between _i,j and _i,j it remains to give a bijection
 between _i,j^syn and _i,j. The bijection, shown in Figure <ref>, is very simple. 
For T∈_i,j^syn, the corresponding U∈_i,j is obtained as follows: 
orient all the plain edges of T from black to white nodes, then contract the blue plain edges, 
and finally delete the two legs at each 
black node of type NN or EE (the black nodes of type EE become outgoing red leaves,
those of type NN become outgoing green leaves). 






Acknowledgement. 
The authors are grateful to the two anonymous referees for very helpful comments and suggestions to improve the presentation, and 
 thank Frédéric Chapoton, Guillaume Chapuy, Wenjie Fang, and Mathias Lepoutre for interesting discussions.  
ÉF acknowledges the support of ANR-16-CE40-0009-01 “GATO”, and AH the support of ERC-2016-STG 716083 “CombiTop”.






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