Fully Packed Loops in a triangle: matchings, paths and puzzles
Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the study of ordinary Fully Packed Loop configurations (FPLs) on the square grid where they were used to show that the number of FPLs with a given link pattern that has nested arches is a polynomial function in . It soon turned out that TFPLs possess a number of other nice properties. For instance, they can be seen as a generalized model of Littlewood-Richardson coefficients. We start our article by introducing oriented versions of TFPLs; their main advantage in comparison with ordinary TFPLs is that they involve only local constraints. Three main contributions are provided. Firstly, we show that the number of ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs and thus it suffices to consider the latter. Secondly, we decompose oriented TFPLs into two matchings and use a classical bijection to obtain two families of nonintersecting lattice paths (path tangles). This point of view turns out to be extremely useful for giving easy proofs of previously known conditions on the boundary of TFPLs necessary for them to exist. One example is the inequality where are -words that encode the boundary conditions of ordinary TFPLs and is the number of cells in the Ferrers diagram associated with . In the third part we consider TFPLs with ; in the first case their numbers are given by Littlewood-Richardson coefficients, but also in the second case we provide formulas that are in terms of Littlewood-Richardson coefficients. The proofs of these formulas are of a purely combinatorial nature.
Fully Packed Loop configurations (FPLs) are subgraphs of a finite square grid such that each internal vertex has degree , and the boundary conditions are alternating, see Figure 1. These objects made their first appearance in statistical mechanics, and were later realized to be in bijection with the famous Alternating Sign Matrices, as well as numerous other structures; cf. [Pro01]. In particular, the total number of FPLs on a grid with vertices is given by the famous formula first proven by Zeilberger; the story of this problem is told in the book by Bressoud [Bre99].
What distinguishes FPL configurations from other structures in bijection are the paths that join two of the external edges. Therefore each FPL configuration is associated with a link pattern, which encodes the pairs of endpoints that are joined by paths. The quantity of interest then becomes: given a link pattern , how many FPL configurations have associated pattern ? These integers attracted some interest from mathematicians thanks to the Razumov-Stroganov (ex-)conjecture [RS04], which says that there exists a simple Markov chain on the set of link patterns of size such that its stationary distribution is given by . This was proven by Cantini and Sportiello in 2010 [CS11]. The proof proceeds by pretty combinatorial arguments, making heavy use of Wieland gyration [Wie00] which is a particular operation on FPLs that was originally invented to prove a certain rotational invariance of the numbers .
Another line of research was developed in the works of Di Francesco and Zinn-Justin, see [ZJ10b] and the articles cited in there. They relate the numbers to certain quantities through a change of basis, and give formulas for the latter in terms of multiple integrals. Thanks to the result of [CS11], these become formulas for the numbers themselves. The drawback of these formulas is that they lack combinatorial interpretations. That is, one would like to understand FPLs well enough to be able to write formulas directly from a combinatorial analysis, which reflects their structure.
From the combinatorial point of view, the study of FPLs was advertised in the paper [Zub04], which gathers several conjectures around the enumeration of FPL configurations. For instance, it deals with FPLs refined according to link patterns with “nested arches”, written : the conjecture is that is a polynomial function with an explicit dominant term. This was proven in [CKLN04]. Experimentally, these polynomials appear to have a number of surprising properties, notably regarding their roots: several conjectures were made in the work of the second author with Fonseca [FN11].
For big enough, FPLs with link pattern admit a combinatorial decomposition in which Fully Packed Loops in a triangle (TFPLs) naturally arise (it has in particular the enumerative consequence summarized later in the expression (1.3)). These TFPLs as well as oriented versions of them (see Definition 1.8) are the focus of the present article, their precise definition being given in Section 1.1. An oriented example is displayed in Figure 1, right. Here we shall just say that the internal vertices have degree ; only on the three boundaries of the triangle there are vertices of degree or . TFPLs are enumerated according to a refinement encoded by three binary words of the same length , so that we speak of TFPLs with boundary .
Aside from their appearance in the expression (1.3) for FPLs, there are at least two other motivations to study TFPLs: The first one is that they allow us to compute nice coefficients in the following polynomial identity:
Here are link patterns of size . These relations were conjectured in [Tha07] and proven in [Nada]; notice that they are different from the relations in the Razumov-Stroganov (ex-)conjecture, because they involve FPLs on two different grid sizes.
The second motivation comes a posteriori, in the sense that, when one delves into the structure of TFPLs, some intriguing and beautiful combinatorics come up naturally. This was already the case in the two articles just mentioned, where in particular a strong link with semistandard tableaux is established. It was then shown in [Nadb] that, under the constraint , TFPLs with boundary are enumerated by Littlewood–Richardson coefficients; here is the number of inversions of a binary word. The proof in [Nadb] proceeds by a convoluted argument. In Section 5 of the present paper, this result will be proven again, in a straightforward way, and in a more general setting.
We distinguish three main contributions in this article:
The article can be divided roughly into three parts: Sections 1 and 2 deal with the relations between ordinary and oriented TFPLs, Sections 3 and 4 with the structure of oriented TFPLs and path tangles, and Sections 5 and 6 with the enumeration of TFPLs, ordinary and oriented, in some special cases. We now detail the content of each of these parts.
TFPLs and oriented TFPLs
The interplay between ordinary TFPLs and oriented TFPLs is the focus of Sections 1 and 2. We first define both objects in slightly greater generality than in previous works: both kinds of TFPLs were used indeed in [Nada, Nadb, Tha07, ZJ10a], but with a specific restriction on the boundary words and a specific orientation of the paths. The obvious reason for this is that this restriction on the boundary holds in Formula (1.3) expressing FPLs in terms of TFPLs, and that, with the specific orientation, ordinary TFPLs and oriented TFPLs are in fact equivalent. So why bother generalizing? First, such generalized TFPLs occurred already (although they were note explicitly defined) in [Nada, Section 5]. Second, when studying TFPLs directly as we will do in this paper, it soon appears that keeping the extra constraints is not useful. Although it makes the definition of ordinary TFPLs a bit more involved since we need to extend the definition of patterns, the definition of oriented TFPLs becomes more natural.
Now given any three words of the same length, the set of TFPLs with boundary can be naturally embedded into the set of oriented TFPLs with the same boundary conditions. In particular, if there is no oriented TFPL with boundary , then there can be no such ordinary TFPL either. Constraints on the boundaries of TFPLs are summarized in Theorem 3.1 and its corollary. The upshot is that oriented TFPLs with boundary are easier to manipulate than TFPLs, essentially because their definition involves only local constraints. But even more so, we can actually derive the enumeration of TFPLs from a certain weighted enumeration of oriented TFPLs: this is the content of Section 2, the final result being the second equation in Corollary 2.9.
As a consequence oriented TFPLs are the primary object of study in the rest of the paper.
Oriented TFPLs, Matchings and Path Tangles
The second contribution of this paper is the introduction of certain path tangles which encode oriented TFPLs nicely, and help prove a combinatorial interpretation of the quantity , see Theorem 4.3. This is the focus of Sections 3 and 4.The key idea is to split an oriented TFPL into two perfect matchings (Theorem 3.3); we can then proceed to study individually each matching, which allows already to prove the first two statements of Theorem 3.1. Each matching can be itself encoded as a configuration of nonintersecting lattice paths.
When the path configurations from the two matchings are reunited, one obtains what we chose to call a path tangle: indeed here the paths intersect in general. The possible ways that they may cross is constrained by the fact that the two perfect matchings are disjoint. This gives a bijection between oriented TFPLs with a given boundary and path tangles with prescribed departure and arrival points for each path: this is Theorem 4.1. The study of the ways the paths intersect in a path tangle leads to the explicit formula of Theorem 4.3, which gives a combinatorial interpretation of the quantity : it is the number of occurrences of certain local patterns in any TFPL with boundary . In particular, it shows that such TFPLs cannot exist if , which completes the proof of Theorem 3.1.
Enumeration of TFPLs when or
The starting point here is Theorem 4.3, which says that the quantity (which we call the excess) is equal to the number of occurrences of various local patterns in any oriented TFPL with boundary . It is therefore natural to look first at the special cases of small excess.
The case where the excess is was dealt with first in [Nadb]: it was proven there that, in the case of ordinary TFPLs and when is restricted to be essentially a Dyck word, the number of TFPLs with boundary is given by the Littlewood–Richardson coefficient , where is a natural way to associate an integer partition with a word. Here we prove this result again in Section 5, removing the restriction on and extending it to oriented TFPLs as well. The bijection at the core of the proof is the same as in [Nadb], i.e. a map to Knutson–Tao puzzles [KT03]; here, though, the proof that this is indeed a one-to-one correspondence is direct and avoids the roundabout approach of the aforementioned article.
We then go one step further and achieve the enumeration of configurations with excess in Section 6: this is the most complex part of this work. However, the line of argument is of combinatorial nature. We show first that TFPL configurations of excess look like a configuration of excess but with one “defect”. We encode such configurations by puzzles which extend those of Knutson and Tao by one extra piece. To enumerate the puzzles, we determine some rules which move the extra piece to the boundary of the puzzle. This shows how the enumeration in the case of excess can be reduced to the case of excess ; the resulting expression is Theorem 6.20(3). To finish, we show how to deduce the enumeration of ordinary TFPLs with excess in Theorem 6.21.
In this section we will define FPLs on a triangle (TFPLs), as well as oriented TFPLs, in a more general setting than in previous works.
Definition 1.1 (The graph ).
Let be a positive integer. We define as the induced subgraph of the square lattice made up of consecutive centered rows with vertices from top to bottom.
The graph is represented in Figure 2. Note that is a bipartite graph, where the bipartition consists of odd and even vertices; by convention the vertices on the left side are odd. In the pictures, we will represent odd vertices by circles while even vertices will be represented by squares. Some vertices play a special role: we let be the set of even vertices on the bottom row of , and (resp. ) be the set of odd vertices which are leftmost (resp. rightmost) in each row of . All vertices are numbered from left to right, cf. Figure 2 again.
In the whole article we call words of length the finite sequences where for all . We denote by (resp. ) the number of occurrences of (resp. ) in the word. Define a partial order on words of length by if and only if for . This is especially nice to see on the Ferrers diagram associated to the word , cf. Figure 3.: if each have occurrences of and occurrences of , then if and only if .
We can now define Fully Packed Loop configurations in a triangle, or TFPLs in short.
A TFPL of size is a subgraph of , such that:
The vertices of have degree or .
The vertices of have degree .
All other vertices of have degree .
A path in cannot join two vertices of , nor two vertices of .
An example of a TFPL for is given on Figure 4.
In Section 2.1, we will need to consider local configurations around each vertex of a TFPL, and for this reason it is necessary that all vertices have degree in a TFPL. Therefore we introduce external edges on the left and right boundary, as well as below all even vertices on the bottom boundary, to ensure that all vertices of have degree . These external edges are represented in Figure 4 by dotted lines.
The first three conditions of Definition 1.2 show that a TFPL configuration is composed of a number of paths, in which the non-closed paths have their extremities in . We will be interested in the structure and enumeration of TFPLs according to certain boundary conditions that depend on the extremities of non-closed paths:
To each TFPL are associated three words of length as follows:
If the vertex has degree then , otherwise .
If the vertex has degree then , otherwise .
Consider the path starting from the vertex , and let be the other endpoint of this path. If then , while if it belongs to then .
We say that the TFPL has boundary . We denote the set of configurations with boundary by , and its cardinality by .
The words attached to the TFPL of Figure 4 are represented on the same figure. We first note an evident symmetry of TFPLs. Given a word , define as the word where . Also note that is the conjugate of .
Vertical symmetry exchanges and ; in particular, .
Define and . Given a configuration in , the set of all endpoints of its paths is then (when ignoring the external edges). To encode the pairs of endpoints linked by a path in , we need the notion of extended link patterns.
1.2. Extended link patterns
Define a link pattern of size as a partition of in pairwise noncrossing pairs , which means that there are no integers such that and are both in . We will represent link patterns as noncrossing arches between aligned points, see Figure 5. We denote by the set of words of length , such that and each prefix of verifies ; these are called Dyck words.
There is then a bijection between link patterns and Dyck words, defined simply by associating to the word such that if and only if is the smaller element in the pair of . As an example, is associated to the pattern of Figure 5. This correspondence can be extended to all words (and not only Dyck words) by introducing the notion of extended link patterns:
An extended link pattern on is the data of integers (left points) and (right points), with , together with a link pattern on each maximal interval of integers not containing any of the points or .
In figures we will represent left and right points by attaching the extremity of an arch to the points and , with the arch going left (resp. right) for a left point (resp. a right point ); see Figure 6, where the left points are and the right point is . Extended link patterns can be in fact equivalently defined as usual link patterns on for certain , such that no two elements in belong to the same pair.
Given an extended link pattern with integers and as above, define a word of length as follows: first set and for all left and right points, and associate with each link pattern appearing in its corresponding Dyck word. As an example, the word associated with the pattern of Figure 6 is .
The function is a bijection from extended link patterns on to words of length .
We show how to construct the inverse of . Let be a word of length : it can be uniquely decomposed as the concatenation
where and the words , and are all Dyck words. Let and be the indices of the s and s respectively which occur in (1.1). Construct an extended link pattern on as follows: the and are left and right points respectively, while to each Dyck word in (1.1) associate a usual link pattern. This gives the desired inverse bijection to , as is readily checked. ∎
To a TFPL is naturally associated an extended link pattern on as follows: if are linked by a path in , then , while if is linked to a vertex of (resp. ) then is a left point of (resp. a right point). The following is now an immediate consequence of Definition 1.3 (3) and the definition of .
For any TFPL with extended link pattern , one has .
So the words describe exactly where each path of starts and ends: the set of endpoints is determined once we know and , and Proposition 1.7 shows that encodes the extended link pattern, which suffices to determine the pairs of endpoints which are connected together.
1.3. Oriented TFPLs
TFPLs with boundary conditions appear naturally in the study of FPLs on the square grid, as shown in [CKLN04, Nada]. The difficulty in enumerating them lies in part in the fact that their definition involves global conditions, since both conditions, (4) in Definition 1.2 and (3) in Definition 1.3, involve figuring out how endpoints are connected two by two. The notion of oriented TFPLs that we study in this section only involves local conditions, and will therefore be easier to deal with. Their relation to TFPLs is studied in Section 2, where we will see the important fact that one can recover the enumeration of TFPLs from a certain weighted enumeration of oriented TFPLs.
An oriented TFPL of size is a TFPL on together with an orientation of each edge with the following conditions: each degree vertex has one incoming and one outgoing edge; the edges attached to are outgoing; the edges attached to are incoming.
We introduce the same external edges as in non-oriented TFPLs, represented by dotted lines in Figure 7; their orientation is chosen such that each vertex has one incoming and one outgoing edge. Note also that the global Condition (4) in Definition 1.2 can be omitted when dealing with oriented TFPLs, since the orientations on the left or right boundaries automatically prevent paths from returning to these boundaries; therefore the constraints on an oriented TFPL configuration are indeed local.
We say that the oriented TFPL has boundary if the following hold:
if the vertex has out-degree then , otherwise ;
if the vertex has in-degree then , otherwise ;
if the vertex has in-degree then , while if it has out-degree then .
We denote the set of oriented configurations by and their number by .
Notice the important fact that while and have the same interpretation as in Definition 1.3 for the underlying TFPL, this is not the case for which concerns the local orientation of the edges and not the global connectivity of the paths.
The following oriented version of Proposition 1.4 is immediate:
Vertical reflection together with the reorientation of all edges exchanges and ; in particular, .
Also the concept of extended link patterns has the following natural analog in this context:
A directed extended link pattern on is an extended link pattern on , such that in each pair from one of the link patterns we let an integer be the source and the other be the sink; by definition, left points are sinks, while right points are sources. We let be the number of pairs where the larger integer is the source. We assign the source-sink word on to an extended directed link pattern as follows: we set if and only if is a source in .
It is clear that each oriented TFPL is naturally associated with an extended directed link pattern; we represent by orienting each linked pair from its source to its sink, while left and right points have their attached half arch oriented to the right, cf. Figure 7, right. In this representation counts the number of arrows going from right to left, and is equal to in this example.
There is a natural injection from to : given a TFPL with boundary , orient all its closed paths clockwise, and each path between two vertices from to if . The other paths have a forced orientation by Definition 1.8. Note that the chosen orientation ensures that is indeed the bottom boundary word of the resulting oriented TFPL, therefore this is an injection from to , so that we have
In the other direction, with each oriented TFPL we can associate a non-oriented TFPL by ignoring the direction of the edges, but this operation does not preserve the bottom words in general. In Section 2, we will explain how from a certain weighted enumeration of oriented TFPLs one can deduce the numbers .
1.4. FPLs and TFPLs
We recall here the definition of FPLs and their connection to TFPLs, and refer to [Nada] for a detailed explanation of interactions between FPLs and TFPLs. We fix a positive integer , and let be the square grid with vertices. We impose periodic boundary conditions on , which means that we select every other external edge on the grid, starting by convention with the topmost on the left side; we number these external edges counterclockwise. A Fully Packed Loop (FPL) configuration of size is a subgraph of such that each vertex of is incident to two edges of . An example of an FPL configuration is given in Figure 8 (left).
An FPL configuration on naturally defines non-crossing paths between its external edges, so we can define the link pattern as the set of pairs where label external edges which are the extremities of the same path in : see Figure 8, right. If is a link pattern, we denote by the number of FPL configurations of size such that . Given an integer , define as the link pattern on given by the nested pairs for , and the pairs for each . Note that FPLs such that are of size .
Here is the number of semistandard tableaux of shape and entries in ; it is a polynomial in given by the hook-content formula.
2. Recovering TFPLs from oriented TFPLs
In this section we study more precisely the connection between TFPLs and oriented TFPLs. The main result is that we can deduce the number of TFPLs from a certain weighted enumeration of oriented TFPLs, see Corollary 2.9.
2.1. A relation between the orientation of closed paths, paths oriented from right to left and turns
We consider directed polygons in the plane. The signed curvature of a turn is the angle in between the extension of the incoming edge and the outgoing edge, where we take the negative angle if the turn is to the left, see Figure 9. The turning number of a directed polygon is the sum of the signed curvatures of its turns. The following is equivalent to the well-known fact that the sum of exterior angles of an undirected simple closed polygon (convex or concave) is ; see for instance [Mei75] for a proof of the latter.
The turning number of a directed closed self-avoiding polygon is if it is oriented clockwise and it is otherwise.
We now restrict our considerations to directed lattice paths on the square grid with step set with the additional assumption that our paths do not contain two consecutive steps going in opposite directions. We say that a step is of type u if it is a -step; similar for r, d, l. The eight possible turns are displayed and named in Figure 10. For a given directed path , let denote the number of turns of type , denote the number of turns of type , etc., and set and .
The succession of steps is encoded by the simple graph of Figure 10 (right), which has u,r, d, l as vertices and the possible turns as edges.
Let be a directed path on the square grid.
There exists an integer with the property that
belongs to .
The vector is determined as follows: let the first and last steps of be a and b respectively, and consider the shortest clockwise path from a to b in the graph of Figure 10, right. The coordinate equals if and only if labels an edge in .
Now suppose is closed and self-avoiding. Then if the orientation is clockwise and if the orientation is counterclockwise.
The first and the second part are a direct consequence of the fact that the possible sequences of successive turns in a path correspond to the sequences of edge labels of paths in the graph represented in Figure 10, right. As for (3), it is a direct consequence of Lemma 2.1, because the signed curvature of the turns in the first row of Figure 10 is and it is for the turns in the second row. ∎
In the following consequence of the proposition, let (resp. ) denote the set of turns displayed in the first (resp. second) row of Figure 10. Also, if is any type of turn and a collection of directed paths, we denote by the number of occurrences of turns of type in .
Let be a directed closed self-avoiding path on the square grid and . Then is equal to (resp. ) if is oriented clockwise (resp. counterclockwise).
This enables us to provide an interpretation for the difference of the number of turns of type and the number of turns of type in an oriented TFPL if and .
Let . For any oriented TFPL , let be the associated directed extended link pattern, and denote . Also let resp. , denote the number of closed paths in which are oriented clockwise, resp. counterclockwise. Then
By Corollary 2.3, it is enough to show the following for a non closed path: if goes from a bottom vertex to a vertex with (“from right to left”), and otherwise.
By inspection, can start with a step of type r or u and can end with a step of type r or d: here the dotted edges in Figure 7 are considered part of the paths. By Proposition 2.2 in this case, in particular the coordinates corresponding to the turns , , and are vanishing. Now we complete such a path to a directed closed self-avoiding path by adding a path below the configuration with the least possible number of turns, see Figure 11 in a particular case. The closed path is oriented clockwise if and only if the original path was directed from left to right. By Proposition 2.2(3), we have in this case and otherwise. ∎
This motivates the definition of the weighted enumeration of oriented TFPLs: fix and .
2.2. Reorienting paths oriented from right to left
The goal of this section is to express the number of ordinary TPFLs in terms of the weighted enumeration of oriented TFPLs which was introduced in the previous section. To this end, we let denote the subset of oriented TFPLs in where the associated directed extended link pattern verifies , which means that all bottom paths are oriented from left to right. Let also be the corresponding weighted enumeration, cf. (2.1). The following proposition relates to .
Let be a primitive sixth root of unity, so that verifies . Then .
By Proposition 2.4, we have
for all elements in . Thus
The assertion follows as . ∎
A word of size is feasible for a word of length if there exists a directed extended link pattern with underlying extended link pattern such that is the source-sink word of . Such a is unique, and we can then define for all words such that is feasible for .
Every word is feasible for itself, by orienting all arches in a link pattern from right to left, and clearly one has . For another example, the word is feasible for . Indeed, where is represented in Figure 6. If one orients the pairs and from right to left, and the remaining pairs from left to right, then the source-sink word of the directed pattern thus obtained is precisely . In this case .
Consider now the transformation which takes a TFPL in and reorients all its bottom paths from left to right. By definition the resulting configuration belongs to for a certain (which depends on ) which is feasible for . Note that the weight is decreased by in the transformation, so we obtain the following:
Our goal is to invert the last relation, so that together with the help of Proposition 2.5 we will be able to express the number of TFPLs in terms of the weighted enumeration of oriented TFPLs. Note that if is feasible for then and have the same number of s (and therefore of s also).
Definition 2.7 (Matrix ).
Given such that , the square matrix has rows and columns indexed by words with s and s, and the entry is given by if is feasible for , and otherwise.
This is a square matrix of size .
The matrix is invertible.
It is easy to see that is feasible for if and only if there exist ordered pairs , verifying and is a Dyck word for any , such that is given by for any and for all other indices. Then the description of feasibility just given shows that if is feasible for , then necessarily ; also clearly . Otherwise said, given any linear ordering on words extending and using that order for rows and columns of , we get that is lower triangular with s on the diagonal, and is thus invertible. ∎
Let be three words of length , and let . Then
and in particular
This expresses the number of TFPLs in terms of the number of oriented TFPLs, with the proviso that we have to keep track of the number of turns of type and . We will use this inversion formula in Section 6.
The matrices have made a recent appearance in the literature, in an apparently unrelated context. Indeed, for , they are essentially given by certain upper triangular submatrices of the matrix in [KW11]. In this paper, the entries of the inverse matrix are given a combinatorial interpretation: they enumerate (up to sign) certain “Dyck tilings” of the skew shape . Furthermore, as is noted at the end of [Kim12], this combinatorial interpretation is easily extended to the case of general : this shows that the coefficients in Corollary 2.9 can be directly computed without resorting to matrix inversion.
3. Perfect matchings and nonintersecting lattice paths
In the previous section we proved that one can express TFPLs in terms of oriented TFPLs. We focus now therefore on the latter, since they are easier to deal with as we argued in Section 1. In this section we will show that an oriented TFPL is essentially a pair of disjoint perfect matchings on some graphs, and that such matchings can be encoded by certain families of nonintersecting lattice paths. One goal is to provide easy proofs of necessary conditions on the boundary conditions of TFPLs (ordinary and oriented).
3.1. Necessary conditions for the existence of TFPLs
For a word we define as the number of inversions in , that is, the number of pairs such that and . Also note that , the number of cells of the diagram . We can then state the following theorem:
Let be three words of length . Then implies the following three constraints:
The conclusions of Theorem 3.1 hold also if .
The proofs of the different items of Theorem 3.1 will be given in Sections 3 and 4. More precisely, part (1) will be proven at the end of Section 3.2, part (2) in Section 3.3, and part (3) is Corollary 4.4.
These results were already partly known in the special case of ordinary TFPLs and where is such that is a Dyck word, which was the only case considered in previous papers due to its direct connection with FPLs on a square grid, as explained in Section 1.4. Part (2) was proven first in [CKLN04] by a tedious argument, and later in [Nada] in a manner similar as the one given here in Section 3.3. Finally, part (3) was proven in [Tha07] but only for non oriented TFPLs; also, the proof given there was algebraic, whereas we will show that enumerates occurrences of certain patterns in any configuration of (cf. Formula (4.2)), which automatically proves the inequality in (3).
3.2. Perfect matchings and oriented TFPLs
For two words of length , we define the graph as the induced subgraph of obtained by removing the rightmost vertices , the vertices such that , and the vertices such that . We are interested in perfect matchings of : an example is given on the left of Figure 13. Given an edge in such a perfect matching , orient it from its odd vertex to its even vertex: then the corresponding direction is up, down, left or right. Denote the respective corresponding sets of edges by and their cardinalities by .
We define similarly a graph as follows: start from and remove the leftmost vertices , the vertices for which and respectively. Given an edge in a perfect matching of , orient it from its even vertex to its odd vertex: then the corresponding direction is up, down, left or right. Denote the respective corresponding sets of edges by and their cardinalities by .
The introduction of these graphs and matchings is motivated by the following result:
Let be a positive integer, be three words of length . For any oriented TFPL configuration in , denote by (respectively ) the subset of its edges oriented from an odd vertex to an even vertex (resp. from an even vertex to an odd vertex).
Then is a bijection between:
oriented TFPL configurations in , and
ordered pairs where (resp. ) is a perfect matching on (resp. on ), and such that and are disjoint as subsets of edges of the graph .
In an oriented TFPL, all indegrees and outdegrees are at most one; from this observation it follows that and form matchings on . The vertices of which do not belong to any edge of are those which are either odd with outdegree or even with indegree . By definition of oriented TFPLs, these are exactly the vertices removed when passing from to . Therefore is indeed a perfect matching on ; by symmetry, one has that is a perfect matching on . In an oriented TFPL, one cannot have two directed edges corresponding to the two possible orientations of the same underlying nonoriented edge; this implies immediately that and are disjoint matchings, and therefore the map is well-defined between the sets described in and .
It is then immediate to verify that it is indeed a bijection between these two sets; its inverse consists in orienting edges of from odd vertices to even ones, edges of from even vertices to odd ones, and then considering the union of these two sets of edges on the full graph . ∎
We can already prove the first part of Theorem 3.1:
Consider ; by Theorem 3.3 it gives rise to a perfect matching on . Now perfect matchings on a bipartite graph can only exist if there are the same number of even and odd vertices. A quick computation shows that this is the case for if and only if .
Now also gives rise to a perfect matching on , which implies by symmetry. Since and have the same length, this is equivalent to and concludes the proof. ∎
From now on, we will assume that and , since we have just proven that is empty unless these conditions are fulfilled.
3.3. Invariants of perfect matchings
The next theorem shows that in the perfect matchings under consideration, certain enumerations of edges only depend on the graph and not the matching itself. The proof will rely on the following lemma.
For any word with and , one has