Triangular fully packed loop configurations of excess 2

Triangular fully packed loop configurations (TFPLs) came up in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. To a TFPL is assigned a triple $(u,v;w)$ of $01$-words encoding its boundary conditions which must necessarily satisfy that $d(u)+d(v)\leq d(w)$, where $d(u)$ denotes the number of inversions in $u$. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers $A_\pi$ of FPLs corresponding to a given link pattern $\pi$. Later, Wieland drift - a map on TFPLs that is based on Wieland gyration - was defined. The main contribution of this article is a linear expression for the number of TFPLs with boundary $(u,v;w)$ where $d(w)-d(u)-d(v)=2$ in terms of numbers of stable TFPLs, that is, TFPLs invariant under Wieland drift. This linear expression is consistent with already existing enumeration results for TFPLs with boundary $(u,v;w)$ where $d(w)-d(u)-d(v)=0,1$.


Introduction
The basis for this article is the fully packed loop model that has its origin in the six-vertex model (which is also called square ice model) of statistical mechanics; a fully packed loop configuration (FPL) of size n is a subgraph F of the n × n-square grid together with 2n external edges such that each of the n 2 vertices is of degree 2 in F and every other external edge is occupied by F starting with the topmost horizontal external edge on the left side. See Figure 1 for an example. FPLs are significant to algebraic combinatorics due to their one-to-one correspondence to alternating sign matrices (ASMs). This is why FPLs of size n are enumerated by the famous formula for the number of ASMs of size n proved in [13].
In contrast to alternating sign matrices, FPLs allow a refined study in dependency on the connectivity of the occupied external edges (these connections are encoded as a link pattern). The study of FPLs having a link pattern with nested arches is an example of one such refined study; it was conjectured in [14] and later proved in [4] that the number of FPLs having a fixed link pattern π ∪ m consisting of a link pattern π of size n and m nested arches is polynomial in m. In the course of the proof of this conjecture triangular fully packed loop configurations (TFPLs) came up. To be more precise, the following expression for the number A π (m) of FPLs having link pattern π ∪ m including numbers t w u,v of TFPLs satisfying certain boundary conditions encoded by a triple (u, v; w) of 01-words was shown: where the sum runs over all Dyck words u, v of length 2n, u denotes the 01 word obtained from a Dyck word u by deleting the first 0 and the last 1, w(π) denotes the Dyck word corresponding to the link pattern π, λ(u) denotes the Young diagram associated with a 01 word u, λ denotes the conjugate of a Young diagram λ and with c(C) being the content of the cell C and h(C) the hook length of C. Apperantly, Equation (1.1) motivates the study of TFPLs and the numbers t w u,v . Another motivation for their study comes from the many nice properties of TFPLs which have been discovered since the emergence of TFPLs, see [11], [8] and [5]. An example of one such property is that the boundary (u, v; w) of a TFPL has to fulfill that d(u) + d(v) ≤ d(w), where d(ω) denotes the number of inversions in a word ω; the integer is said to be the excess of u, v, w. To study TFPLs with respect to the excess of their boundary turned out to be fruitful; in [5] enumeration results for TFPLs with boundary (u, v; w) where exc(u, v; w) = 0, 1 were proved. Wieland gyration, on the other hand, is an operation on FPLs that was invented in [12] to prove the rotational invariance of the numbers A π of FPLs corresponding to given link patterns π. Later it was heavily used by Cantini and Sportiello [3] to prove the Razumov-Stroganov conjecture. In connection with TFPLs, Wieland gyration first appeared in [8] following work of [11], after which Wieland drift was introduced in [1] as the natural definition of Wieland gyration for TFPLs. In contrast to Wieland gyration, Wieland drift is not an involution. It was shown in [1] that Wieland drift is eventually periodic with period 1.
This article will focus on TFPLs with boundary (u, v; w) where exc(u, v; w) = 2. The main contribution of this paper will be a linear expression for t w u,v in terms of numbers of stable TFPLs, that is, TFPLs invariant under the application of Wieland drift. This linear expression is consistent with the already existing enumeration results for TFPLs with boundary (u, v; w) where exc(u, v; w) = 0, 1.
By a result in [5] it holds For that reason, the linear expression for the number of TFPLs with boundary (u, v; w) where exc(u, v; w) = 1 in terms of TFPLs of excess 0 proved in Theorem 6.16 (5) in [5] can be written as follows: Summing up, the linear expression stated in Theorem 1 is consistent with the already existing enumeration results for TFPLs with boundary (u, v; w) where exc(u, v; w) = 0, 1. This suggests a study of TFPLs with boundary (u, v; w) where exc(u, v; w) ≥ 3 based on the methods presented in this article in order to obtain expressions for the numbers t w u,v in terms of stable TFPLs.
A poster about this work will be presented at FPSAC 2015.

Words and Young diagrams.
Given a word ω the number of occurrences of 0 (resp. 1) in ω is denoted by |ω| 0 (resp. |ω| 1 ). Furthermore, it is said that two words ω, σ of length N with the same number of occurrences of 1 satisfy ω ≤ σ if |ω 1 · · · ω n | 1 ≤ |σ 1 · · · σ n | 1 holds for all 1 ≤ n ≤ N . Finally, the number of inversions of ω that is pairs 1 ≤ i < j ≤ N satisfying ω i = 1 and ω j = 0 is denoted by d(ω). Throughout this article, in a Young diagram empty columns and empty rows are allowed. With a word ω a Young diagram λ(ω) will be associated as follows: to a given word ω a path on the square lattice is constructed by drawing a (0, 1)-step if ω i = 0 and a (1, 0)-step if ω i = 1 for i from 1 to n. Additionally, a vertical line through the path's starting point and a horizontal line through its ending point are drawn. Then the region enclosed by the lattice path and the two lines is a Young diagram which shall be the image of ω under λ. In Figure 3, an example of a word and its corresponding Young diagram is given. For two words ω and σ of length N it then holds ω ≤ σ if and only if λ(ω) is contained in λ(σ). Furthermore, the number of cells of λ(ω) equals d(ω). There are skew shaped Young diagrams which play an important role in the context of Wieland drift: a skew shape is said to be a horizontal strip (resp. a vertical strip) if each of its columns (resp. rows) contains at most one cell. Consider two words ω and σ satisfying |ω| 1 = |σ| 1 , |ω| 0 = |σ| 0 and ω ≤ σ. Then the skew shape λ(σ)/λ(ω) is a horizontal strip (resp. a vertical strip) if and only if for each j ∈ {1, . . . , |ω| 1 } (resp. for each j ∈ {1, 2, . . . , |ω| 0 }) the following holds: If ω i is the j-th one ( resp. zero) in ω then σ i−1 or σ i ( resp. σ i or σ i+1 ) is the j-th one ( resp. zero) in σ. In the following, if the skew shaped Young diagram λ(σ)/λ(ω) is a horizontal strip (resp. a vertical strip) it will be written Semi-standard Young tableaux of skew shape λ(σ)/λ(ω) with entries 1, 2, . . . , m are in bijection with sequences of Young diagrams To be more precise, the horizontal strip λ(τ i )/λ(τ i−1 ) gives the cells of the semi-standard Young tableau of skew-shape λ(σ)/λ(ω) that have entry i for 1 ≤ i ≤ m. For instance, the semi-standard Young tableau of skew shape λ(011011100)/λ(001011011) in Figure 3 corresponds to the sequence λ(001011011) ⊆ λ(010101110) ⊆ λ(011011010) ⊆ λ(011011100).

Triangular fully packed loop configurations.
To give the definition of triangular fully packed loop configurations the following graph is needed: Let N be a positive integer. The graph G N is defined as the induced subgraph of the square grid made up of N consecutive centered rows of 3, 5, . . . , 2N + 1 vertices from top to bottom together with 2N + 1 vertical external edges incident to the 2N + 1 bottom vertices.  In Figure 4, the graph G 7 is depicted. From now on, the vertices of G N are partitioned into odd and even vertices in a chessboard manner where by convention the leftmost vertex of the top row of G N is odd. In the figures, odd vertices are represented by circles and even vertices by squares. There are vertices of G N that play a special role: let L N = {L 1 , L 2 , . . . , L N } (resp. R N = {R 1 , R 2 , . . . , R N }) be the set made up of the vertices which are leftmost (resp. rightmost) in each of the N rows of G N and let B N = {B 1 , B 2 , . . . , B N } be the set made up of the even vertices of the bottom row of G N . The vertices are numbered from left to right. Furthermore, the N (N + 1) unit squares of G N including external unit squares that have three surrounding edges only are said to be the cells of G N . They are partitioned into odd and even cells in a chessboard manner where by convention the top left cell of G N is odd. Definition 2.2 (Triangular fully packed loop configuration). Let N be a positive integer. A triangular fully packed loop configuration (TFPL) of size N is a subgraph f of G N such that: (1) Precisely those external edges that are incident to a vertex in B N are occupied by f .  An example of a TFPL is given in Figure 5. A cell of f is a cell of G N together with those of its surrounding edges that are occupied by f . To each TFPL of size N is assigned a triple of words of length N . 3. Let f be a TFPL of size N . The triple (u, v; w) of words of length N is assigned to f as follows: (1) For i = 1, . . . , N set u i = 1 if the vertex L i ∈ L N has degree 1 and u i = 0 otherwise.
(2) For i = 1, . . . , N set v i = 0 if the vertex R i ∈ R N has degree 1 and v i = 1 otherwise.
(3) For i = 1, . . . , N set w i = 1 if in f the vertex B i ∈ B N is connected with a vertex in L N or with a vertex B h for an h < i and w i = 0 otherwise. The triple (u, v; w) is said to be the boundary of f . Furthermore, the set of TFPLs with boundary (u, v; w) is denoted by T w u,v and its cardinality by t w u,v . For example, the triple (0101111, 0011111; 1101101) is the boundary of the TFPL depicted in Figure 5. The definitions of both a TFPL and its boundary contain global conditions. Those can be omitted when adding an orientation to each edge of a TFPL. Definition 2.4 (Oriented triangular fully packed loop configuration). An oriented TFPL of size N is a TFPL of size N together with an orientation of its edges such that the edges attached to L N are outgoing, the edges attached to R N are incoming and all other vertices of G N are incident to an incoming and an outgoing edge.
In Figure 5, an example of an oriented TFPL of size 7 is given. In the underlying TFPL of an oriented TFPL condition (4) can be omitted because the required orientations of the edges attached to a vertex of the left or right boundary prevent paths from returning to the respective boundary.
Definition 2.5. An oriented TFPL f has boundary (u, v; w) if the following hold: (1) If the vertex L i ∈ L N has out-degree 1 then While u and v coincide with the respective boundary word in the underlying ordinary TFPL this is not the case for w. Instead of the connectivity of the paths w encodes the local orientation of the edges. Only in the case when in an oriented TFPL all paths between two vertices B i and B j of B N are oriented from B i to B j if i < j the boundary word w coincides with the respective boundary word of the underlying TFPL. Hence, the canonical orientation of a TFPL is defined as the orientation of the edges of the TFPL that satisfies the conditions in Definition 2.4 and in addition that each path between two vertices B i , B j ∈ B N is oriented from B i to B j if i < j and that all closed paths are oriented clockwise.
A triple (u, v; w) that is the boundary of an ordinary or an oriented TFPL has to fulfill the following conditions: |u| 0 = |v| 0 = |w| 0 , u ≤ w, v ≤ w and d(w) − d(u) − d(v) ≥ 0. These conditions were proved in [4,11,5]. The last condition gives rise to the following definition: Definition 2.6 ( [5]). Let u, v, w be words of length N . Then the excess of u, v, w is defined as If exc(u, v; w) = k then both an ordinary and an oriented TFPL with boundary (u, v; w) are said to be of excess k.
In [5], the following interpretation of the excess of u, v, w in terms of numbers of occurrences of certain local configurations in an oriented TFPL with boundary (u, v; w) is proved: Theorem 4.3]). Let f be an oriented TFPL with boundary (u, v; w). Then 2.3. Blue-red path tangles. In this subsection an alternative representation of oriented TFPLs is introduced, namely blue-red path tangles. They came up in [5] and are crucial for the proofs given in this article. Throughout this subsection, let u, v, w be words of length N such that |u| 0 = |v| 0 = |w| 0 = N 0 , A blue-red path tangle consists of an N 0 -tuple of non-intersecting blue lattice paths and an N 1 -tuple of non-intersecting red lattice paths. The blue lattice paths use steps (−1, 1), (−1, −1) and (−2, 0), whereas the red lattice paths use steps (1, 1), (1, −1) and (2, 0). Furthermore, neither a blue nor a red lattice path goes below the x-axis. The k-th blue lattice path of an N 0 -tuple of non-intersecting blue lattice paths starts in a certain fixed vertex D k and ends in a certain fixed vertex E k . The definitions of the vertices D k and E k solely depend on the positions of the k-th zeroes in w and u and are omitted here. Instead, the vertices D 1 , . . . , D N0 and E 1 , . . . , E N0 are indicated with an example in Figure 6. In the following, the set of N 0 -tuples of non-intersecting blue lattice paths (P 1 , P 2 , . . . , P N0 ) where P k is a path from D k to E k is denoted by P(u, w). On the other hand, the -th red path of an N 1 -tuple of non-intersecting red lattice paths starts in a certain fixed vertex D and ends in a certain fixed vertex E . The definitions of the vertices D and E solely depend on the positions of the -th ones in w and v and are omitted here. Instead, D and E are indicated with and example in Figure 6. In the following, the set of N 1 -tuples of non-intersecting red paths (P 1 , P 2 , . . . , P N1 ) where P is a path from D to E is denoted by P (v, w). (2) Each middle point of a horizontal step in B (resp. R) is used by a step in R (resp. B).
The set of such configurations is denoted by BlueRed(u, v; w) and a configuration in BlueRed(u, v; w) is said to be a blue-red path tangle with boundary (u, v; w).
An example of an oriented TFPL and its corresponding blue-red path tangle is given in Figure 6.
Proof. Here the bijection in [5] is repeated: let f be an oriented TFPL of size N and with boundary (u, v; w). As a start blue vertices are inserted in the middle of each horizontal edge of G N which has an odd vertex to its left and red vertices are inserted in the middle of each horizontal edge of G N which has an even vertex to its left. Next, blue edges are inserted as indicated in the left part of Figure 7 and red edges are inserted as indicated in the right part of Figure 7. Then the blue vertices together with the blue edges give rise to an N 0 -tuple of non-intersecting paths B = (P 1 , P 2 , . . . , P N0 ) in P(u, w) and the red vertices together with the red edges give rise to an N 1 -tuple of non-intersecting paths R = (P 1 , P 2 , . . . , P N1 ) in P (v, w). The fact that no diagonal step of R crosses a diagonal step of B is equivalent to that there is a unique orientation of each vertical edge in f . On the other hand, the fact that each middle point of a horizontal step in B (resp. R) is used by a step in R (resp. B) is equivalent to that there is a unique orientation of each horizontal edge in f . Thus, (B, R) ∈ BlueRed(u, v; w).

Wieland drift
The starting point of this section is the definition of Wieland gyration for fully packed loop configurations (FPLs) as introduced in [12]. Wieland gyration is composed of local operations on all active cells of an FPL: the active cells of an FPL can be chosen to be either all its odd cells or all its even cells. Given an active cell c of an FPL two cases have to be distinguished, namely whether c contains precisely two edges of the FPL on opposite sides or not. If this is the case, Wieland gyration W leaves c invariant. Otherwise, the effect of W on c is that edges and non-edges of the FPL are exchanged. In Figure 8, the action of W on an active cell is illustrated.
Also left-and right-Wieland drift will be composed of local operations on all active cells of a TFPL. Similar to FPLs active cells of a TFPL are either chosen to be all its odd or all its even cells. Choosing all odd cells as active cells will lead to what will be defined as left-Wieland drift, whereas choosing all even cells as active cells will lead to what will be defined as right-Wieland drift. In the figures, the active cells of a TFPL will be indicated by gray circles. (1) Insert a vertex L i to the left of L i for 1 ≤ i ≤ N . Then run through the occurrences of ones in (2) Apply Wieland gyration to each odd cell of f . In Figure 9, an example for left-Wieland drift is given. The image of a TFPL with boundary (u, v; w) under left-Wieland drift with respect to u − is again a TFPL and has boundary (

encodes what happens along the right boundary of a TFPL with right boundary v and is denoted by WR
It is defined in an obvious way as the symmetric version of left-Wieland drift and it shall simply be illustrated with an example in Figure 10.
The image of a TFPL with boundary (u, v; w) under right-Wieland drift with respect to v − is a TFPL Given a TFPL with right boundary v the effect of left-Wieland drift along the right boundary of the TFPL is inverted by right-Wieland drift with respect to v. On the other hand, given a TFPL with left boundary u the effect of right-Wieland drift along the left boundary is inverted by left-Wieland drift with respect to u. Since Wieland gyration is an involution on each cell it follows:  (1) Let f be a TFPL with boundary (u + , v; w) and u be a word Such an edge is said to be a drifter.
Note that by Proposition 2.7 a TFPL of excess k exhibits at most k drifters.
Given a TFPL f the sequence (WL m (f )) m≥0 is eventually periodic since there are only finitely many TFPLs of a fixed size. The length of its period is in fact always 1.
The same holds for right-Wieland drift. In Figure 11 an example of a TFPL and its images under left-Wieland drift is given. There a stable TFPL is obtained after the third iteration of left-Wieland drift. From now on, for an instable TFPL f denote by L = L(f ) the positive integer L such that WL (f ) is instable for each 0 ≤ ≤ L and WL L+1 (f ) is stable and by R = R(f ) the positive integer such that WR r (f ) is instable for each 0 ≤ r ≤ R and WR R+1 (f ) is stable.
The path of f -denoted by Path(f ) -is the sequence of all TFPLs that can be reached by an iterated application of left-respectively right-Wieland drift to f that is When v denotes the right boundary of WL (f ) for each 0 ≤ ≤ L + 1 and λ denotes the conjugate of a Young diagram λ then the sequence gives rise to a semi-standard Young tableau of skew shape λ(v L+1 )/λ(v) with entries 1, 2, . . . , L + 1. On the other hand, when u r denotes the left boundary of WR r (f ) for each 0 ≤ r ≤ R + 1 then the sequence gives rise to a semi-standard Young tableau of skew shape λ(u R+1 )/λ(u). It will be shown that for an instable TFPL f with boundary (u, v; w) of excess at most 2 precisely one of the following cases applies: (1) the sequence in (3.1) corresponds to a semi-standard Young tableau in G λ(v) ,λ(v + ) ; (2) the sequence in (3.2) corresponds to a semi-standard Young tableau in G λ(u),λ(u + ) ; (3) neither the sequence in (3.1) corresponds to a semi-standard Young tableau in G λ(v) ,λ(v + ) nor the sequence in (3.2) corresponds to a semi-standard Young tableau in G λ(u),λ(u + ) . In the bijective proof of Theorem 1 an instable TFPL f with boundary (u, v; w) of excess at most 2 will be associated with the triple consisting of the empty semi-standard Young tableau of skew shape λ(u)/λ(u), the stable TFPL Left(f ) and the semi-standard Young tableau corresponding to the sequence in (3.1) if the latter is an element of G λ(v) ,λ(v + ) . If the semi-standard Young tableau corresponding to the sequence in (3.2) is an element of G λ(u),λ(u + ) then f will be associated with the triple consisting of the semi-standard Young tableau in G λ(u),λ(u R+1 ) corresponding to the previous sequence, the stable TFPL Right(f ) and the empty semi-standard Young tableau of skew shape λ(v) /λ(v) . Finally, if neither the sequence in (3.1) corresponds to a semi-standard Young tableau in G λ(v) ,λ(v + ) nor the sequence in (3.2) corresponds to a semi-standard Young tableau in G λ(u),λ(u + ) then to f moves are applied which transform and ultimately turn it into a stable TFPL with boundary (u + , v + ; w) for a u + > u and a v + > v. These moves will be extracted from the effect of Wieland drift on instable TFPLs of excess at most 2. The triple which will be associated with f then consists of this stable TFPL, a semi-standard Young tableau in G λ(v) ,λ(v + ) and one in G λ(u),λ(u + ) .
In the next section, the effect of Wieland drift on instable TFPLs of excess at most 2 is studied.
4. An alternative description of Wieland drift for TFPLs of excess at most 2 The main contribution of this section is a description of the effect of Wieland drift on TFPLs of excess at most 2 as a composition of moves. In Figure 12, the moves which form the basis for that description are depicted. Recall that a TFPL of excess k contains at most k drifters.  (1) if R i in R N is incident to a drifter delete that drifter and add a horizontal edge incident to R i+1 for i = 1, 2, . . . , N − 1; denote the so-obtained TFPL by f ; (2) consider the columns of vertices of G N that contain a vertex, which is incident to a drifter in f : If u ij −1 is the j-th one in u delete the horizontal edge incident to L ij −1 and add a vertical edge incident to L ij for j = 1, 2, . . . , N 1 .
In Figure 13   Since the proofs in this section work by studying the cells of a TFPL it is convenient to fix notations for all the odd and even cells that can occur in a TFPL. In total, there are 16 different odd and 16 different even internal cells -that are cells which are not external -that can occur in a TFPL. By Lemma 4.2 fourteen of those odd and fourteen of those even internal cells that can occur in a TFPL of excess at most 2. The odd respectively even cells that can occur in a TFPL of excess at most 2 will be numbered by 1 up to 14 and are listed in Figure 14, whereas the two excluded odd respectively even internal cells will be numbered by 15 and 16 as indicated in Lemma 4.2.
Proof. First, let f be a TFPL that contains a cell c that coincides with o 15 . In f together with its canonical orientation the oriented edges of c then give rise to two configurations that are counted by the excess, see Proposition 2.7. Additionally, the right vertex of the horizontal edge of c which is oriented from right to left either is adjacent to the vertex to its right or is incident to a drifter. Thus, the TFPL f together with its canonical orientation contains at least three configurations that are counted by the excess. For the same reasons, a TFPL of excess at most 2 cannot contain the third cell in the list.
Now, let f be a TFPL that contains a cell c that coincides with o 16 . Then both the top and the bottom rightmost vertex of c have to be incident to a drifter. Therefore, f contains at least three drifters and therefore has to be of excess at least 3. By the same argument, the fourth cell in the list cannot occur in a TFPL of excess at most 2.  In particular, e = e in that case.
To study the effect of left-Wieland drift on the whole TFPL it suffices to study its effect on the even cells of a TFPL. That is because edges of a TFPL that are not edges of an even cell have to be incident to a vertex in L N and the effect of left-Wieland drift on these edges immediately follows from the definition of left-Wieland drift. To be more precise, in the image of a TFPL under left-Wieland drift all edges incident to a vertex in L N have to be horizontal edges. By Lemma 4.3, to determine the effect of left-Wieland drift on a TFPL it suffices to determine its effect on the one hand on all even cells of the TFPL whereof a vertex is incident to a drifter and on the other hand on all even cells where the odd cells to their left contain a drifter. Now, given a drifter d in an instable TFPL there are at most three even cells whereof a vertex is incident to d and there is at most one even cell such that the odd cell to its left contains d. In Figure 15  In the following, separate proofs for each case in Proposition 4.1 will be given. Proof of Proposition 4.1(2b). Let f be a TFPL of excess 2 that contains two drifters d and d * whereof d is incident to a vertex R i in R N and d * is not incident to a vertex in R N . Note that f contains neither a cell of type o 11 nor of type e 11 . That is because when adding the canonical orientation to f such a cell would give rise to two local configurations that are counted by the excess which would imply that f is of excess greater than 2. r , e * r , e r , e b and e * r is given in Table 1. Next, the case when the drifter d * is contained in e r is studied. In that case the x-coordinate of d * is larger than the one of d.
If v ij +1 is the j-th zero in v delete the horizontal edge incident to R ij +1 and add a vertical edge incident to R ij for j = 1, 2, . . . , N 0 .

The path of a drifter under Wieland drift for TFPLs of excess at most 2
The focus of this section is on how many iterations of left-Wieland drift (resp. right-Wieland drift) are needed to move a drifter in an instable TFPL of excess at most 2 to the right (resp. left) boundary. The results of the previous section facilitate the study of the effect of Wieland drift on a drifter in an instable TFPL of excess at most 2. When looking at the moves that describe the effect of Wieland drift on instable TFPLs of excess at most 2 one immediately sees that in the preimage of the move M 4 there is one drifter whereas in its image there two and that in the preimage of the move M 5 there are two drifters whereas in its image there is one. Thus, in order to pursue a drifter one has to decide which drifter to pursue after applying the move M The proof of Proposition 5.3 will be the content of the rest of this section. The crucial idea is to regard TFPLs together with their canonical orientation. Before starting with the proof a crucial corollary of Proposition 5.3 is stated.  In Figure 5 an instable TFPL of excess 2 and the path of one of its drifters which shall in the following be denoted by d are depicted. The drifter d satisfies R(d) = 2, HeightR(d) = 1, u R(d) = 01101, R 1 (u R(d) ) = 2, L(d) = 2, HeightL(d) = 2, v L(d) = 01011 and L 0 (v L(d) ) = 1. semi-standard Young tableau of skew shape λ(u)/λ(u) and T (f ) is the semi-standard Young tableau of skew shape λ(v + ) /λ(v) corresponding to the sequence where v denotes the right boundary word of WL (f ) for each 0 ≤ ≤ L(f ) + 1. (4) If in f there are two drifters d r and d l such that R(d r ) ≤ R 1 (u R(dr) ) and L(d l ) ≤ L 0 (v L(d l ) ), then g(f ) is the TFPL with boundary (u + , v + ; w) for a u + such that λ(u) ⊂ λ(u + ) and a v + such that λ(v) ⊂ λ(v + ) obtained from f as follows: the drifter d r is moved to the left boundary using the moves M −1 and there replaced by a horizontal edge and the drifter d l is moved to the right boundary using the moves M 1 , M 2 , M 3 and there replaced by a horizontal edge. Furthermore, S(f ) is the semi-standard Young tableau of skew shape λ(u + )/λ(u) with entry R(d r ) + 1 and T (f ) is the semi-standard Young tableau of skew shape λ(v + ) /λ(v) with entry L(d l ) + 1. In Figure 18, the TFPL of excess 2 displayed in Figure 11 and the triple (S, g, T ) associated with it are depicted. In the following, denote by G λ,λ + the set of semi-standard Young tableaux of skew shape λ + /λ with entries in the i-th column, if counted from right, restricted to 1, 2, . . . , i for Young diagrams λ ⊆ λ + which may have empty columns or rows. Note that in a TFPL of excess at most 2 that contains the preimage of the move M 5 (resp. M −1 4 ) for both drifters d and d * holds that L(d) = L(d * ) (resp. R(d) = R(d * )). Therefore, d and d * have to satisfy the preconditions of either (1) or (2). The proof of Proposition 6.2 is based on the following two lemma.  In summary, it follows that Therefore, L(d) − L(d * ) − L 0 (v L(d) ) + L 0 (v L(d * ) ) = 1.
Proposition 6.2 follows immediately from Lemma 6.4.
Proof of Proposition 6.2. Let f be an instable TFPL of excess 2 that contains two drifters d l and d r such that R(d r ) ≤ R 1 (u R(dr) ) and L(d l ) ≤ L 0 (v L(d l ) ). Without loss of generality, suppose that d l cannot be moved to the right boundary using the moves M 1 , M 2 and M 3 . Then, by Lemma 6.4 L(d l ) ≥ L(d r ) + L 0 (v L(d l ) ) − L 0 (v L(dr) ) + 1.
Thus, L 0 (v L(dr) ) > L(d r ) and equivalently R(d r ) > R 1 (u R(dr) ) + 1. That is a contradiction. Therefore, d r can be moved to the right boundary using the moves M 1 , M 2 and M 3 . In Figure 20, the instable TFPL of excess 2 of Figure 5 and the triple (S, g, T ) associated with it are depicted.  Figure 5 and the triple (S, g, T ) it is associated with: S is a semi-standard Young tableau of skew shape λ(10110)/λ(01101), g is a stable TFPL with boundary (10110, 00111; 10110) and T is the empty semi-standard Young tableau of skew shape λ(00111) /λ(00111) .
Proof of Theorem 1. Let u, v, w be words of length N satisfying exc(u, v; w) ≤ 2. Furthermore, let f ∈ T w u,v be instable and denote by (S, g, T ) the image of f under Φ. As a start, S ∈ G λ(u),λ(u + ) for the following reason: let c be a cell of the Young diagram of skew shape λ(u + )/λ(u) then its entry in S has to be R(d) + 1 for a drifter d in f and it has to hold R(d) ≤ R 1 (u R(d) ) by the definition of Φ. Since R 1 (u R(d) ) is the number of columns to the right of c the entry of c can at most be the number of columns to the right of c plus one. By analogous arguments, T ∈ G λ(v) ,λ(v + ) .
(3) If u + > u and v + > v, then Ψ(S, g, T ) is the TFPL obtained from g as follows: since u + > u and v + > v the skew shaped Young diagrams λ(u + )/λ(u) and λ(v + ) /λ(v) both consist of precisely one cell. Hence, denote by j − 1 the number of columns to the right of the one cell λ(u + )/λ(u) contains and by i j the index of the j-th one in u + . On the other hand, denote by j − 1 the number of columns to the right of the one cell λ(v + ) /λ(v) contains and by i j the index of the j -th zero in v + . Now, a drifter d l incident to L ij +1 in g is inserted whereas the horizontal edge incident to L ij is deleted, a drifter d r incident to R i j −1 in g is inserted, whereas the horizontal edge incident to R i j is deleted, d l is moved L times by a move M 1 , M 2 or M 3 and d r is moved R times by a move M −1 1 , M −1 2 or M −1 3 . The so-obtained TFPL is the image of (S, g, T ) under Ψ. (4) If u + = u and v + = v, then Ψ(S, g, T ) = g. It can easily be seen that Ψ is the inverse map of Φ.