Matthieu Josuat-Vergès

Stammering tableaux

The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic model of moving particles, which is of great interest in combinatorics, since it was realized that its partition function counts a certain kind of tableaux. These tableaux have several variants such as permutations tableaux, alternative tableaux, tree-like tableaux, Dyck tableaux, etc. We introduce in this context certain walks in Young's lattice, that we call stammering tableaux (by analogy with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some natural bijections make a link with rook placements in a double staircase, chains of Dyck paths obtained by successive addition of ribbons, Laguerre histories, Dyck tableaux, etc.

1 Introduction

The PASEP

The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic model of moving particles which motivated a lot of recent combinatorial developments. Our description of the model is very brief since we are more interested in the combinatorial objects, so we refer to [START_REF] Derrida | Exact solution of a 1D asymmetric exclusion model using a matrix formulation[END_REF] for a definition, and [START_REF] Corteel | Tableaux combinatorics for the asymmetric exclusion process[END_REF] for the related combinatorics. A state in this model is a binary word of length N over • and •, and can be interpreted as a sequence of N sites, each being empty (•) or occupied by a particle (•). The possible transitions are :

-a factor •• becomes •• (a particle moves to the right), -a factor •• becomes •• (a particle moves to the left), -an initial • becomes • (a new particle arrives from the left), -a final • becomes • (a particle exits on the right). These 4 events occur with probabilities depending on 4 real positive parameters p, q, α, β, and by a rescaling argument we can assume p = 1. For each state τ ∈ {•, •} N , its probability in the stationary distribution is denoted p τ .

A recipe to compute the stationary probability p τ is the Matrix Ansatz of Derrida et al. obtained in [START_REF] Derrida | Exact solution of a 1D asymmetric exclusion model using a matrix formulation[END_REF]. Suppose that we have two operators F and E satisfying a commutation relation:

F E -qEF = F + E. (1)
We can identify a state τ with a binary word w in F and E by the rule • → F , • → E. Using (1), the resulting binary word can be put in normal form, i.e., a linear combination of E i F j :

w = i,j≥0 c i,j E i F j
where c i,j are polynomials in q (and a finite number of them are nonzero). Then, the result is as follows:

p τ = pτ Z N
where the nonnormalized probability pτ is i,j≥0 c i,j α -i β -j , and the normalizing factor is Z N = pτ summed over τ ∈ {•, •} N (it can also be obtained from the normal form of (F + E) N since this expands as the sum of all binary words). A consequence is that p τ and Z N are polynomials with nonnegative coefficients in α -1 , β -1 , and q.

Tableau combinatorics

The connection of the PASEP model with permutations and statistics on them is done in [START_REF] Corteel | Tableaux combinatorics for the asymmetric exclusion process[END_REF], where Corteel and Williams used so-called permutation tableaux, defined as 0-1 fillings of Young diagrams with certain rules. Indeed, since each pτ is a polynomial, it is a natural to look for its combinatorial meaning, and the main result of [START_REF] Corteel | Tableaux combinatorics for the asymmetric exclusion process[END_REF] shows that this quantity is a generating function of permutation tableaux whose shape is a particular Young diagram depending on τ . In particular the partition function Z N is a 3-variate refinement of (N + 1)! with a combinatorial interpretation on permutations.

Later, other kind of tableaux have been introduced: alternative tableaux [START_REF] Viennot | Alternative tableaux, permutations and partially asymmetric exclusion process[END_REF], tree-like tableaux [START_REF] Aval | Dyck tableaux[END_REF], and Dyck tableaux [START_REF] Aval | Tree-like tableaux[END_REF]. They are all variants of each other, but each has its own combinatorial properties. For example, Viennot [START_REF] Viennot | Alternative tableaux, permutations and partially asymmetric exclusion process[END_REF] showed that the relation (1) leads to alternative tableaux, bypassing the probabilistic model. Tree-like tableaux [START_REF] Aval | Dyck tableaux[END_REF] have a nice tree structure, and their is a nice insertion algorithm that permits to build these object inductively. Particularly relevant to the present work, Dyck tableaux [START_REF] Aval | Tree-like tableaux[END_REF] make a connection with labelled Dyck paths called Laguerre histories (see also [START_REF] Josuat-Vergès | Combinatorics of the three-parameter PASEP partition function[END_REF]). These various refecences make clear that all this tableau combinatorics is interesting in itself, and the link with the PASEP is sometimes not apparent.

Essentially, our stammering tableaux (tableaux bégayants in French) can be seen as another variant of these objects. However, the word "tableaux" has a different meaning here: they are not fillings of Young diagrams, and should rather be considered as a variant of oscillating tableaux, vacillating tableaux, or hesitating tableaux (see [START_REF] Chen | Crossings and nestings of matchings and partitions[END_REF] and below). These objects are essentially obtained from the Matrix Ansatz in (1).

Realizations of the Matrix Ansatz operators

Although it is not a priori needed to compute the stationary probabilities p τ , it is useful to have explicit realizations of operators E and F satisfying (1). For example, Derrida et al. provide tridiagonal semi-infinite matrices realizations of E and F that were exploited in [START_REF] Josuat-Vergès | Combinatorics of the three-parameter PASEP partition function[END_REF].

A natural idea to realize the relation in ( 1) is to start from the much more common relation

DU -qU D = I ( 2
)
where I is the identity (sometimes called the q-canonical commutation relation). Then, let

F = D(U + I), E = D(U + I)U.
It can be checked that the relation in (2) implies the one in (1), indeed:

F E = D(U + I)D(U + I)U = D(U + I)DU + D(U + I)DU U, EF = D(U + I)U D(U + I) = D(U + I)U D + D(U + I)U DU,
and we get

F E -qEF = D(U + I) + D(U + I)U = F + E.
Thus, explicit realizations of E and F can be obtained from the ones of D and U .

Walks in Young's lattice

In the particular case q = 1, a classical way to realize the relation DU -U D = I is to define

U (λ) = µ λ µ and D(λ) = µ λ µ, (3)
where is the cover relation in Young's lattice Y, and D, U are seen as linear operators on the vector space based on Y. We refer to Stanley's notion of differential poset [START_REF] Stanley | Differential posets[END_REF] (see next section). In this setting, -oscillating tableaux naturally appear in the expansion of (U + D) n ∅,

-vacillating tableaux naturally appear in the expansion of ((U + I)(D + I)) n ∅,

-hesitating tableaux naturally appear in the expansion of (DU + ID + U I) n ∅.

See [START_REF] Chen | Crossings and nestings of matchings and partitions[END_REF] for definitions and more on this subject. So, motivated by the Matrix Ansatz of the PASEP, it is natural to consider the tableaux appearing in similar expansions, with the powers of F + E = D(U + I) 2 . These are our stammering tableaux, formally defined in the next section.

These new objects allow one to see more clearly some properties that have been investigated before in permutation tableaux and their variants. The various bijections here are rather simple and natural. In particular, there is the question of a recursive construction to make clear that there are n! such objects of size n, which can involve insertion algorithms [START_REF] Aval | Dyck tableaux[END_REF][START_REF] Aval | Tree-like tableaux[END_REF]. As we will see through this article, the recursive construction in our case is extremely simple. Another property is the link with earlier results by Françon and Viennot in [START_REF] Franc ¸on | Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'Euler et nombres de Genocchi[END_REF]. We will see that this link becomes clear through our notion of chains of Dyck shapes (see Section 5). But perhaps the most important point is that these objects make a bridge between the tableaux combinatorics of the PASEP and the tableaux combinatorics of Young's lattice.

Organization

Sections 2 to 6 give definitions and several bijections (except Section 4 that shows that a poset of Dyck paths naturally appearing in Section 3 is a lattice). In Section 7, we consider a variant of stammering tableaux that we know how to enumerate, and Section 8 contains a few open problems.

Acknowledgement

This work was first presented at the "Journées Viennot" in June 2012, at the University of Bordeaux. I thank the organizers for the invitation, and I thank Xavier for all I learned from him.

Also, I thank Tom Roby for drawing my attention to the work in his thesis [START_REF] Roby | Applications and extensions of Fomin's generalization of the Robinson-Schensted correspondence to differential posets[END_REF]. This work is supported by ANR-12-BS01-0017 (CARMA).

Stammering tableaux and the double staircase

We begin with a few definitions and reference to the literature. A general reference is [START_REF] Stanley | Enumerative Combinatorics[END_REF]. A Young diagram is a bottom left aligned set of unit squares, for example:

. Our definition respects the French convention but others exist. Such a diagram is characterized by its sequence of row lengths, from bottom to top. The previous example gives 4, 2, 1. Such a sequence is an integer partition, i.e., a nonincreasing sequence of positive integers. This gives a bijection between Young diagram and integer partition. Each of the unit square in the Young diagram is called a cell. Each cell has four corners, and a point being a corner of at least one cell is called a vertex of the Young diagram.

Let Y denote Young's lattice, which is the set of all Young digrams endowed with the inclusion order (seeing Young diagrams as set of cells). Let ∅ denote its minimal element, the empty Young diagram. Let denote the cover relation on Y (and the reverse relation).

An important property of Y is that it is a differential poset [START_REF] Stanley | Differential posets[END_REF].

Definition 2.1. A stammering tableau of size n is a sequence (λ (0) , . . . , λ (3n) ) ∈ Y 3n+1 such that λ (0) = λ (3n) = ∅, and:

-if i ≡ 0 or 1 mod 3 then either Roby [START_REF] Roby | Applications and extensions of Fomin's generalization of the Robinson-Schensted correspondence to differential posets[END_REF] and Krattenthaler [START_REF] Krattenthaler | Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes[END_REF] showed that oscillating tableaux and their variants correspond via Fomin's growth diagrams [START_REF] Fomin | The generalized Robinson-Schensted-Knuth correspondence[END_REF] to some fillings of Young diagrams. We can do the same with stammering tableaux, and we are led to the following definition: Definition 2.2. Let 2δ n denote the double staircase diagram of n rows, defined as the Young diagram with row lengths (2n, 2n -2, . . . , 2). A partial filling of 2δ n with some dots, such that there is exactly one dot per row and at most one per column, is called a rook placement of size n.

λ (i) λ (i+1) or λ (i) = λ (i+1) , -if i ≡ 2 mod 3 then λ (i) λ (i+1
See Figure 1 for examples. Note that it is easy to build a rook placement T of size n inductively from a rook placement T of size n -1. When adding a new row to the bottom of T , this gives a priori 2n possible choices for the location of the dot of this new row, but n -1 of them are forbidden by the rule that there is at most one dot per column. So there remains n + 1 choice for T starting with a given T . In particular this shows that there are (n + 1)! rook placements of size n. Let us present the description of Fomin's growth diagrams. See [START_REF] Krattenthaler | Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes[END_REF][START_REF] Roby | Applications and extensions of Fomin's generalization of the Robinson-Schensted correspondence to differential posets[END_REF]. The idea is to assign a Young diagram to each vertex of the double staircase. So in the pictures, we will have a big Young diagram, namely the double staircase, and small Young diagrams at its vertices. Note that the terminology "growth diagram" is rather standard even if there is possible confusion with the notion of Young diagram.

• • • • • • • • • • •
In each cell of the double staircase, the Young diagram in the North-East corner is determined by the three Young diagrams in the other corners, according to the local rules given below. Starting with empty diagrams ∅ along the West and South border of the double staircase, we can assign diagrams to all other vertices. The rules are as follows. Consider the four diagrams at the corners of a cell, then, with the notation of Figure 2:

-if µ = ν, then ρ = µ ∪ ν; -if λ = µ = ν and the cell is empty, ρ = λ;
-if λ = µ = ν and the cell contains a dot, then ρ is obtained from λ by adding 1 to its first part;

-if λ = µ = ν, then ρ is obtained from µ by adding 1 to its (k + 1)st part, where k > 0 is minimal such that λ k = µ k . (•) λ µ ρ ν Figure 2: Generic cell of a growth diagram. • • • • • ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ • • • • • • ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Figure 3: Growth diagrams.
The growth diagrams obtained from the rook placements in Figure 1 are in Figure 3. By following the North-East boundary, from the top-left corner to the bottom-right corner, we read a sequence of Young diagrams. The two growth diagrams in Figure 3 give our two running examples Λ 1 and Λ 2 defined at the beginning of this section.

The same bijection can be described using the Schensted insertion instead of the growth diagrams. This algorithm is described in [START_REF] Stanley | Enumerative Combinatorics[END_REF]Chapter 7]. An incomplete standard tableau is a filling of a Young diagram with integers such that:

-the integers are increasing along rows (from left to right) and columns (from bottom to top), -there are no repetition among them.

Let T be such a tableau. If k > 0 is not an entry of T , let T ← k denote the tableau obtained after row insertion of k (see [START_REF] Stanley | Enumerative Combinatorics[END_REF]Chapter 7]). If k is an entry of T located at a corner, let T → k denote the tableau obtained after removing k.

From a rook placement R of size n, we can form a sequence T 0 , . . . , T 3n of incomplete standard tableaux in the following way. We label the dots in R so that the dot in the kth row (from bottom to top) has label k, see Figure 4. We read the North-East border of R from the top-left corner to the bottom-right corner. This border contains 3n steps, and starting from the empty tableau T 0 , we define T 0 , . . . , T 3n in such a way that:

-if the ith step is horizontal, at the top of a column which contains no dot, then T i = T i-1 , -if the ith step is horizontal, at the top of a column which contains the dot with label k, then

T i = T i-1 ← k,
-if the ith step is vertical, at the right of a row which contains the dot with label k, then

T i = T i-1 → k.
For example, the rook placements in Figure 4 respectively give the sequences 4 ,4 6 ;4 ,4 5 ,4 5 ;4 ,4 ,4 1 ;1 ,1 3 ,1 3 ;1 ,1 ,1 2 ;1 ,1 ,1 ; ∅.

∅, 2 , 2 5 ; 2 , 2 4 , 4 2 3 ; 2 3 , 2 3 , 2 1 3 ; 2 1 , 2 1 , 2 1 ; 1 , 1 , 1 ; ∅ and ∅,
If we forget the integer entries, the sequence of shapes of the incomplete standard tableaux T 0 , . . . , T 3n form a stammering tableau λ (0) , . . . , λ (3n) , and it is precisely the one associated to R via the growth diagram. This is a general fact and follows from known properties of these constructions [START_REF] Fomin | The generalized Robinson-Schensted-Knuth correspondence[END_REF][START_REF] Krattenthaler | Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes[END_REF][START_REF] Roby | Applications and extensions of Fomin's generalization of the Robinson-Schensted correspondence to differential posets[END_REF]. Finally, let us present a third equivalent construction via Viennot's shadow lines [START_REF] Viennot | Une forme géométrique de la correspondance de Robinson-Schensted[END_REF]. We give only one example of this construction, in Figure 5 (page 7).

Start from a rook placement R in the double staircase. The idea is that each dot projects a quarterplane-like shadow to its North-East. The boundary of the union of all these shadows gives a curve, the first shadow. We can do the same thing with the dots left alone by the first shadow, this gives the second shadow. The other shadows are constructed in the same way. In the first picture of the example, we see that only two shadows are necessary to cover all dots of the rook placement. Now, the idea is that these shadow lines determine the first rows of the partitions in a stammering tableau as follows. Starting from 0, we read the 3n steps of the North-East boundary of the diagram, add 1 when reading a horizontal step crossing a shadow line, remove 1 when reading a vertical step crossing a shadow line, doing nothing otherwise.

The next step is to put a dot at each -shaped corner of the shadows. We obtain a new set of dots, which has to be a rook placement as well. The shadows of these dots gives the second rows in the stammering tableau. See the second picture in the example of Figure 5.

Going on with the successive shadows of the successive new sets of dots give the full stammering tableau. We don't give the proof that this construction is equivalent to the previous ones, and refer to [START_REF] Viennot | Une forme géométrique de la correspondance de Robinson-Schensted[END_REF] for more details.

Remark 2.3. We have introduced stammering tableaux using the interpretation of the operators D and U acting on Young diagrams, and given bijections with the rook placements in the double staircase using various methods. It is also possible to obtain the rook placements from the operators D and U (as abstract operators satisfying a commutation relation DU -U D = I) using the methods in [START_REF] Varvak | Rook numbers and the normal ordering problem[END_REF].

Remark 2.4. Although we do not give the details here, it is possible to give reverse local rules for growth diagrams. See [14, Section 2.1.8]. In the notation of Figure 3, it means that if we know µ, ν, and ρ, we can recover λ and whether the cell contains a dot or not. So if we have a stammering tableaux along the North-East border of the double staircase diagram, we can apply these rules to complete the rest of the picture, and get a rook placement. This is the reverse bijection of the one presented in this section.

• • • • • • • • • • • • ∅, , ; , , ; , , ; , , ; , , ; , , ; , , ; ∅

Chains of Dyck shapes

A Dyck path of length 2n is a path in N 2 from (0, 0) to (2n, 0) with steps (1, 1) and (1, -1). The two kind of steps will be denoted = (1, 1) and = (1, -1). The height of (x, y) ∈ N 2 is its second coordinate y. Accordingly, each step in a Dyck path has an initial height and a final height, namely the heights of its starting point and ending point, respectively. We can also see a Dyck path of length 2n as a binary word of length 2n over { , } such that in each prefix, there are more than (in the weak sense). These words are called Dyck words. A prefix of a Dyck word is called a Dyck prefix.

A property of stammering tableaux or rook placements that will be important in the sequel is that we can naturally associate a Dyck path to each of them. Definition 3.1. Let R be a rook placement in the double staircase 2δ n . We define a Dyck path d(R) of length 2n + 2 as follows:

-the first step is , the last step is , -if 2 ≤ i ≤ 2n+1, the ith step is if the (i-1)st column of R contains a dot and otherwise.

See Figure 6 for an example. Proof. Let 1 ≤ i ≤ 2n. Among the first i columns of R, say that x contains a dot and y are empty (so x + y = i). In the double staircase, the first i columns entirely contain the topmost i/2 rows, so x ≥ i/2 , and y = i -x ≤ i/2 . The first i + 1 steps of d(R) contains 1 + x steps and y steps , so the height after the (i + 1)st step is 1 + x -y. We have 1 + x -y ≥ 1 + i/2 -i/2 ≥ 0 so that the path d(R) stays above the horizontal axis.

Instead of Dyck paths, it will convenient to use the equivalent notion given below.

Definition 3.3. The staircase partition of size n is δ n = (n, n -1, . . . , 1) ∈ Y. A Dyck shape of length 2n is a skew shape δ n /λ where λ ∈ Y is such that λ ⊂ δ n-1 .
We will use here an unusual notation for Young diagrams, that we call Japanese notation. It is obtained from the French notation by performing a 135 degree clockwise rotation. By following the upper border of a Dyck shape in Japanese notation, we recover a Dyck path. See Figure 7 for an example with n = 5 and λ = (3, 1, 1, 1). Note that the parts of the partition correspond to the number of cells in each South-East to North-West diagonal.

Each (unit) square cell of a Dyck shape has some coordinates (x, y). They are normalized so that the bottom left cell has coordinates (0, 0), the one to its North-East has coordinates (1, 1), the bottom right cell has coordinates (2n -2, 0), and so on. A column of a Dyck shape is a set of cells having the same x-coordinate, for example the Dyck shape in Figure 7 has 8 non-empty columns, one of them having 2 cells and the others only 1 cell. Similarly, a row of a Dyck shape is a set of cells having the same y-coordinate. In the example of Figure 7, there are three non-empty rows. The notion of diagonals should be clear, to distinguish the two different kinds, we refer to them as -diagonals and -diagonals. In this definition, we assume that the two Dyck shapes are arranged so that their leftmost cells coincide (which is coherent with the way we defined the coordinates in the skew shape).

Remark 3.5. This definition is easily translated in terms of binary words over and . Let D a dyck word, then D < E if and only if E is obtained from D by changing a into a (and each step of D can be changed except the last one, so that there are n + 1 possibilities for a path of length 2n).

Figure 8 shows the ribbons that can be added to the Dyck path of Figure 7. Note that the number of -diagonals of these ribbons are exactly the integers from 1 to 6. This is a general fact: Proposition 3.6. Let D be a Dyck shape of length 2n and 1 ≤ i ≤ n + 1. Then there is a unique way to add to D a ribbon whose number of -diagonals is i.

The result is clear upon inspection and does not deserve a detailed formal proof. Note that from Proposition 3.6, the number of n-chains of Dyck shapes is n!.

When we have an

n-chain D 1 < D 2 < • • • < D n , the ribbons D i /D i-1 with 1 ≤ i ≤ n (
where we take the convention D 0 = ∅) form a partition of D n as a set of cells. We can thus represent an n-chain ending at D as a partition of D in n ribbons, see Figure 9 for an example. Conversely, a Dyck shape D of length 2n partitioned in n ribbons arise in this way if it satisfies the following condition: the right extremities of the ribbons are exactly the n cells in the bottom row of D. Proposition 3.8. Let R be a rook placement in 2δ n-1 . Let R i be the rook placement in 2δ i obtained by keeping only the i top rows of R (by convention, R 0 is empty so that d(R 0 ) = δ 1 /∅, the unique Dyck shape of length 2, and R n-1 = R). Then d(R 0 ), . . . , d(R n-1 ) is an n-chain of Dyck shapes. This defines a bijection between rook placements in 2δ n-1 and n-chains of Dyck shapes.

Proof. Let us compare d(R i-1 ) and d(R i ). If the dot in the ith row of R i is on the right extremity, d(R i ) is obtained from d(R i-1 ) by adding at the end. In the other cases, since only one dot is added from R i-1 to R i , d(R i ) as a Dyck word is obtained from d(R i-1 ) by replacing a with a and adding two at the end. This shows d(R i-1 ) < d(R i ), so d(R 0 ), . . . , d(R n-1 ) is indeed a chain of Dyck shapes.

We can recover the rook placement R from the chain of Dyck shapes. Indeed, let j be the index of the first step where the two paths d(R i-1 ) and d(R i ) differs, or

j = 2i + 1 if d(R i-1 ) is a prefix of d(R i ).
Then there is a dot in the (j -1)st cell of the ith row of R.

Since both sets have the same cardinality, what we have defined is a bijection.

Figure 9: The chains of Dyck shapes corresponding to rook placements in Figure 1.

The poset of Dyck paths

The transitive and reflexive closure of the relation < defines the structure of a graded poset on the set of Dyck paths of all possible lengths. The rank of a path is half its length, for example the minimal Dyck path (i.e., the empty path) has rank 0, and has rank 1. In terms of this poset, what we call an n-chain of shape D is a saturated chain of n elements starting at and arriving at D. Let (D, ≤) denote this poset, the goal of this section is to prove that it is lattice.

Let D be a Dyck path. Its Dyck word is the infinite binary word (d i ) i≥1 obtained from D by replacing with 1, with 0, and appending an infinite sequence of 0's. Note that the rank of a Dyck path is the number of 1's in the corresponding Dyck word. Though seemingly useless, appending the final sequence of 0's slightly simplifies some statements such as the next proposition.

The height in a Dyck path D after i steps is the number of 1's minus the number of 0's among d 1 , . . . , d i . We have d i ≥ 0 for i ≤ 2n where n is the rank of the path and d i < 0 for i > 2n. We also define the height similarly for a binary word which is not a priori a Dyck word. Proof. It is clear that if D 1 , D 2 are Dyck words, D 1 < D 2 means that D 2 can be obtained from D 1 by changing a 0 into a 1. So D < E implies d i ≤ e i for all i. We deduce that D ≤ E also implies this condition.

It remains to show the other implication. So we suppose now d i ≤ e i for all i, and we will show D ≤ E by induction on the difference of ranks.

The initial case is clear: if D and E have the same rank, we get d i = e i for all i so D = E and in particular D ≤ E. Otherwise, let M be the maximal integer such that d M < e M . In particular, d M = 0 and e M = 1. We define E = (e i ) by changing this e M = 1 into a 0. If E is indeed a Dyck word, we have D ≤ E by using the induction hypothesis. Also E < E by construction, so E ≤ E, and it follows D ≤ E.

So it remains to show that E is a Dyck word. If the difference of ranks is 1, we have E = D so it is a Dyck word. So we can assume that the difference of ranks is at least 2. From d i ≤ e i , we get that h i (E) -h i (D) is increasing with i. We have that h M (E) -h M (D) is twice the difference of rank of E and D, i.e., at least 4. So h M (E ) -h M (D) is at least 2. So h i (E ) = h i (D) + 2 for i ≥ M , so it is nonnegative as long as h M (D) ≥ -2. Now let M be the minimal integer with h M (E ) = 0. We easily obtain d i = e i = 0 for i ≥ M and the fact that E is a Dyck word. We have f i ≤ c i , so the path F goes below height 0 after |c | steps. We deduce that f i = 0 when i is greater than |c |. So F ≤ D ∧ E.

Permutations and their profiles

As mentioned above, the number of n-chains of Dyck shapes is n!, and the goal of this section is to give a bijection with permutations. It is related with the notion of the profile of a permutation as defined by Françon and Viennot [START_REF] Franc ¸on | Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'Euler et nombres de Genocchi[END_REF], i.e., a Dyck path encoding the location of peaks, valleys, double ascents and double descents, as defined below (the term shape was maybe more close to the french "forme", but it was already in use in this paper). This notion of profile gives another interpretation of the relation < (addition of a ribbon) in Dyck paths, giving a natural bijection between chains of Dyck shapes and permutations.

Definition 5.1. Let σ ∈ S n . We will take the convention that σ 0 = 0 and σ n+1 = n + 1. The integer σ i ∈ {1, . . . , n} is called:

-a peak if σ i-1 < σ i > σ i+1 , -a valley if σ i-1 > σ i < σ i+1 , -a double ascent if σ i-1 < σ i < σ i+1 , -a double descent if σ i-1 > σ i > σ i+1 .
For example, if σ = 42135, 1 is a valley, 2 is a double descent, 3 and 5 are double ascents, 4 is a peak. Note that in this definition, the value σ i , not the index i, is a peak if σ i-1 < σ i > σ i+1 (and similarly for the other statistics). This is not coherent with the usual definition of a descent and ascent (an index i such that σ i > σ i+1 or σ i < σ i+1 ), but it is nonetheless standard. Definition 5.2. The profile of a permutation σ ∈ S n is a Dyck path ∆(σ) of length 2n, defined as follows. We see it as a binary word w = w 1 • • • w 2n over the alphabet { , }, such that the factor w 2i-1 w 2i is :

- if i is a valley of σ, - if i is a peak of σ, - if i is a double ascent of σ, - if i is a double descent of σ.
In the previous example σ = 42135, we get ∆(σ) = (which is as well the Dyck path represented in Figure 7). Remark 5.3. This notion of profile of a permutation and the associated Dyck path ∆(σ) has also been introduced by Brändén and Leander [START_REF] Brändén | Multivariate P -Eulerian polynomial[END_REF], in slightly different terms.

Let σ ∈ S n be a permutation, considered as a word σ 1 • • • σ n . We define σ ∈ S n-1 as what we obtain after removing n. For example, (1457236) = 145236. More generally, by mimicking the notation for derivatives, we define σ (0) = σ and σ (i+1) = (σ (i) ) .

The goal of this section is to prove:

Theorem 5.4. The map

σ → ∆(σ (n-1) ), . . . , ∆(σ ), ∆(σ ), ∆(σ) (4)
defines a bijection between S n and n-chains of Dyck shapes.

The inverse bijection will be described explicitly below. Anticipating a little, let us announce that the permutations τ = 513462 and υ = 5471326 correspond to the chains of Dyck shapes in Figure 9. We can check for example that

∆(τ (0) ) = ∆(513462) = , ∆(τ (1) ) = ∆(51342) = , ∆(τ (2) ) = ∆(1342) = , ∆(τ (3) ) = ∆(132) = , ∆(τ (4) ) = ∆(12) = , ∆(τ (5) ) = ∆(1) = ,
which corresponds to the first chain of Dyck paths in Figure 9. The proof of Theorem 5.4 and the definition of the inverse bijection will follow from the lemma below.

Lemma 5.5. Let σ ∈ S n , then we have ∆(σ ) < ∆(σ). If ∆(σ ) is a prefix of ∆(σ), then σ is obtained from σ by inserting n in the last position. Otherwise, let k be the index of the first step where the paths ∆(σ ) and ∆(σ) differ, then:

-if k is odd, σ is obtained from σ by inserting n just after k+1 2 ,
-if k is even, σ is obtained from σ by inserting n just before k 2 . Proof. Let us compare ∆(σ ) and ∆(σ). If n is at the end of σ, then ∆(σ) is obtained from ∆(σ ) by adding at the end, corresponding to the double ascent n. So ∆(σ ) < ∆(σ) (this is the addition of a ribbon with one cell).

Otherwise, n is between two other integers, i.e., there is in σ a factor i, n, j (with the convention that i = 0 if n is at the beginning of σ). We suppose first that i < j. It means that either -j is a double ascent in σ and a valley in σ, or -j is a peak in σ and a double descent in σ.

On the Dyck paths, it means that the jth pair of steps is either in σ and in σ, or in σ and in σ, i.e., in each case the (2j)th step is in σ and in σ. It is easily checked that the other steps do not change. So ∆(σ ) < ∆(σ). Suppose then that i > j. It means that either -i is a double descent in σ and a valley in σ, or -i is a peak in σ and a double ascent in σ.

On the Dyck paths, it means that the ith pair of steps is either in σ and in σ, or in σ and in σ, i.e., in each case the (2i -1)st step is in σ and in σ. It is easily checked that the other steps do not change. So ∆(σ ) < ∆(σ).

Note that the cases i < j and i > j we considered can be distinguished via the parity of the changed step between ∆(σ ) and ∆(σ). Thus we can finish the proof of the proposition: in the first case, we can find j from the index of the changed step and we know that n is just before j, and in the second case, we can find i from the index of the changed step and we know that n is just after i.

The inverse bijection of (4) can be described explicitly. Let (D 1 , . . . , D n ) be an n-chain of Dyck paths. We build a permutation σ ∈ S n by inserting successively 1, 2, . . . , n. Start from 1 ∈ S 1 , corresponding to the 1-chain of Dyck paths (

). Suppose we have already inserted 1, . . . , j, then we insert j + 1 as follows. Let i ≥ 1 be such that the leftmost cell of the ribbon D j+1 /D j is in the ith column. If i = 2j + 1, which means the ribbon D j+1 /D j contains a single square, then j + 1 is inserted in last position. Otherwise:

-if i is even and i = 2i , then j + 1 is inserted to the left of i , -if i is odd and i = 2i -1, then j + 1 is inserted to the right of i .

Let us give the details for the first example of Figure 9. In this case, we have 6 ribbons, the leftmost cells of the successive ribbons are in columns 1,3,4,5,2,7. So, starting from the permutation 1:

-first i = 3, so insert 2 in the last position and get 12, -then i = 4, so insert 3 to the left of 2 and get 132, -then i = 5, so insert 4 to the right of 3 and get 1342, -then i = 2, so insert 5 to the left of 1 and get 51342, -then i = 7, so insert 6 to the right of 4 and get 513462, Thus we get τ = 513462.

Laguerre histories and Dyck tableaux

One result of Françon and Viennot [START_REF] Franc ¸on | Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'Euler et nombres de Genocchi[END_REF] is that the number of permutations with a given profile D is a product of integers easily described in terms of the Dyck path D. This is done via a bijection between permutations and Laguerre histories (see [START_REF] Viennot | Une théorie combinatoire des polynômes orthogonaux[END_REF]), the name being justified by the link with Laguerre orthogonal polynomials. To relate this with Theorem 5.4 in the previous section, we give a bijection between chains of Dyck shapes and these objects. Note that Laguerre histories were previously related to the PASEP partition function and permutation tableaux in [START_REF] Josuat-Vergès | Combinatorics of the three-parameter PASEP partition function[END_REF]. Definition 6.1. A column of a Dyck path is below a step if its topmost cell is not contiguous to another cell to its North-West. Alternatively, the kth column is below a step if the kth step of the path is . Let D be a Dyck path of length 2n, then a Laguerre history of shape D is a filling of D with n dots such that there is one dot in each column below a step (and no dot in the other columns).

• • • • • • • • • • • • • Figure 10: Laguerre histories.
See Figure 10 for some examples of Laguerre histories. If a step in a Dyck path is from height h to h + 1, the column below it contains h 2 + 1 cells (here the height of the step is seen as the y-coordinate in the path in N 2 with two kinds of steps, not in the skew shape). This means that a Laguerre history is a Dyck path together with a choice among h 2 + 1 possibilities for each step from height h to h + 1, i.e., we recover the notion of subdivided Laguerre history (see Viennot [20, Chapter 2] and de Médicis and Viennot [START_REF] De Médicis | Moments de polynômes de q-Laguerre et la bijection de Foata-Zeilberger[END_REF]Section 4]). In particular, it follows from [15, Section 4] that there are n! Laguerre histories of length 2n. by changing some step (say, this is the kth step) into a . By construction, it means that the ith ribbon has its left extremity in the kth column of the Dyck path. We can deduce that there is exactly one dot in each column below a step in D at the end of the process.

Then, we define the inverse bijection. Let H be a Laguerre history of shape D. To find what are the ribbons of the corresponding chain of Dyck paths, we proceed as follows. Start from the rightmost cell of the Dyck path, and follow the upper border of cells from right to left until arriving to a cell containing a dot; the path we just followed is the last added ribbon D, remove D and do the same thing to find the other ribbons. It is easy to show that the two bijections are inverses of each other.

For example, the bijection described in the previous proposition sends the chains in Figure 9 to the Laguerre histories in Figure 10. Definition 6.3 (Aval et al. [START_REF] Aval | Dyck tableaux[END_REF]). Let D be a Dyck shape of length 2n. A Dyck tableau of shape D is a filling of D with n dots such that there is one dot in each odd column.

See Figure 11 for some examples of Dyck tableaux.

• • • • • • • • • • • • • Figure 11: Dyck tableaux.
To see the equivalence between Dyck tableaux and our definition of Laguerre histories, we need the following lemma. Lemma 6.4. Let D be a Dyck shape. Then there is a bijection between the columns of D below a step , and its odd columns. It is such that c and κ(c) contain the same number of cells. Proof. The bijection κ is constructed as follows. If a column c is both odd and below a step , then κ(c) = c. This is always the case for the first column. As for the other case, let c be an even column below a step . It will be associated to κ(c), an odd column of the same height as c and which is not below a step .

We use the notion of facing steps. If we have a step in a Dyck path, from (x, y) to (x+1, y+1), its facing step is the step from (x , y + 1) to (x + 1, y) where x > x is minimal. Note that in this situation, x and x have different parity, which follows because a Dyck path only visits pairs (i, j) with i + j even. Reciprocally, each step is the facing step of a unique step , than we can also call its facing step.

To define κ(c), we follow the illustration in Figure 12. The column c is the below a step , say (x, y) to (x + 1, y + 1). Note that x + 1 is even, by the assumption that c is an even column. And the number of cells of c is (y + 1)/2. In the example, x = 7, y = 5. The step from (x, y) to (x + 1, y + 1) has a facing step from (x , y + 1) to (x + 1, y). Then κ(c) is defined as the (x + 1)st column, i.e., that below the end point of this facing step .

Since the notion of facing step gives a pairing between steps and , it is easily seen that we can recover c from κ(c). By construction, κ(c) is not below a step , and x + 1, y are odd. It follows that κ(c) has the same number of cells as c. See Figure 12.

The previous lemma can be used to give a bijection between Laguerre histories of shape D and Dyck tableaux of the same shape. Let H be a Laguerre history of shape D, we can define a Dyck tableau T of shape D by the following condition :

-if a column c in H contains a dot in its ith cell (from bottom to top, for example), then the column κ(c) in T also contains a dot in its ith cell. This is clearly bijective. For example, from the Laguerre histories in Figure 10

A generalization

In this section, we consider a generalization of stammering tableaux where we do not require

λ (0) = λ (3n) = ∅ anymore. Let T (n)
µ,ν be the number of such tableaux λ (0) , . . . , λ (3n) with λ (0) = µ and λ (3n) = ν. We don't know how to compute this number in general, but we have a simple answer when either µ or ν is empty.

Also we need to consider a generalization of rook placements in 2δ n , so in this section we do not require that there is exactly one dot per row.

Lemma 7.1. The number of fillings of 2δ n with k dots, such that there is at most one dot per row and at most one per column, is

(n + 1)! (n -k + 1)! n k .
Proof. These numbers satisfy the recursion a n,k = a n-1,k + (2n -k + 1)a n-1,k-1 , which can be checked either on the definition with rook placements or on the given formula. The initial conditions are a 0,0 = 1, a n,k = 0 if n < 0 or k < 0 or k > n. Proof. The idea is to use growth diagrams to build a bijection between these generalized stammering tableaux and rook placements in the double staircase together with some additional data. Note that we will here use the reverse local rules of growth diagrams ([14, Section 2.1.8]). Using the notation of Figure 3, it means that if we know µ, ν, ρ, we can find λ and whether the cell contains a dot or not.

Let us begin with the first equality. Let λ (0) , . . . , λ (3n) with λ (0) = ∅ and λ (3n) = λ. We can put this sequence of partitions along the North-East border of 2δ n from the top left corner to the bottom right corner, and use the reverse local rules of growth diagrams to complete the picture, i.e., fill some cells with a dot and assign a Young diagram to each vertex. We obtain a rook placement R in 2δ n with n dots, and the bottom border contains a sequence ρ (0) , . . . , ρ (2n) such that ρ (i-1) = ρ (i) if the ith column of R contains a dot, ρ (i-1) = ρ (i) or ρ (i-1) ρ (i) otherwise. This ρ (0) , . . . , ρ (2n) can be encoded by a standard tableau of shape λ (describing the sequence of shapes without repetition) and the choice of k columns among the n columns without a dot (describing what are the indices i such that ρ (i-1) ρ (i) ). So there are n k f λ such sequences for a given rook placement. Since there are (n + 1)! rook placements in 2δ n , we obtain T (3) where U is defined with the cover relation of ≤ and D is defined with the cover relation of . See [START_REF] Fomin | Duality of graded graphs[END_REF] for details. Note that must be a tree in this situation. If there is such a construction, it might be related with a Hopf algebra of Dyck paths. So, defining a relevant coproduct on the algebra of Dyck path of Brändén and Leander [START_REF] Brändén | Multivariate P -Eulerian polynomial[END_REF] is a related question.

(n) ∅,λ = (n + 1)! n k f λ . • • ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅
It is well known that standard tableaux are related with representations of the symmetric groups. Oscillating tableaux are related with that of Brauer algebras [START_REF] Barcelo | Combinatorial representation theory[END_REF], and vacillating tableaux are related with that of partition algebras [START_REF] Halverson | Partition algebras[END_REF]. It would be interesting to see if our stammering tableaux have any similar algebraic meaning.

On the purely combinatorial level, there are a lot of enumerative or bijective results on Dyck tableaux, tree-like tableaux, permutation tableaux, and links with permutation statistics, q-Eulerian numbers, etc. See for example [START_REF] Aval | Dyck tableaux[END_REF][START_REF] Aval | Tree-like tableaux[END_REF][START_REF] Josuat-Vergès | Combinatorics of the three-parameter PASEP partition function[END_REF][START_REF] Viennot | Alternative tableaux, permutations and partially asymmetric exclusion process[END_REF]. It would be interesting to see if the combinatorial objects presented in this paper are helpful on these topics.

Figure 1 :

 1 Figure 1: Rook placements in the double staircase.

Figure 4 :

 4 Figure 4: Rook placements in double staircase, with labeled dots.

Figure 5 :

 5 Figure 5: Shadows of a rook placement.

Figure 6 :

 6 Figure 6: The Dyck paths associated to the rook placements in Figure 1.

Figure 7 :

 7 Figure 7: A Dyck shape.

Figure 8 :Definition 3 . 7 .

 837 Figure 8: Addition of a ribbon to a Dyck path.

Proposition 4 . 1 .

 41 Let D, E ∈ D and (d i ) i≥1 , (e i ) i≥1 their Dyck words. We have D ≤ E if and only if d i ≤ e i for all i.

Proposition 4 . 2 .

 42 D is a lattice. Proof. Let D and E be two Dyck path. To show the existence of the join operation, let us define a Dyck path D ∨ E by: (D ∨ E) i = max(d i , e i ). Using the previous lemma, it is clear that D ≤ D ∨ E, E ≤ D ∨ E, and D ≤ F , E ≤ F implies D ∨ E ≤ F . So we only have to show that D ∨ E is indeed a Dyck path with this definition. Suppose for example that the length of D is bigger than that of E. We see from the definition that D ∨ E is obtained from D by changing some steps into , so it is clear that the new path cannot go below height 0. Now, we show the existence of the meet operation D ∧E. First, consider the sequence c defined by c i = min(d i , e i ). It might not be a Dyck word, for example if D = 110010... and E = 101010... (where the dots represent the infinite sequence of 0's). So, let us consider the smallest prefix c of c containing more 0's than 1's. We define D ∧ E as the Dyck word c ... (where the dots represent the infinite sequence of 0's). It is clear that D ∧ E ≤ D, D ∧ E ≤ E. Suppose that F ≤ D and F ≤ E for some Dyck path F = (f i ).

Proposition 6 . 2 .

 62 Let D 1 < • • • < D n be a chain of Dyck paths of shape D = D n . We define a filling of D by putting a dot in the left extremity of each ribbon D i+1 /D i . Then what we obtain is a Laguerre history, and this defines a bijection between n-chains of Dyck shapes and Laguerre histories of the same shape D.Proof. We first prove that what we obtain is indeed a Laguerre history. Let 2 ≤ i ≤ n. The Dyck path D i is obtained from D i-1

Figure 12 :

 12 Figure 12: Definition of the bijection κ.

  , we obtain the Dyck tableaux in Figure 11. More precisely, the correspondence between columns can be illustrated by numbering dots,

Proposition 7 . 2 .

 72 Let λ ∈ Y with |λ| = k, and let f λ be the number of standard tableaux of shape λ. Then we have T

Figure 14 :

 14 Figure 14: A growth diagram illustrating reverse local rules.

  ) .

	For example,
		∅, ,	; ,	,	; , , ; , , ; ∅
	is a stammering tableau of size 4. Note that for readability, there is a ";" at every three steps,
	i.e., at each step going down. Through the whole article, we will have two running examples to
	illustrate the various bijections:
	Λ 1 = ∅, ,	; ,	,	;	,	,	; , , ; , , ; ∅ .
	and
	Λ 2 = ∅, ,	; ,	,	; , , ; ,	,	; , ,	; , , ; ∅ .

The second result is similar but starting with the assumption λ (0) = λ and λ (3n) = ∅. We put this sequence along the North-East border of 2δ n , and the reverse local rule of growth diagrams are performed. We obtain a partial rook placement R in 2δ n with n -k dots, and the left border of 2δ n is a sequence ρ (0) , . . . , ρ (n) such that ρ (i-1) = ρ (i) if the ith row of R contains a dot, and ρ (i-1) ρ (i) otherwise. Using the previous lemma, we obtain T

To illustrate the first identity, if we start from

, the growth diagram is in Figure 13. Here n = 4 and k = 3. We get a rook placement in 2δ 4 , and the bottom line

This sequence of Young diagrams can be encoded in the standard tableau

which can be encoded in the standard tableau

Note that we do not need to keep track of when repetitions occur in this sequence, because these correspond to rows containing a dot in the growth diagram.

Open problems

The poset introduced in Section 4 is reminicent of the notion of Fomin's dual graded graphs [START_REF] Fomin | Duality of graded graphs[END_REF], since there are n! paths from the minimal element to some element of rank n. So the problem is to find some dual order relation on the set of Dyck paths of all possible lengths, so that we have the commutation relation DU -U D = I (or some variation) for the up and down operators as in


Aydar Rakhmatullin

Kelly Machado

Didier Zanghi

Ilya B Polovov

Rinat Bakirov

Konstantin V Maksimtsev

Catherine Bessada

Study of the NaF-ScF3 system as a molten bath for production of Sc alloys: A combination of NMR and molecular dynamics simulations

Keywords: High temperature NMR, scandium fluorides, molten salts, Molecular Dynamics, DFT 1. Introduction

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

One of the main fields of application of scandium is the development of lightweight, high-strength aluminum alloys [START_REF] Kononov | High-temperature electrochemical synthesis and properties of intermetallic compounds of the Ni-Sc system Part 1. Electrochemical behaviour of Sc(III) in chloride-fluoride melts[END_REF]. Scandium in aluminum-based alloys is introduced during their manufacture as an aluminum-scandium master alloy where scandium content is varied from 1.5 to 5 wt%. The structure of the Al-Sc master alloy represents a solid solution of scandium in aluminum with dispersed particles of the Al3Sc intermetallic compound [START_REF] Zakharov | Kinetics of Decomposition of the Solid Solution of Scandium in Aluminum in Binary Al-Sc Alloys[END_REF]. The maximal solubility of scandium in aluminum-based solid solutions does not exceed 0.8 wt% [START_REF] Fujikawa | Solid solubility and residual resistivity of scandium in aluminum[END_REF]. Thus, А13Sс is formed during both initial and eutectic crystallization, and as result of decomposition of a supersaturated solid solution. This intermetallic phase, with a face centered cubic lattice, has very close geometrical parameters to the aluminum lattice that results in the unique influence of scandium on structure and properties of aluminum alloys [START_REF] Sawtell | Mechanical properties and microstructures of Al-Mg-Sc alloys[END_REF].

The main manufacturers of Al-Sc master alloys use the following methods for their synthesis: direct alloying of scandium metal with aluminum, electrowinning from molten salts containing scandium salts or oxide, and aluminothermic reduction of scandium salts or Sc2O3 [START_REF] Suzdaltsev | Modern ways for obtaining Al-Sc master alloys: A review[END_REF]. The last technique allows technological use of scandium concentrate based on NaScF4 that was obtained from waste solutions arising from uranium underground leaching according to the original method [START_REF] Rychkov | Method of reprocessing of wasted scandium-containing solutions from uranium extraction[END_REF].

Therefore, the study and comprehension of the ionic structure of the electrolyte, consisting of NaF and ScF3, represents an important contribution to understanding the mechanism of the formation of Al3Sc. Chemical and physical properties of this melt strongly depend on scandium speciation.

The information concerning scandium speciation in halide systems is relatively poor.

Moreover, most authors have used limited number of techniques in their works. Several electrochemical studies indicated that complex Sc 3+ ions are the only soluble scandium species in chloride and chloride-fluoride molten salts [START_REF] Kononov | High-temperature electrochemical synthesis and properties of intermetallic compounds of the Ni-Sc system Part 1. Electrochemical behaviour of Sc(III) in chloride-fluoride melts[END_REF][START_REF] Stangrit | Electrochemical behaviours of scandium in chloride melts[END_REF][START_REF] Castrillejo | Electrochemistry of scandium in the eutectic LiCl-KCl[END_REF], however their structure in the melts is unknown. CsI-ScI3 and CsCl-ScCl3 mixtures were studied in detail in both molten and solid state by Raman spectroscopy over the entire composition range at temperatures up to 1000 °C [START_REF] Zissi | Changes of vibrational modes upon melting solid Cs2NaScCl6, Cs3ScCl6, Cs3Sc2Cl9 and ScCl3[END_REF][START_REF] Zissi | Composition and temperature induced changes on the structure of molten ScCl3-CsCl mixtures[END_REF][START_REF] Metallinou | Thermal analysis and Raman spectroscopic measurements on the scandium iodide-cesium iodide system[END_REF]. In the molten CsI-ScI3 system with scandium iodide mole fractions below 0.6, the presence of ScI6 3-and ScI4 -species in the equilibrium was proposed. In the CsCl-ScCl3 system, a different ionic species, i.e. ScCl7 4-, ScCl6 3-, Sc2Cl9 3-, and ScCl4 -, were established. A clusterlike model for the structure of pure molten ScCl3 was also proposed, where fragments of scandium octahedra bridged by chlorides were terminated with scandium tetrahedra having terminal chlorides.

The aim of this contribution is to investigate the structure of molten NaF-ScF3 binary systems by a multinuclear ( 19 F, 45 Sc, and 23 Na) in situ NMR spectroscopy at high temperature and by computations combining Molecular Dynamics (MD) simulations and density functional theory (DFT) calculations. To perform MD simulations over a large dynamic time interval (1 -5 ns), a classical interaction atomic potential for NaF-ScF3 melts has been specially developed.

The interatomic potential was validated over a wide range of compositions, by computing the NMR chemical shifts of each nucleus in the framework of the density functional theory. DFT calculations were performed on configurations generated along the MD simulations without further optimization. Statistical analysis of ion trajectory from MD simulations allowed to describe accurately the structure of the melts and to establish the relation between the physicochemical and structural properties.

Methods and Experiments

Materials.

For the preparation of the samples, the following chemicals were used: NaF (Fluka, min. 99.9 %; dried in vacuum at 773 K for 4 h), ScF3 (Sigma-Aldrich, 99.9 %), without further purification. All fluorides were stored in a dry box under argon atmosphere maintained below 0.3 ppm of moisture and 0.1-0.5 ppm of oxygen. Different compositions of ScF3 and NaF were prepared by mixing and grinding the corresponding fluorides. For high temperature (HT) NMR measurements, approximately 60 mg of the mixtures were put into tightly closed high purity boron nitride crucibles to avoid any contamination by the surrounding atmosphere.

2.2.NMR spectroscopy.

The HT NMR spectra were acquired using the laser heated NMR system developed in CEMHTI Orleans [START_REF] Lacassagne | Structure of High-Temperature NaF-AlF3-Al2O3 Melts: A Multinuclear NMR Study[END_REF]. The BN crucible was hermetically closed, containing the sample in powder form is placed inside the RF-coil, in the center of the cryomagnet. To insure a good homogeneous heating, the top and the bottom of the crucible are heated by a CO2 laser (Coherent Diamond 250W), passing axially through the NMR probe. The temperature was controlled by the laser power following a preliminary calibration curve. During the experiment, the temperature was held at 10 °C above the corresponding sample melting point [START_REF] Thoma | The Sodium Fluoride-Scandium Trifluoride System[END_REF]. The melting point of the sample is clearly detected by the modification of the shape and width of the NMR line. The transition from solid to liquid state can be followed by 19 F NMR spectra during the heating of the sample. The desired temperature should be reached in the sample after 4-5 minutes. During the NMR experiments, the peaks do not change with time once the sample is in the liquid state. NMR spectra of 19 F, 23 Na, and 45 Sc have been acquired for NaF-ScF3 compositions ranging from 0 to 70 mol% ScF3. It was not possible to obtain HT NMR spectra at concentrations above 70 mol%, due to the very high melting point [START_REF] Thoma | The Sodium Fluoride-Scandium Trifluoride System[END_REF]. ScF3 has a maximum melting temperature among all simple fluorides (MFx) Tm = 1550 ± 3 °C [START_REF] Spedding | High temperature enthalpies and related thermodynamic functions of the trifluorides of Sc[END_REF].

2.3.Molecular dynamics simulations

Molecular dynamics simulations were performed using the polarizable ion model (PIM) [START_REF] Madden | Covalent' effects in 'ionic' systems[END_REF][START_REF] Hutchinson | A unified description of MCI3 systems with a polarizable ion simulation model[END_REF]. This model uses a classical interatomic potential which is a sum of four pairwise additive interactions. The analytical expression is made up of four terms: The first term corresponds to the electrostatic interactions. The two following terms are expressed by a Born-Huggins-Mayer type potential; they consist of an exponentially decaying term accounting for the overlap repulsion of the electronic clouds at short distances, and a term, which represents the dispersion effects. The B ij and a ij represent the repulsion coefficients and the Cij 6 and Cij 8 parameters are the dipole-dipole and dipole-quadrupole dispersion coefficients.

𝑉 = ∑ (
The fn are Tang-Toennies dispersion damping functions [START_REF] Tang | An improved simple model for the van der Waals potential based on universal damping functions for the dispersion coefficients[END_REF] describing the short range penetration correction to the asymptotic multipole expansion of dispersion.

Finally, the fourth term, called polarization term reflects the distortion of the electronic density in response to the electric fields due to all the other ions, which results in induced dipole on each ion [START_REF] Salanne | Polarization effects in ionic solids and melts[END_REF]. This polarization term include charge-dipole and dipole-dipole interactions as well as the polarizability , corresponding to the energy cost of deforming the charge density of the ion i to create the induced dipole:

𝑉 𝑝𝑜𝑙 = ∑ [(𝑞 𝑖 𝜇 𝛼 𝑗 𝑔 𝑖𝑗 (𝑟 𝑖𝑗 ) -𝑞 𝑗 𝜇 𝛼 𝑖 𝑔 𝑗𝑖 (𝑟 𝑖𝑗 )) 𝑇 𝛼 𝑖𝑗 -𝜇 𝛼 𝑖 𝜇 𝛽 𝑗 𝑇 𝛼𝛽 𝑖𝑗 ] 𝑖,𝑗 + ∑ ( 1 2𝛼 𝑖 |𝜇 𝑖 | 2 )
𝑖 where µ i are induced dipoles on each ion i, T and T are the charge-dipole and dipole-dipole interaction tensors,  i is the polarizability of ion i and 𝑔 𝑗𝑖 are Tang-Toennies functions.

The parameters listed in Table 1 for each atomic pair are specific of the studied system and force field used.

Their determination was made by a 'force-matching' method [START_REF] Machado | Study of NaF-AlF3 Melts by Coupling Molecular Dynamics, Density Functional Theory, and NMR Measurements[END_REF][START_REF] Salanne | Calculation of Activities of Ions in Molten Salts with Potential Application to the Pyroprocessing of Nuclear Waste[END_REF] based on ab initio DFT calculations of forces, dipoles and stress tensor chosen as reference. These first-principles electronic structure calculations were performed on nine high temperature ionic configurations to cover a wide range of compositions going from 5 to 70 mol% of ScF3. The DFT calculations were carried out by using a generalized gradient approximation (GGA) with a Perdew, Burke, and Ernzerhof (PBE) exchange-correlation functional [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] as implemented in the Vienna Ab-Initio Simulation Package (VASP) [START_REF] Kresse | Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF] with the cut-off energy (Ecut) fixed at 800 eV. The dispersion interactions are explicitly calculated through the use of the DFT-D3 correction [START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF],

which gives a better representation of the interaction. The forces and dipoles on each ion were determined from the results of this calculation. The forces on each species and the stress tensors were obtained directly from each DFT calculation using the Hellmann-Feynman theorem. As for the dipoles, they were extracted by making use of the transformation of the Kohn-Sham orbitals to a Maximally Localized Wannier Function set [START_REF] Marzari | Maximally localized generalized Wannier functions for composite energy bands[END_REF]. Then the parameters in the polarizable potential were optimized by matching the dipoles, forces and stress tensors from the potential on the same ionic configuration to the Ab-Initio values [START_REF] Heaton | A First-Principles Description of Liquid BeF2 and Its Mixtures with LiF: 1. Potential Development and Pure BeF2[END_REF]. A visual representation of the matching obtained from DFT and PIM is presented in the supporting information (Fig. 1S). In this work, a simple cubic box corresponding to the unit cell was used with periodic boundary conditions that impose a specific symmetry on the system under study. For the MD simulations, the systems were first equilibrated using the isothermal-isobaric (NPT) ensemble for several compositions ranging from 5 to 70 mol% of ScF3. The temperature and the average internal pressure were held at 10 °C above melting point (Table 2) and 0 GPa, respectively, by coupling the simulation cell to Nosé-Hoover thermostat chains and a barostat [START_REF] Martyna | Nosé-Hoover chains: The canonical ensemble via continuous dynamics[END_REF][START_REF] Martyna | Constant pressure molecular dynamics algorithms[END_REF]. The melting temperatures for the different compositions were extracted from the phase diagram established by Thoma and Karraker [START_REF] Thoma | The Sodium Fluoride-Scandium Trifluoride System[END_REF].

The relaxation time that sets the strength of the coupling of the system to the thermostats or to the barostat was fixed to 20 ps and the time step for integrating the equations of motion was fixed to 1 fs using the Verlet algorithm [START_REF] Verlet | Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules[END_REF]. The aim of this equilibration period, whose duration was typically 200 ps, was to allow the system to reach equilibrium. The cell length was fixed to the stable average value of the NPT run. In our case, this equilibration time corresponds to the time required for the total energy of the system, the temperature and the density were calculated from length of the cell, to reach a steady state around an average value.

We considered that this stability was achieved when the ratio between the average absolute deviation and the average value is less 5 %.

After the first stage of equilibration, the system was simulated for 1 ns in the canonical ensemble (NVT) by fixing the length of the cell to its average value. During this production phase, the atomic coordinates, the energies, the dipoles and the velocities of the different ions were recorded every 100 fs. The parameters of the cubic cells (compositions, number of atoms and volumes) used for simulations are provided in Table 2. The volume of the box was calculated by considering a cubic box. Using a confidence interval of 68 %, the uncertainty about the length cell was determined from the calculation of its mean value and the corresponding standard deviation on the last 100 ps of the equilibration phase.

NMR DFT computational details

The chemical shielding tensors for all atoms ( 19 F, 23 Na, 45 Sc) in the molten system NaF-ScF3 have been calculated using the gauge including projector augmented wave (GIPAW) method [START_REF] Pickard | All-electron magnetic response with pseudopotentials: NMR chemical shifts[END_REF] implemented with the NMR-CASTEP code [START_REF] Stewart | First principles methods using CASTEP[END_REF][START_REF] Yates | Calculation of NMR chemical shifts for extended systems using ultrasoft pseudopotentials[END_REF]. The PBE functional [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] was used for the exchange-correlation energy and the core-valence interactions were described by ultrasoft pseudopotentials (USPP) [START_REF] Yates | Calculation of NMR chemical shifts for extended systems using ultrasoft pseudopotentials[END_REF]. The USPP were generated during "the on the fly" generator (OTFG_USPP) included in CASTEP. The wave functions were expanded on a plane-wave basis set with a cutoff-energy of 610 eV. The Brillouin zone was sampled using a Monkhorst-Pack grid [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] spacing of 0.05 Å -1 .

In order to take into account, the thermal agitation of the ions due to the temperature into the NMR parameters calculation, 10 snapshots were extracted every 100 ps along the MD trajectory simulated in the NVT ensemble. For each snapshot, the individual chemical shielding tensor is calculated for all the nuclei of the system. The isotropic shielding values are then averaged to give a mean value for each atomic configuration. For each composition, the isotropic chemical shielding was obtained by averaging the values obtained from the 10 snapshots [START_REF] Machado | Study of NaF-AlF3 Melts by Coupling Molecular Dynamics, Density Functional Theory, and NMR Measurements[END_REF][START_REF] Machado | Anionic Structure in Molten Cryolite-Alumina Systems[END_REF][START_REF] Machado | Structural, Dynamic, and Thermodynamic Study of KF-AlF3 Melts by Combining High-Temperature NMR and Molecular Dynamics Simulations[END_REF]. Therefore, to convert the calculated isotropic chemical shielding into isotropic chemical shift, the definition of calibration curve relating isotropic chemical shift and shielding is necessary. For 45 Sc and 19 F the relationship reported in our previous study [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF] and for 23 Na the relationship reported on [START_REF] Machado | Study of NaF-AlF3 Melts by Coupling Molecular Dynamics, Density Functional Theory, and NMR Measurements[END_REF] were applied.

Results and Discussions

3.1.HT NMR in NaF-ScF3 melts

The NMR spectra acquired in the molten state consist in a single Lorentzian line, characteristic of a rapid exchange at the timescale of NMR between the different atomic configurations around the observed nucleus. The measured chemical shift is thus the average of the individual chemical shifts of the different species.

The 19 F spectra obtained in NaF-ScF3 melts at high temperature, over the whole range of compositions are presented in Figure 1. 19 F chemical shifts obtained in our system ranged from -233.5 ppm to -7.9 ppm. The evolution of chemical shift was monotonous and nonlinear with composition (Fig. 2 left). This evolution is very similar to the trend already observed in the case of rare earth fluoridealkali fluoride systems AF-ReF3 [START_REF] Bessada | In situ NMR approach of the local structure of molten materials at high temperature[END_REF][START_REF] Rollet | Local Structure Analogy of Lanthanide Fluoride Molten Salts[END_REF], LiF-ZrF4 [START_REF] Pauvert | Ion Specific Effects on the Structure of Molten AF-ZrF4 Systems (A + = Li + , Na + , and K + )[END_REF], and for the AF- Hence, the average coordination number around fluorine atoms tends to monotonically increase.

The 45 Sc chemical shift decreases linearly from 9 ppm to -2 ppm with increasing concentration of ScF3 and it remains constant after 50 mol% (Fig. 2 right). In alkali fluoroscandate solid compounds, scandium is present in octahedral (Li3ScF6 [START_REF] Tyagi | Syntheses and structures of Li3ScF6 and high pressure LiScF4, luminescence properties of LiScF4, a new phase in system LiF-ScF3[END_REF], Na3ScF6 [43],

KScF4 [START_REF] Champarnaud-Mesjard | KScF4: A New ABX4 Octahedral cis-trans Layered Structure[END_REF], and K5Sc3F14 [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF]), seven (KSc2F7 [START_REF] Turrell | Spectroscopic studies of mixed hexa-hepta scandium fluorides[END_REF] and NaScF4 [START_REF] Ai | Lanthanide-doped NaScF4 nanoprobes: crystal structure, optical spectroscopy and biodetection[END_REF]) and eight (LiScF4 [START_REF] Tyagi | Syntheses and structures of Li3ScF6 and high pressure LiScF4, luminescence properties of LiScF4, a new phase in system LiF-ScF3[END_REF]) coordinations with fluorine. The scandium chemical shifts domains are correlated with the different possible coordination state of scandium atoms. The 45 Sc chemical shift values for six and seven coordinated scandium environments range from -20 to 14 ppm and from -53.5 to -40 ppm, respectively [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF]. Therefore, we can conclude from scandium NMR that in our melts, the average coordination number around scandium is six over the whole range of compositions.

High temperature 23 Na measurements were performed for the different compositions.

We observe a decrease of the chemical shifts values over the whole range of ScF3 content (Fig.

right

). This reduction with ScF3 addition corresponds to the increased shielding of the alkali cations. Therefore, the electronic cloud around the Na nucleus becomes more symmetric. This means that the Na-F interaction decreases with ScF3 amount and sodium becomes free. This conclusion is important in terms of conductivity as the Na + ion would become the main charge carrier at high ScF3 content.

3.2.Experimental and calculated Chemical shifts

The chemical shift is one of the most sensitive markers of the local structure. It is sensitive to the first neighbors, i.e. to the coordination numbers and the corresponding bond lengths. The chemical shifts, computed using the coupled MD/DFT approach, are compared with experimental data on Figure 2. The uncertainty of the calculated chemical shift was estimated to be ±5 ppm for 45 Sc, ±5 ppm for 23 Na and ±3 ppm for 19 F and; it is comparable with the uncertainty in NMR experimental measurements: ±2 ppm for 19 F, ±3 ppm for 45 Sc and ±1 ppm for 23 Na. A good agreement between the calculated and experimental shift values is observed, which indicates that the polarizable ion model used in these simulations is capable of reproducing accurately the local structure around each nucleus in the NaF-ScF3 molten salt.

3.3.Coordination Number

The coordination numbers are obtained using the first minimum of the radial distribution function as the radius of the solvation shell to count the number of neighbors around a given ion. Figure 3 shows the proportion of each species, extracted from ion trajectory obtained by MD simulations. As mentioned above, in solid alkali fluoroscandates, the Sc 3+ ions can usually occur in a 6-, 7-, or rarely 8-fold coordination. In the molten state, the dominant fluoroscandate species is ScF6 3-in all compositions. In contrast to the solid state, the four-fold and five-fold coordinated ScF5 2-ions were found. It has to be noted that the presence of ScF8 5-species was detected, but their relative concentration is very small (less than 1 %). The second species present over the whole range of concentrations is ScF5 2-. Up to 40 mol%, the concentration of ScF5 2-varies slightly, then it increases up to 30 mol%. It appears that ScF7 Then the polymerization process continues: 1) amount of bridging F increases rapidly which leads to the formation of bonded chain or network -F-Sc-F-Sc-; 2) three coordinated fluorine atoms start to form with the formation of bonds between the chains (Fig. 4). It is worth noting that the formation of fluorine atoms bonded with three scandium atoms is accompanied by the formation of five-and seven-fold coordinated scandium ions, whereas the concentration of sixcoordinated scandium decreases. At melt compositions rich in scandium fluoride, the melt becomes fully polymerized. We can therefore deduce that the polymerization process is the origin of the variation of the density shown below.

In the case of Li-and Na-cryolites, the formation of AlFm 6-m dimers bridged via common F atoms was detected by Raman, NMR spectroscopies [START_REF] Lacassagne | Structure of High-Temperature NaF-AlF3-Al2O3 Melts: A Multinuclear NMR Study[END_REF][START_REF] Robert | Study of NaF-AlF3 Melts by High-Temperature 27 Al NMR Spectroscopy: Comparison with Results from Raman Spectroscopy[END_REF] and computing methods [START_REF] Machado | Study of NaF-AlF3 Melts by Coupling Molecular Dynamics, Density Functional Theory, and NMR Measurements[END_REF][START_REF] Machado | Anionic Structure in Molten Cryolite-Alumina Systems[END_REF][START_REF] Bučko | Effect of alkaline metal cations on the ionic structure of cryolite melts: Ab-initio NpT MD study[END_REF]. In contrast to these systems, in our case, formation of dimeric species was not observed. either the cation can reduce the number of anions in its coordination shell to remain isolated, or the individual cations might begin to share anions and have overlapping shells. In other words, there is an increasing degree of corner and edge-sharing between coordination polyhedra as the YCl3 or ScF3 concentration increases, leading ultimately to a network structure. They show that the tendency to increase coordination number is linked to the propensity to form linkages of different types.

The average coordination number around fluorine atoms increases monotonically in case of both methods. Our calculation results are fully consistent with our earlier conclusions deduced from the NMR experimental data.

Density and electrical conductivity

Density and electrical conductivity are very important properties for industrial applications of molten salts. Such systems are rarely amenable to experimental measurements due to the high price of scandium fluoride. To the best of our knowledge, there are no data in the literature on these physical properties of NaF-ScF3 melts. Thus, to obtain a quantitative estimation of such systems, we resorted to calculation. The good agreement shown above between calculations and experiment allows us to present computed parameters here with confidence in the accuracy of the results.

The electrical conductivity is calculated from MD simulations by taking into account the correlations of the displacements between the different species, according to the formula:

𝜎 = 𝑒 2 𝑘 𝐵 𝑇𝑉 lim 𝑡→∞ 1 6𝑡 (|∑ 𝑞 𝑖 𝛿𝑟 𝑖 (𝑡) 𝑖 | 2 )
where, e is the elementary charge, T is the temperature, V is the volume of the cubic cell, kB is the Boltzmann constant, qi is the formal charge of the atom i, and δri(t) is the displacement of ion i in time t. where ρi, qi and μi are respectively the density number, the charge and the conventional mobility of the ion of type i. The density number is given by the ratio of the Avogadro number and the molar volume: NA/Vm. As shown in the above formula, the conductivity obeys to the ratio 𝜇 𝑖 (𝑇, 𝑥) 𝑉 𝑚 (𝑇, 𝑥) ⁄ and it is given by the comparative growth, in temperature and composition, between both the molar volume Vm and ionic mobility µi. However, the temperature dependence of the ionic mobility is in general greater than the thermal expandability, thus, the conductivity tends to increase with temperature.

In our future work, we plan to investigate the mechanism of the Al metal dissolution in the NaF-ScF3 melt and to analyze how different anionic species affect this process.

Conclusions

The local structure of scandium and fluoride ions was investigated in the molten NaF-ScF3 system by combining HT NMR measurements and molecular dynamics simulations.

ThF4 [ 41 ]

 41 system (A= Li, Na, and K; Re=Y, La, Ce, Lu). This evolution is explained by the existence of three different fluoride types. At high NaF content, fluoride anions are mainly free.The signals for the terminal (non-bridging) fluorines appear in the chemical shift range from -63 to -114 ppm and the chemical shift range for the bridging fluorines is from -23 to -75 ppm[START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF]. Therefore, from fluorine NMR results, when ScF3 is added, fluorine atoms start to bond with Sc atoms and to be involved in scandium based complexes [ScFx]3-x. For higher amounts of ScF3, connections are formed between scandium complexes [ScFx]3-x by bridging fluorides.

  4-entities starts to form at 25 mol%. The average coordination number around scandium obtained from calculations remained almost unchanged and its value was 5.8±0.1.

Figure 4

 4 Figure 4 shows the fluorine environments extracted from the MD calculations. As expected, at low ScF3 concentrations fluorine is present mainly as the free anion, whereas with an increase in ScF3 concentration the fluorine anion takes part in the formation of anionic species containing the Sc-F bond (Sc-F•••Na + ). At around 25 mol% its concentration reaches a maximum. More than 90 % of the fluorine atoms are connected with only one scandium atom, forming the terminal fluorines. Bridging fluorides (Sc-F-Sc) appear for a concentration of ScF3 larger than 25 mol% and it becomes the dominant fluoroscandate species for concentrations of ScF3 exceeding 55 mol%. It should be noted that the free fluorine atoms disappear from the melt at the eutectic composition (40 mol% ScF3) and there are only terminal and bridging fluorines.

  The average coordination number around scandium obtained from calculations and NMR experiments agrees closely and it does not change with the concentration of ScF3. Corradini et al.[START_REF] Corradini | Coordination numbers and physical properties in molten salts and their mixtures[END_REF] compared experimental and simulation data of the molten YCl3-(LiCl-KCl)eut and LaCl3-(LiCl-KCl)eut mixtures. The average coordination number around a lanthanum ion chloride melt ranges from 6 to 8. As in our system, the yttrium coordination number remains close to 6 across the concentration range. They propose two possible behaviors of the melt:

Figures 5 and 6

 6 Figures 5 and 6 show the variations of the density and electrical conductivity of the NaF-ScF3 melts with the composition. The density increases with increasing concentration of ScF3 and after 40 mol% it remains almost constant. The conductivity shows an important decrease up to 40 mol%, and then it rapidly increases. Both physical quantities change their trend at the eutectic point due to changes of the speciation in the melt. This is consistent with the structural

  From45 Sc high-temperature NMR spectra an average coordination number of 6 has been determined for the scandium on all domains of composition of 5-70 mol%. The 19 F chemical shift evolution coincides with the existence of at least three types of fluorines: free at infinite dilution, fluorines connected with one scandium, and bridging fluorines at high ScF3 contents.The anionic structure was calculated for the first time for a wide range of compositions of NaF-ScF3, and was in agreement with experimental results. The agreement between experimental and calculated chemical shifts for the three nuclei ( 19 F,45 Sc and 23 Na) allows us to validate the new interatomic potential specially developed for simulating the molten binary NaF-ScF3 in a wide range of composition (0-70 mol%). The validity of this polarizable ion model allows us to calculate reliable thermophysical data, such as density and electrical conductivity. Their evolutions with the ScF3 content are consistent with the structural description of the molten NaF-ScF3 systems. Analysis of computed data made it possible to identify the network formation in the melt. This work will open new perspectives for the study of structural, dynamic and thermodynamic properties of NaF-ScF3 mixtures for the production of Sc alloys, for which experimental data are lacking.

Figure Captions:Figure 1 .

 1 Figure Captions:Figure 1. The 19 F NMR spectra of NaF-ScF3 melts.

Figure 2 .

 2 Figure 2. NMR chemical shift evolution for 19 F (left) and for 23 Na, 45 Sc (right) in NaF-ScF3. The calculated (full markers) and experimental (empty markers) chemical shifts are shown.

Figure 3 .

 3 Figure 3. Scandium atomic fraction of: ScF4 -(); ScF5 2-(•); ScF6 3-(▲); ScF7 4-(▼), ScF8 5- (♦), and average coordination (★) as a function of concentration of ScF3 in the molten NaF-

Figure 4 .

 4 Figure 4. Fluor atomic fraction of: free fluorine (), terminal fluorine F-Sc (•), bridged fluorine Sc-F-Sc (▲), three bridged fluorine (▼), and average coordination (★) varying concentrations of ScF3 at molten state.

Figure 5 .

 5 Figure 5. Evolution of calculated (■) and experimental (•) [50] density as a function of mol% ScF3 in the molten state.

Figure 6 .

 6 Figure 6. Electrical conductivity of molten the NaF-ScF3 mixtures as a function of the mol% of ScF3 at 10 °C above melting point obtained by MD simulations.

Fig. 1

 1 Fig.1

Fig. 2

 2 Fig.2

Fig. 3

 3 Fig.3

  Fig.4

Fig. 5

 5 Fig.5

Fig. 6

 6 Fig.6

Table 1 .

 1 Pair

	Ion pair	𝐵 𝑖𝑗	𝛼 𝑖𝑗	𝐶 𝑖𝑗 6	𝐶 𝑖𝑗 8
	F --F -	392.57	2.48	3.11	104.93
	F --Sc 3+	51.45	1.77	4.68	33.76
	F --Na +	38.73	1.89	0.65	8.07
	Sc 3+ -Sc 3+	3.00	0.99	761.16	1454.85

potential parameters (all of the parameters are given in atomic unit (au)).

Where, Bij and αij are parameters from the repulsion term; 𝐶 6 𝑖𝑗 and 𝐶 8 𝑖𝑗 are respectively dipoledipole and dipole-quadrupole terms from the dispersion component between two ions at a longer range. The dipole polarizabilities of fluoride, Scandium and sodium ions obtained in this work are 8.158 au, 2.318 au and 0.847 au, respectively.

Table 2 .

 2 Molecular dynamics simulations conditions of NaF-ScF3: compositions, number of

	mol% ScF3	NF-	NNa+	NSc3+	Volume, Å 3	Melting point, K

atoms (NF-, NNa+, and NSc3+), volume of the cubic cell, and melting point.

Acknowledgment

This work was supported by the Ministry of Education and Science of the Russian Federation No. 02.G25.31.0210 of 27.04.2016. For the calculations, we thank the "Centre de Calcul Scientifique en region Centre" (Orleans, France). We thank also Dr. M. Salanne and Dr. M.

Pitcher for useful discussions.