Cyril Banderier

Pawe L Hitczenko

Philippe Flajolet

Enumeration and asymptotics of restricted compositions having the same number of parts

Keywords: integer composition, pairs of combinatorial structures, local limit theorem, asymptotics of D-finite sequences, diagonal of algebraic generating function

We study pairs and m-tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of parts. Under the uniform probability model, we obtain the asymptotics for the probability that two or, more generally, m randomly and independently chosen compositions of n have the same number of parts. For a large class of compositions, we show how a nice interplay between complex analysis and probability theory allows to get full asymptotics for this probability. Our results extend an earlier work of Bóna and Knopfmacher. While we restrict our attention to compositions, our approach is also of interest for tuples of other combinatorial structures having the same number of parts.

Introduction

In this note, we study tuples of compositions of positive integers having the same number of parts, and the asymptotics of related generating functions satisfying some differential equations. Let us recall that a composition of a positive integer n is any k-tuple (κ 1 , . . . , κ k ), k ≥ 1, of positive integers that sum up to n. The κ j 's are called the parts (or summands) of a composition. It is elementary and well-known (see, e.g. [START_REF] Andrews | The theory of partitions[END_REF]) that there are n-1 k-1 compositions of n with k parts, and thus there are 2 n-1 compositions of n. By restricted compositions we mean compositions whose parts are confined to be in a fixed subset P of N.

The main motivation for this work is a recent paper [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF] in which the authors studied pairs of compositions with the same number of parts. Our extension of this work is directly connected to the question of obtaining the asymptotics of coefficients of functions satisfying a linear differential equation which, despite the deep work by Fabry, Frobenius, Fuchs, Picard and other analysts more than one century ago, remains open and is conjectured to be undecidable. We present here a new way to use probability theory in addition to complex analysis in order to solve this problem for a large class of functions.

In their paper [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF], Bóna and Knopfmacher studied the asymptotic probability that two randomly and independently chosen compositions of n have the same number of parts. Furthermore, relying on the generating function approach, for a few specific subsets P they addressed the same question for pairs of restricted compositions. In each of these cases this probability is asymptotic to C/ √ n with C depending on P. Our main aim here is to extend these results. First, we show that this asymptotics is universal. That is, we show that for an arbitrary subset P containing two relatively prime elements the probability that two independently chosen random compositions of n with parts restricted to P have the same number of parts is asymptotic to C/ √ n. The value of C depends, generally, on P and is explicit. (See our Theorem 5.1 and subsequent remarks, which include e.g. a correction of a constant appearing in [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF].) Secondly, we consider the same question for m > 2 and we show that in this case the sought probability is asymptotic to C/ √ n m-1 for an explicitly given constant C whose value depends on P and m only. (See our Theorem 5.3.)

Bóna and Knopfmacher's approach relied on complex analysis; the universality of using a more probabilistic technique was then noticed by Bóna and Flajolet [START_REF] Bóna | Isomorphism and symmetries in random phylogenetic trees[END_REF], where certain types of random trees were studied. Our approach is in one sense a mixture of complex analysis (which gives the full asymptotics expansion, up to a multiplicative constant, and with the price of heavy computations), and probability theory (a local limit theorem which gives without any heavy computation the first asymptotic term, and therefore gives access to the multiplicative constant, but intrinsically no access to further asymptotic terms). Bóna and Flajolet obtained, in particular, a general statement indicating how local limit theorem can help in evaluating probabilities that two independently chosen random structures of the same size have the same number of components (this is their Lemma 6 in [START_REF] Bóna | Isomorphism and symmetries in random phylogenetic trees[END_REF], which corresponds to our Lemma 5.2 for Gaussian density with a slightly different proof. Our Lemma 5.2 was obtained independently, but later). As we will see, these statements remain true if one considers more than two random structures.

In Section 2, we present our model. We proceed in Section 3 with some examples (and en passant, some nice questions in computer algebra) and argue on the intrinsic limitations of an approach relying only on complex analysis. This serves as a motivation for introducing the local limit law result in Section 4, which finds application in Section 5, thus solving the initial problem of the asymptotic evaluation of the probability that tuples of compositions have the same number of parts. We conclude with some perspectives in Section 6.

Generating functions for pairs of compositions having the same number of parts

Let us consider compositions with parts in a set P (a fixed subset of N). To avoid trivial complications caused by the fact that there may be no compositions of a given n with all parts from P, we assume that P has at least two elements that are relatively prime (except when explicitly stated otherwise).

We introduce the generating function of the parts p(z) = j∈P p j z j , (p j is not necessarily 0 or 1, it can then be seen as the possible colors or the weight of part j). We thus assume that the p j 's are non-negative real numbers such that j∈P p j > 1. This last condition is to ensure supercriticality of our scheme (see Section 4 below for more details). In the classical situation when p j is 0 or 1, this condition holds automatically. Denote by

P (z, u) = n≥0,k≥0 P n,k u k z n = 1 1 -up(z) (1)
the bivariate generating function of compositions of n where k encodes its number of parts, and where the "size" of the composition is n.

With a slight abuse of notation, the corresponding univariate generating function is

P (z) = n≥0 P n z n = 1 1 -p(z) . (2)
This terminology is classical. For example, here are all the compositions of 5 with 3 parts from the set P = {1, 2, 3, 4, 10}: 5 = 1 + 1 + 3 = 1 + 3 + 1 = 3 + 1 + 1 = 1 + 2 + 2 = 2 + 1 + 2 = 2 + 2 + 1. Accordingly, P 5,3 = 6.

Let X P n be the random variable giving the number of parts in a random composition of n with parts belonging to P. Random means that we consider the uniform distribution among all compositions of n with parts belonging to P.

Given two subsets P 1 and P 2 of N, we consider the probability π n := Pr(X P1 n = X P2 n ) that a random composition of n with parts in P 1 has the same number of parts as a random composition with parts in P 2 . We assume throughout that, whenever two such compositions are chosen, they are chosen independently and from now on we will not be explicitly mentioning it. We then introduce the generating function D(z) of the number of pairs of compositions (the first one with parts in P 1 , the second one with parts in P 2 ) having the same size and the same number of parts. (D stands for "double" or "diagonal", as D(z) can be obtained as a diagonal of multivariate function.)

That is, we consider all k-tuples of elements of P 1 and all k-tuples of elements of P 2 such that their sum is n. For a fixed n, let D n be the total number of such configurations (i.e., we sum over all k).

In the next section, we deal with some interesting examples for which we get explicit formulas.

3 Some closed-form formulas

An example on tuples of domino tilings

Consider the classical combinatorial problem of tiling a 2 × n strip by dominoes. Any tiling is thus a sequence of either one horizontal domino or 2 vertical dominoes. The generating function of domino tilings is thus

P (z) = Seq(z + z 2 ) = 1 1-z-z 2 , which is the generating function of P n = F n+1 , where F n is the Fibonacci number F n = F n-1 + F n-2 , F 0 = 0, F 1 = 1.
(Equivalently, the Fibonacci recurrence reflects the fact that removing a horizontal domino on the top of an existing 2 × n tiling leads to a 2 × (n -1) tiling, while removing 2 vertical dominoes on the top leads to a 2 × (n -2) tiling.) Let us now consider a less trivial question, which is archetypal of the problem we consider in this article (note that it has a closed-form solution but we will address later in this article similar problems having no such nice closed-form solution): Puzzle 3.1 Each of m children makes a tiling of a 2 × n strip. What is the probability π n that these m tilings all have the same number of vertical dominoes, when n gets large? For m = 2, the number of pairs is given by D(z) = [t 0 ] k≥0 p k (zt)p k (1/t), where p(z) = z + z2 , and the Cauchy formula gives

D(z) = 1 2iπ 1 1 -p(zt)p(1/t) dt t = 1 2iπ N um(z, t) Den(z, t) dt , (3)
where N um and Den are polynomials in z, t. Let Z(z) be any root of Den, i.e. Den(z, Z) = 0, such that Z is inside the contour of integration for z ∼ 0. Then, a residue computation gives:

D(z) = Z N um(z, Z) ∂ t Den(z, Z) = 1 √ z 4 -2z 3 -z 2 -2z + 1 = 1 + z + 2z 2 + 5z 3 + 11z 4 + 26z 5 + 63z 6 + 153z 7 + 376z 8 + 931z 9 + O(z 10 ) .
This is the sequence A051286 from [START_REF] Neil | The On-Line Encyclopedia of Integer Sequences[END_REF]

D n = n k=0 n-k k
2 , Bóna and Knopfmacher [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF] gives a bijective proof that it is also the Whitney number of level n of the lattice of the ideals of the fence of order 2n. The probability that 2 tilings of a 2 × n strip have the same number of vertical dominoes is therefore (via singularity analysis, which can be done automatically with some computer algebra systems, e.g. via the equivalent command of Bruno Salvy, in the Algolib Maple package available at http://algo.inria.fr/libraries):

π n = D n /P 2 n ∼ 5 3/4 2 √ π √ n + 5 1/4 ( 11 32 - √ 5
4 ) √ πn 3/2 + O( 1 √ n 5 ) ≈ .9432407854 √ n . (4)
Note that this is consistent with the constant C given in Equation (2.10) in [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF]. Our computations are available on-line in a Maple session1 . Note that as Maple does not always simplify algebraic numbers like humans would do (some denesting options are missing), we used here some of our own denesting recipes so that these nested radicals become more readable for human eyes. For m = 3, it is possible to compute the diagonal D(z) via creative telescoping (as automated in Maple via the MGfun package of Frédéric Chyzak or in Mathematica via the package HolonomicFunctions of Christoph Koutschan). This leads to the following differential equation:

0 = 4z 7 + 7z 6 + 7z 5 + 15z 4 + 41z 2 + z + 1 D (z) + 5z 8 + 12z 7 + 7z 6 + 62z 5 + 88z 3 + z 2 + 6z -1 d dz D (z) +z z 2 + 1 z 4 -z 3 + 5z 2 + z + 1 z 2 + 4z -1 d 2 dz 2 D (z) .
Here, the so-called Frobenius method gives the basis of the vector space of solutions of this ODE, under the form of local formal solutions around any singularity, by using the associated indicial polynomial (see [START_REF] Flajolet | Analytic combinatorics[END_REF]Chapter VII.9]). In full generality, the dominating singularity of D(z) is z = ρ m , where ρ is the radius of convergence of 1/(1 -P (z)); this can be proven via our Theorem 5.3. In our case, the Frobenius method gives around the dominating singularity ζ := √ 5 -2:

D(z) = λ 1 80 + 41 √ 5 90 ln(z -ζ) + 8 + 5 √ 5 9 ln(z -ζ) + O((z -ζ) 2 ) + λ 2 1 - 8 + 5 √ 5 9 (z -ζ) + 207 + 89 √ 5 81 (z -ζ) 2 + O((z -ζ) 3 )
for some unknown coefficients λ 1 , λ 2 (related to the so-called Stokes constants or connection constants) 2 . However, only the first summand contributes to the asymptotics of D n and a numerical scheme of our own allows to determine (with the help of the heuristic LLL algorithm) the value of λ 1 . Using singularity analysis then leads to

π n ∼ 5 √ 15 6 1 πn + 5(10 √ 3 -9 √ 15) 54 1 πn 2 + O(1/n 3 ) ≈ 1.027340740 n .
This asymptotics also proves that D(z) is not an algebraic function (the local basis of the differential equation involves a logarithmic term). For m = 4, creative telescoping leads to the following differential equation:

2 132z 16 -3563z 15 + • • • + 110 D (z) + 2 209474z 14 + • • • -1581z d dz D (z) + 704z 18 + • • • -10143z 2 d 2 dz 2 D(z) + z 165z 18 + • • • -55 d 3 dz 3 D (z) +z 2 (z -1) (z + 1) z 2 + z + 1 z 2 -7z + 1 z 2 -z + 1 z 4 + • • • + 1 11z 6 + • • • + 11 d 4 dz 4 D (z) .
Using the Frobenius method and a numerical scheme of ours, this leads to

π n ∼ 25 8 5 1/4 √ 2 √ πn 3 + 5 256 5 1/4 √ 2(47 √ 5 -240) √ π 3 n 5 + O( 1 √ n 7 ) ≈ 1.186814138 √ n 3 .
It is noteworthy that this asymptotics is compatible with the fact that D(z) could be an algebraic function. However, a guess based on Padé approximants fails to find any algebraic equation. What is more, the index of nilpotence mod 2, 3, 5, 7, 11 of D(z) is 3 (i.e. the smallest i such that (d/dz) i = L mod p is i = 3 for primes p = 2, 3, 5, 7, 11... and L is the above irreducible unreadable linear differential operator cancelling D(z)). Therefore, according to a conjecture of Grothendieck on the p-curvature (see [START_REF] Bostan | Fast algorithms for differential equations in positive characteristic[END_REF]), the function is not algebraic. For m = 5, D(z) is a non algebraic function satisfying a differential equation of order 6 and of degree 38, which leads to

π n ∼ 25 √ 5 4 1 π 2 n 2 ≈ 1.416006588 n 2 .
The closed form of the coefficients is

D n (m) = n k=0 n-k k m
, as can also be obtained via a bijective proof approach. It is possible to get their asymptotics via the Laplace method or our Theorem 5.3, this leads to

π n ∼ C m / (πn) m-1 with C m = (5 3/4 ) m-1 / √ 2 m-1 m.
This allows us to give a proof of the following claim (which was a conjecture by Paul D. Hanna, see [12, A181545]):

Proposition 3.1 D n+1 (m)/D n (m) ∼ (F m √ 5 + L m )/2
, where L m are the Lucas numbers, defined by the same recurrence as the Fibonacci numbers F m , but with different initial conditions, namely L 0 = 2 and L 1 = 1.

Proof. D n+1 (m) D n (m) = π n+1 (m)P m n+1 π n (m)P m n = C m / (π(n + 1)) m-1 P m n+1 C m / (πn) m-1 P m n = n n -1 m-1 2 P n+1 P n m ∼ 1 + (1/2 -m/2) 1 n + O( 1 n 2 ) 1 ρ (1 + O(ε n )) m ∼ 1 ρ m
where ρ = p -1 (1) and the asymptotics for P n is explained in detail in the next section (Equation 6). In the case of p(z) = z + z 2 , the claim then follows from ρ = 1/φ and the exact formula

φ m = (F m √ 5 + L m )/2.
Note that for all odd values of m > 2, the presence of an integer power of π in the asymptotics of D n (m) implies that the function D(z) can not be algebraic, whereas for all even values of m > 2, the asymptotics match the patterns appearing in the asymptotics of coefficients of algebraic functions. However, we expect the following conjectures to be true. m (non algebraicity of Franel numbers of order m). Nota bene: We gave here several ways to prove the non-algebraicity for some value of m, and we proved it for all odd m > 2, we are however unaware of any way of proving this for all even m > 2 at once, except, in some cases, an evaluation at some z leading to a transcendental number, or the Christol-Kamae-Mendès-France-Rauzy theorem on automatic sequences.

Definition 3.2 (Closed-form sequence) A sequence of integers D n is said to have a closed-form expression if it can be expressed as nested sums of hypergeometric terms, with natural boundaries (i.e. the intervals of summation are 0 and n). N.b: the number of nested sums has to be independent of n.

Typical examples of closed-form expression are nested sums of binomials; without loss (or win!) of generality, it is possible to allow more general intervals of summation or internal summands. Conjecture 3.2 Let D(z) be like in equation 5 below (for any rational functions p i (z) ∈ N(z)), then its coefficients D n can be expressed in closed form.

An effective way of finding this nested sum could be called a "reverse Zeilberger algorithm". It then makes sense to give the following broader conjecture: Conjecture 3.3 The coefficients of any D-finite function (i.e. a solution of a linear differential equation with polynomial coefficients) can be expressed in closed form.

Note that it follows from the theory of G-series that this does not hold for closed-forms of the type "one sum of hypergeometric terms" [START_REF] Garoufalidis | G-functions and multisum versus holonomic sequences[END_REF]. The formulas we will give in the rest of this section are somehow illustrating these conjectures.

Other nice explicit formulas

It is clear from the previous subsection that we could play the same game for any m-tuple of compositions with parts restricted to m different sets, encoded by p 1 (z), . . . , p m (z).

Proposition 3.3

The generating function for the number of m-tuples of compositions having the same number of parts is given by

D(z) = 1 (2iπ) m-1 1 1 -p 1 (zt 2 . . . t m )p 2 (1/t 2 ) . . . p m (1/t m ) dt 2 t 2 . . . dt m t m . (5)
Therefore, one should not expect any nice closed-form solution for D(z) whenever m > 2; while for m = 2, whenever all the p i (z)'s are polynomials or rational functions, D(z) will be an algebraic function whose coefficients can be expressed by nested sums of binomial coefficients (using Lagrange inversion).

For example, if p 1 (z) = p 2 (z) = 2z + z 2 (which can be considered as tilings with bicolored horizontal dominoes), one gets

D n+1 (2) = n k=0 k j=0 k j k+j j . If p i (z) = z
1-z (i.e., we consider compositions with any parts), then D n is the sequence of Franel numbers of order m:

D n+1 (m) = n k=0 n k
m , and we will see in Section 5 that the probability that m unrestricted compositions of n have the same number of parts is thus

π n ∼ C m / (πn) m-1 , with C m = 2 m-1 /m.
Note that if we replace p i (z) (for i > 1) by (1 + p i (z)) in the integral formula of Proposition 3.3, then this gives the generating function of tuples of compositions such that the number of parts is in decreasing order.

Let us add a few examples for which parts are in two different sets P 1 and P 2 . If p 1 (z) = z + z 2 and p 2 (z) = z + 2z 2 , then one gets an interesting case as we have here

π n ∼ 72 + 42 √ 3( √ 5 -5)( √ 2 -2) 12 √ πn 1 - √ 2 - √ 5 + √ 10 2(2 - √ 3) n ≈ 1.62 .95 n √ πn ,
which is therefore exponentially smaller that the order of magnitude of our previous examples. We will comment later on this fact. Going to a slightly more general case p i (z) = α i z + β i z 2 , one has for m = 2:

D(z) = 1 1 -2α 1 α 2 z + (α 2 1 α 2 2 -2β 1 β 2 )z 2 -2α 1 α 2 β 1 β 2 z 3 + β 2 1 β 2 2 z 4 .
Therefore the generating function only depends on the products α 1 α 2 and β 1 β 2 . This implies e.g. that p 1 (z) = 2z + 3z 2 and p 2 (z) = 3z + 5z 2 will lead to the same D(z) as p 1 (z) = 6z + z 2 and p 2 (z) = z + 15z 2 .

Note that D(z) factors nicely when β 1 = β 2 = 1:

D(z) = 1/ (α 1 α 2 z -1 -2z -z 2 )(α 1 α 2 z -1 + 2z -z 2 ) .
If, additionally, α 1 α 2 = ±4 this gives the sum of central Delannoy numbers [START_REF] Banderier | Why Delannoy numbers?[END_REF]:

D(z) = 1/(1 -z) × 1/ 1 + (2 + |α 1 α 2 |)z + z 2 .
When the allowed parts are only a and b, i.e. P i (z) = α i z a + β i z b , then all the compositions in the constrained tuples have necessarily the same number of parts "a" (this also holds for the number of parts "b"). Choosing the order of the n 1 parts "a" and the n 2 parts "b" leads to the formula

D n (m) = n1a+n2b=n n 1 + n 2 n 1 m (α 1 . . . α m ) n1 (β 1 . . . β m ) n2 .
There is no longer such a simple formula as soon as one has more than two allowed parts, because the parts can then compensate each other in many ways, e.g., assume that the allowed parts contain 3 integers 0 < a < b < c, then one can always create a composition P 1 having n 1 "a", n 2 "b", n 3 "c" and a composition P 2 having m 1 "a", m 2 "b" and m 3 "c" such they have the same number of parts

n 1 + n 2 + n 3 = m 1 + m 2 + m 3 , but (n 1 , n 2 , n 3 ) = (m 1 , m 2 , m 3 ). To achieve this, consider n 1 = c -b, n 3 = b -a, n 2 = n 1 + n 3 , m 2 = 0, m 1 = 2n 1 , m 3 = 2n 3 , thus one gets two different compositions of n: n = n 1 a + n 2 b + n 3 c = m 1 a + m 2 b + m 3 c. If p 1 (z) = αz + βz 2 and p 2 (z) = z 2 /(1 -z 2 ), then D 2n = β n , while if p 1 (z) = αz + βz 2 and p 2 (z) = z/(1 -z), then D(z) = 1 2 + 1 2 1 + αz 1 -2αz + z 2 (α 2 -4β) .
So, a nice surprise is given by p 1 (z) = z+z 2 and p 2 (z) = z/(1-z) , for which we get D(z) = 1/2+1/2 1+z 1-3z , which is known to be the generating function of directed animals [12, A005773]. This sequence also counts numerous other combinatorial structures: variants of Dyck paths, pattern avoiding permutations, base 3 n-digit numbers with digit sum n, . . . It also counts prefixes of Motzkin paths and this leads to an alternative formula

D n+1 = M n = 3 n - n-1 k=0 3 n-k-1 E k ,
where M n and E n stand for meanders and excursions of length n, following the definitions and notations from [START_REF] Banderier | Basic analytic combinatorics of directed lattice paths[END_REF].

We leave to the reader the pleasure of finding a bijective proof of all of this. (Some of them go via a bijection with lattice paths, as done in [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF], and then via the bijection between heaps of pieces and directed animals, see Fig. 1.) Note that some of the bijections can lead to efficient uniform random generation algorithms. In summary, it may seem possible to compute everything in all cases; however, for a generic P, in order to compute the constant

C m involved in π n ∼ C m 1 √
(πn) m-1 , we need heavier computations if the degrees of the p i (z)'s get large or if m is large. Current state of the art algorithms will take more than one day for m = 6, and gigabytes of memory, so this "computer algebra" approach (may it be via guessing or via holonomy theory) has some intrinsic limitations. What is more, for a given P, it remains a nice challenge to get a rigorous (Zeilbergerian computer algebra) proof for all m at once.

In the next sections, we show that the technical conditions to get a local limit law hold, and that this allows to get the constant C, for any P, for all m.

We conclude this section with Tables 1 and2 (see next page) summarizing our main closed-form formulas.

Local limit theorem for the number of parts in restricted compositions

The discussion in this section pretty much gathers what has been developed in various parts of the compendium on Analytic Combinatorics by Flajolet & Sedgewick [START_REF] Flajolet | Analytic combinatorics[END_REF]. Our main generating function (see Equation ( 1)) is a particular case of a more general composition3 scheme considered in Flajolet and Sedgewick, namely F (z, u) = g(uh(z)). In our case g(y) = 1/(1-y) and h(z) = p(z). According to terminology used in [9, Definition IX.2, p. 629, Sec. IX.3], under our assumption that j∈P p j > 1 the scheme is supercritical (i.e., when z increases, one meets the singularity y = 1 of g(y) before any other potential singularity of p(z)). As a consequence, the number of parts X P n is asymptotically normal as n → ∞, with both the mean and the variance linear in n. We now briefly recapitulate the statements from [START_REF] Flajolet | Analytic combinatorics[END_REF]. The equation p(z) = 1 has a unique positive root ρ ∈ (0, 1). As a consequence, F (z, 1) has a dominant simple pole as its singularity and thus the number P n of compositions of n with all parts in P is

[z n ]F (z, 1) ∼ 1 ρp (ρ) ρ -n (1 + O(ε n )) , (6)
where ε is a positive number less than 1, see [9, Theorem V.1, p. 294]. The probability generating function of X P n is given by The notation a α i means that the part a is considered (m = 2, 3, 4, 5: A051286, A181545, A181546, A181547).

f n (u) = [z n ]F (z, u) [z n ]F (z, 1
{a α i , b β i } Dn(m) =
{n > 1} Dn+2(m) = n k=0 n -k k m
n k=0 n -k k 2 2 k A108488 Expansion of 1/ √ 1 -2x -3x 3 -4x 3 + 4x 4 . N N ∪ {0} n k=0 n -1 k n + k k = n-1 k=0 n -k n n k 2 2 n-k-1 A047781
Convolution of central Delannoy numbers and Schroeder numbers. Chebyshev transform of the central

{1, 2} {1, 1, 2} 1 2 n n k=0 (-1) k 2k k n-2k j=0 n -2k j 2 3 k A101500
binomial numbers (the formulas in this OEIS entry are not correct). Some coefficients are 0, as gcd(parts) = 1.

N dN D dn = (d + 1)n + d -1 n A045721 For d = 2, related to lattice paths, trees, standard tableaux... A000079 {1 α , 2 β } 2N D2n = β n A000244
The notation "2 β " means that the part 2 A000302... comes with a weight (or multiplicity) β.

{1, 2} N Dn = n k=0 n -1 k n -k k A005773
directed animals (and numerous avatars of Motzkin paths, constrained matrices...).

Table 2: Summary of the main closed-form formulas (for m = 2) of our Section 3.

In a sufficiently small neighborhood of u = 1, as a function of z, F (z, u) given in (1) has a dominant singularity ρ(u) which is the unique positive solution of the equation up(ρ(u)) = 1.

Consequently,

f n (u) = [z n ]F (z, u) [z n ]F (z, 1) ∼ p (ρ(1)) p (ρ(u)) • ρ(u) ρ (1)
-n-1

.

It follows from the analysis of supercritical sequences given in [9, Proposition IX.7, p. 652] that the number of parts X P n satisfies

X P n -EX P n var(X P n ) d -→ N (0, 1),
where N (0, 1) denotes a standard normal random variable whose distribution function is given by

Φ(x) = 1 √ 2π x -∞ e -t 2 /2 dt,
and where the symbol " d -→" denotes the convergence in distribution. The asymptotic expressions for the expected value and the variance of X P n are given by

EX P n = n ρp (ρ) + O(1) with ρ ∈ (0, 1) such that p(ρ) = 1, (7) var(X
P n ) = Kn + O(1)
where

K = ρp (ρ) + p (ρ) -ρ(p (ρ)) 2 ρ 2 (p (ρ)) 3 . (8)
[Note that the expression for the coefficient K of the variance given in Proposition IX.7 in [START_REF] Flajolet | Analytic combinatorics[END_REF] is incorrect; the correct version is (ρh (ρ) + h (ρ) -ρh (ρ) 2 )/(ρh (ρ) 3 ), as given in many other places in the book.]

We now note that the central limit theorem can actually be strengthened to the local limit theorem, pretty much as discussed [9, Theorem IX.14 and the remarks following its proof on p. 697]. Let us recall the following notion. Definition 4.1 Let (X n ) be a sequence of integer valued random variables with EX n = µ n and var(X n ) = σ 2 n . Let (ε n ) be a sequence of positive numbers going to 0. We say that (X n ) satisfies a local limit theorem (of Gaussian type) with speed ε n if

sup x∈R σ n Pr(X n = µ n + xσ n ) - e -x 2 /2 √ 2π ≤ ε n .
As was discussed in [9, p. 697], to see that the local limit theorem holds for restricted compositions, it suffices to check that ρ(u) when restricted to the unit circle uniquely attains its minimum4 at u = 1. This is what we prove in the following lemma.

Lemma 4.2 Let p be a power series with nonnegative coefficients, of radius of convergence ρ p > 0 (possibly ρ p = +∞).

Let ρ(u) be as above the positive root5 of up(ρ(u)) = 1. If p is aperiodic6 , then for 0 < 1 R < ρ p and t ∈]0, 2π[, we have

ρ(R) < |ρ(Re it )| ,
i.e. the minimum on each circle is on the positive real axis. In particular, if the radius of convergence of p is larger than 1, then for |u| = 1 and u = 1 we have ρ

= R). As p is increasing on [0, 1/R], we can apply p -1 to p(|ρ(u)|) > 1/R which gives p -1 (p(|ρ(u)|) > p -1 (1/R), that is |ρ(u)| > ρ(R).
Note that the aperiodicity condition is important, e.g. for p(z) = z 2 + z 6 (i.e. ρ(u) is the radius of convergence of P (z, u) = 1/(1 -u(z 2 + z 6 ))), one has ρ(-1) = iρ(1); however some periodic cases have a unique minimum on the unit circle, e.g. p(z) = z 2 + z 4 . Note also that (in either periodic or aperiodic case), the uniqueness of the minimum on the circle |u| = 1 at u = 1 does not hold in general for the other roots of up(ρ(u)) = 1.

5 Asymptotic probability that restricted compositions have the same number of parts

Our motivation for including the results from [START_REF] Flajolet | Analytic combinatorics[END_REF] in the preceding section is the following theorem which considerably extends the main results of [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF]. We single out the case m = 2 since in some cases it has been already studied in the literature.

Pairs of compositions

Theorem 5.1 Let P ⊂ N. The probability that two random compositions with parts in P have the same number of parts is, asymptotically as n → ∞,

π n ∼ C √ π √ n ,
where the value of C is related to the constant K from Equation 8, namely:

C = 1 2 √ K = ρ(p (ρ)) 3/2 2 ρp (ρ) + p (ρ) -ρ(p (ρ)) 2 . ( 9
)
Before proving this theorem let us make some comments.

Remarks and examples:

(i) Some special cases were considered in [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF]. They include unrestricted compositions (P = N), P = {1, 2}, or more generally P = {a, b} with a, b relatively prime, compositions with all parts of size at least d (P = {n ∈ N : n ≥ d}), and compositions with all parts odd and at least d. The arguments of [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF] rely on the analysis of the asymptotics of the bivariate generating functions, which is sometimes difficult and does not seem to be easily amenable to the analysis in the case of a general subset P of positive integers. Our approach is much more probabilistic and relies on a local limit theorem for the number of parts in a random composition with parts in P. This turned out to be a much more universal tool.

(ii) To illustrate the principle behind our approach, consider the unrestricted compositions. As was observed in [START_REF] Pawe L Hitczenko | On the multiplicity of parts in a random composition of a large integer[END_REF], in that case X P n is distributed like 1 + Bin(n -1, 1/2) random variable. Therefore,

π n = Pr(Bin(n -1, 1/2) = Bin (n -1, 1/2))
where Bin and Bin denote two independent binomial random variables with specified parameters. Since the second parameter is 1/2 we have

Bin(n -1, 1/2) d = n -1 -Bin(n -1, 1/2).
Therefore, by independence we get

π n = Pr(Bin(n -1, 1/2) + Bin (n -1, 1/2) = n -1).
Finally, since Bin(n -1, 1/2) + Bin (n -1, 1/2)

d = Bin(2(n -1), 1/2),
we obtain by Stirling's formula that

π n = Pr(Bin(2n -2, 1/2) = n -1) = 2n-2 n-1 2 2n-2 ∼ 1 √ πn .
This is consistent with (9) (and with [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF]) as for unrestricted compositions p(z) = k≥1 z k = z/(1 -z), so that ρ = 1/2, p (z) = 1/(1 -z) 2 , and p (z) = 2/(1 -z) 3 which gives C = 1.

(iii) Although the above argument may look very special and heavily reliant on the properties of binomial random variables, our point here is that it is actually quite general. The key feature is that the number of parts (whether in unrestricted or arbitrarily restricted compositions) satisfies the local limit theorem of Gaussian type, and this is enough to asymptotically evaluate the probability in Theorem 5.1.

(iv) For another example, consider compositions of n into two parts, i.e. P = {a, b} with a, b relatively prime. Then Theorem 5.1 holds with

C = (aρ a + bρ b ) 3/2 2|a -b| ρ a+b , ( 10
)
where ρ is the unique root of z a + z b = 1 in the interval (0, 1). In this case p(z) = z a + z b so that p (z) = az a-1 + bz b-1 and p (z) = a(a -1)z a-2 + b(b -1)z b-2 . Thus, writing the numerator of (9) as

ρ(P (ρ)) 3/2 = 1 √ ρ (ρP (ρ)) 3/2 = 1 √ ρ (aρ a + bρ b ) 3/2 ,
we only need to check that ρP (ρ) + P (ρ) -ρ(P (ρ)

) 2 = (a -b) 2 ρ a+b-1 . (11)
But ρP (ρ) + P (ρ) = a 2 ρ a-1 + b 2 ρ b-1 so that the left-hand side of ( 11) is

a 2 ρ a-1 + b 2 ρ b-1 -a 2 ρ 2a-1 -b 2 ρ 2b-1 -2abρ a+b-1 .
Factoring and using ρ a + ρ b = 1, we see that this is

a 2 ρ a-1 (1 -ρ a ) + b 2 ρ b-1 (1 -ρ b ) -2abρ a+b-1 = (a -b) 2 ρ a+b-1 ,
as claimed.

When a = 1 and b = 2 we have the Fibonacci numbers relation so that ρ = ( √ 5 -1)/2 and (10) becomes

C = (ρ + 2ρ 2 ) 3/2 2ρ 3/2 = 1 2 (1 + 2ρ) 3/2 = 5 3/4 2 ,
which agrees with (4) above and also with the expression given in [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF] (see equation (2.10) therein). However, in the case of general a and b, the value of C was given in the last display of Section 3 in [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF] as

ρ(aρ a-1 + bρ b-1 ) 2 4(a + b)ρ 2a+2b-2 + 2(1 -ρ 2a -ρ 2b )(aρ 2a-2 + bρ 2b-2 ) . (12)
This is incorrect as it is lacking a factor |a-b| in the denominator (so that it gives the correct value of C when |a-b| = 1 but not otherwise). To see this and also to reconcile ( 12) with (10) (up to a factor |a -b|) we simplify [START_REF] Neil | The On-Line Encyclopedia of Integer Sequences[END_REF] by noting that ρ a + ρ b = 1 implies

1 -ρ 2a -ρ 2b = 1 -(ρ a ) 2 -ρ 2b = (1 + ρ a )ρ b -ρ 2b = ρ b (1 + ρ a -ρ b ) = 2ρ a+b
so that the expression under the square root sign in (12) becomes 4ρ a+b-2 ((a + b)ρ a+b + aρ 2a + bρ 2b ) = 4ρ a+b-2 (aρ a + bρ b )(ρ a + ρ b ).

Using again ρ a + ρ b = 1 ( 12) is seen to be

ρ 2 (aρ a-1 + bρ b-1 ) 2 2 ρ a+b (aρ a + bρ b ) = (aρ a + bρ b ) 3/2
2 ρ a+b , which, except for the factor |a -b| in the denominator, agrees with [START_REF] Garoufalidis | G-functions and multisum versus holonomic sequences[END_REF].

(v) Other examples from [START_REF] Bóna | On the probability that certain compositions have the same number of parts[END_REF] can be rederived in the same fashion, but we once again would like to stress universality of our approach. As an extreme example, we can only repeat after [START_REF] Flajolet | Analytic combinatorics[END_REF]: even if we consider compositions into twin primes, P = {3, 5, 7, 11, 13, 17, 19, 29, 31, . . . }, we know that the probability of two such compositions having the same number of parts is of order 1/ √ n. This is rather remarkable, considering the fact that it is not even known whether this set P is finite or not.

Proof of Theorem 5.1. This will follow immediately from the following lemma applied to X n = X P n and formula (8) which gives the expression for σ n . This lemma should be compared with a more general Lemma 6 of [START_REF] Bóna | Isomorphism and symmetries in random phylogenetic trees[END_REF]. We include our proof to illustrate that seemingly very special arguments used in item (ii) are actually quite general. Lemma 5.2 Let (X n ) with EX n = µ n and var(X n ) = σ 2 n → ∞ as n → ∞, be a sequence of integer valued random variables satisfying a local limit theorem (of Gaussian type) with speed ε n as described in Definition 4.1. Let (X n ) be an independent copy of (X n ) defined on the same probability space. Then

π n = Pr(X n = X n ) = 1 2 √ πσ n + O ε n σ n + 1 σ 2 n .
Proof. For X n and X n as in the statement we have

π n = Pr(X n = X n ) = k≥1 Pr(X n = k = X n ) = k≥1 Pr 2 (X n = k) = ∞ =-∞ Pr(X n = µ n + ) Pr(X n = µ n + ). (13) Now, Pr(X n = µ n + ) = Pr(X n = µ n -) + Pr(X n = µ n + ) -Pr(X n = µ n -) .
To estimate the term in the curly brackets take x + and x -such that

µ n + = µ n + x + σ n , and µ n -= µ n -x -σ n . By elementary considerations, -2{µ n }/σ n ≤ x + -x -≤ 2(1 -{µ n })/σ n (where {z} is the fractional part of z), hence |x + - x -| ≤ 2/σ n . Then Pr(X n = µ n + ) -Pr(X n = µ n -) = 1 √ 2πσ n e - x 2 + 2 -e - x 2 - 2 + Pr(X n = µ n + x + σ n ) - e -x 2 + /2 √ 2πσ n -Pr(X n = µ n -x -σ n ) - e -x 2 -/2 √ 2πσ n .
The absolute value of the second term is Coming back to equation ( 13), we see that

1 σ n σ n Pr(X n = µ n + x + σ n ) - 1 √ 2π e - x 2 + 2 ≤ ε n σ n , and
∞ =-∞ Pr(X n = µ n + ) Pr(X n = µ n -) + O ε n σ n + 1 σ 2 n = ∞ =-∞ Pr(X n = µ n + , X n = µ n -) + 1 × O ε n σ n + 1 σ 2 n = Pr(X n + X n = 2 µ n ) + O ε n σ n + 1 σ 2 n .
Since X n + X n is a sum of two i.i.d. random variables, it has mean 2µ n and the variance 2σ 2 n . Furthermore, since each of the summands satisfies the local limit theorem of Gaussian type, so does the sum (its probability generating function is the square of f n (u) and thus falls into quasi-power category, just as f n (u) does). Since 2 µ n = 2µ n + x √ 2σ n for some x = O(1/σ n ), just as before we have

√ 2σ n Pr(X n + X n = 2µ n ) - 1 √ 2π = O ε n + 1 σ n .
Consequently,

π n = Pr(X n = X n ) = Pr(X n + X n = 2µ n ) = 1 2 √ πσ n + O ε n σ n + 1 σ 2 n ,
which completes the proof of Lemma 5.2 and of Theorem 5.1.

3. When the means are not the same, the probability of the same number of parts is generally of much smaller order. This is because if X n has mean cn and X n has mean c n and both have linear variances, then assuming w.l.o.g. c > c and choosing α < c-c 2 we note that if |X n -cn| < αn and |X n -c n| < αn then X n -X n > cn -αn -(c n + αn) = (c -c -2α)n > 0, so that X n = X n . Therefore, π n = Pr(X n = X n ) ≤ Pr(|X n -cn| ≥ αn) + Pr(|X n -c n| ≥ αn).

Since both X n and X n converge to a Gaussian law and σ n = σn, the first probability is roughly (with β = α/ √ σ)

Pr |X n -cn| √ σn ≥ β √ n ∼ 1 √ 2π ∞ β √ n e -t 2 /2 dt ∼ 1 √ 2πβ √ n e -β 2 n 2 ,
by the well-known bound on the tails of Gaussian random variables (see, e.g. [8, Chapter VII, Lemma 2]). This is consistent with an example discussed in Section 3.1. The difficulty with making this argument rigorous is that the error in the first approximation is usually of much bigger (typically 1/ √ n) magnitude than the quantities that are approximated. However, a slightly weaker bound, namely, e -βn (with a generally different value of β) can be obtained by using Theorem IX.15 in [START_REF] Flajolet | Analytic combinatorics[END_REF] which asserts that tail probabilities of random variables falling in the scheme of quasi-powers are decaying exponentially fast. While this theorem is stated for the logarithm of Pr(|X n -cn| > αn), it is clear from its proof that one actually gets exponential bound on the tail probabilities (see Equation (88) on p. 701 in [START_REF] Flajolet | Analytic combinatorics[END_REF] and a few sentences following it). 4. The Gaussian local limit law explains the universality of the 1/(πn) (m-1)/2 appearance for numerous combinatorial problems in which we would force m combinatorial structures of size n to have an extra parameter of the same value. We also wish to point out yet another insight provided by the probabilistic approach. As we mentioned in Section 3.1 (see Footnote 2), it allows to solve the connection constant problem intrinsic to the Frobenius method, and therefore, a combination of these two approaches (local limit law plus Frobenius method) gives access to full asymptotics in numerous cases.

Conjecture 3 . 1

 31 For any rational function p(z) ∈ N(z) (with p(1) > 1), the generating function D(z) is not algebraic for m > 2. It includes the specific case D(z) = n≥0 D n z n with D n = n k=0 n-k k m (non algebraicity of our initial puzzle) or D n = n k=0 n k

Figure 1 :

 1 Figure 1: Pairs of compositions having the same number of parts (e.g. (1,2,2,1,2,1,2,2,2,2,2) and (3,1,1,1,3,2,1,1,4,1,1)) are in bijection with several combinatorial objects, e.g. lattice paths (left) and directed animals (right).

m = 2 , 2 /m

 22 . . αm) (n-kb)/a (β1 . . . βm) k with weight (or multiplicity) αi in the compositions of Pi.(Binomials with fractional entries are considered as zero). A051286: Whitney number of level n of the lattice of the ideals of the fence of order 2n. m = 2, A089165: partial sums of the central Delannoy numbers, 16 k resistance between two nodes of an infinite lattice of unit resistors, # of peaks of odd level in Schroeder path. Same as pairs of compositions of n -1 with parts in {1,2}

  < |ρ(u)|. Proof. First, p has nonnegative real coefficients, therefore the triangle inequality gives p(|ρ(u)|) ≥ |p(ρ(u))|. Equality can hold only if p(ρ(u)) has just nonnegative terms, but this is not possible if ρ(u) ∈ R + as p is aperiodic with nonnegative coefficients. Hence one has a strict triangle inequality: p(|ρ(u)|) > |p(ρ(u))| = |1/u| = 1/R (the middle equality is just the definition of ρ and the last equality comes from the fact we are on the circle |u|

2 ≤

 2 similarly with the third term. Applying the inequality |f (x) -f (y)| ≤ |x -y| sup |f (t)| to the first term gives e - |x + -x -| = O(1/σ n ) and so, the first term is O(1/σ 2 n ). Therefore, Pr(X n = µ n + ) -Pr(X n = µ n -

  ) .

	allowed	number of tuples (P1, . . . , Pm)	Sloane's On-line encyclopedia of integer sequences
	parts in Pi	of compositions of n having	alternative description
		the same number of parts

Table 1 :

 1 Summary of the main closed-form formulas (for any m) of our Section 3.

	allowed	allowed	number of pairs (P1, P2) of	OEIS	alternative
	parts	parts	compositions of n having	reference	OEIS description [12]
	in P1	in P2	the same number of parts
	{1, 2}	{1, 2, 2}

See http://www-lipn.univ-paris13.fr/∼banderier/Pawel/Maple/.

Note that, as typical with the Frobenius method (or also with the Birkhoff-Tritjinski method, see[START_REF] Wimp | Resurrecting the asymptotics of linear recurrences[END_REF]), it is not always possible to decide the connection constant(s); in the next sections, we give a rigorous probabilistic approach which allows to get this constant, and therefore full asymptotics by coupling it with the Frobenius method!

We cannot escape this polysemy: Compositions are enumerated by a composition!

There is a typo in[START_REF] Flajolet | Analytic combinatorics[END_REF] p.697: the inequality direction is wrong.

p(z) has nonnegative real coefficients and is thus increasing in a neighborhood of 0, i.e. on z ∈ [0, + ]. p being analytic near 0, is continuous and for any x ∈ R small enough, p(z) = x will therefore have a real positive root zx, and this root will be analytic in x. This is the root that we call "real positive".

A power series p is said to be periodic if and only if there exists a power series q and an integer g > 1 such that p(z) = q(z g ). Equivalently, the gcd of the support (=the ranks of nonzero coefficients) of the power series p is g = 1. If this gcd g equals 1, then p is said to be aperiodic.

Acknowledgements. Part of this work was done during Pawe l Hitczenko's sojourn at LIPN (Laboratoire d'Informatique de Paris Nord), thanks to the invited professor position funded by the university of Paris 13 and the Institute Galilée. He would like to thank the members of LIPN for their hospitality.

‡ supported in part by the NSA grant #H98230-09-1-0062 § corresponding author.

Tuples of compositions

Here we sketch a proof of the following extension of Theorem 5.1. Theorem 5.3 Let P ⊂ N and let m ≥ 2 be fixed. Then, the probability π n that m randomly and independently chosen compositions with parts in P all have the same number of parts is, asymptotically as n → ∞,

where C m is related to the constant K from Equation 8, namely:

Remark. For unrestricted compositions, the expression in the big parentheses is 2 (see (i) in Remarks above). This gives C m = 2 m-1 /m as stated in Section 3.1. Proof of Theorem 5.3. This follows immediately from the following statement which itself is a straightforward extension of Lemma 6 of [START_REF] Bóna | Isomorphism and symmetries in random phylogenetic trees[END_REF] with essentially the same proof. We will be using it for Gaussian density in which case

Let g be the probability density function and suppose that

. . , m be independent copies of the sequence (X n ) defined on the same probability space. Then

To see this we just follow the argument in [5, Lemma 6] with obvious adjustments: the left-hand side above is

with k-µn σn ≤ x k < k+1-µn σn . This is further equal to

where the first approximation holds by the assumption of the lemma (after having first restricted the range of x k 's to a large compact set) and the second by the Riemann sum approximation of the integral. Since the expression on the right is ∞ -∞ g m (x)dx, the result follows.

6 Concluding remarks 1. In this article, we restricted our attention to compositions (giving first several new closed-form formulas, and then going to the asymptotics), but it is clear that Lemma 5.4 can be applied to many combinatorial structures, e.g. the probability that m random permutations of size n have the same number of cycles (see [START_REF] Wilf | The variance of the Stirling cycle numbers[END_REF] for the case m = 2), or the probability that m permutations have a longest increasing subsequence of the same length, or the probability that m random planar maps have a largest component of same size. This leads to interesting analytic/computational considerations, as it will involve evaluating the integral of g m (x) where g(x) will be the Tracy-Widom distribution density (provided the local limit theorem holds, which has not been proven yet), or the map-Airy distribution density (for which a local limit theorem was established, see [START_REF] Banderier | Random maps, coalescing saddles, singularity analysis, and Airy phenomena[END_REF]).

2. A similar approach can be also applied to tuples of combinatorial structures following m different local limit laws (with m densities having fast decreasing tails), as long as they have the same mean.


L Aldon

J.-C Jumas

Lithium-induced conversion reaction in wüstite Fe 1-x O studied by 57 Fe Mössbauer spectroscopy

Keywords: Wüstite, Mössbauer spectroscopy, Conversion reaction, Lithium, Electrochemical mechanism, Disproportionation, Irreversible capacity

Fe 1-x O wüstite as negative electrode material for Li-ion batteries has been studied. The aim of this work is to get a better understanding of the insertion mechanism involved during reduction/oxidation processes. Electrochemical tests have been done in Swagelock TM cells and shown a high specific capacity of 800 Ah/kg for the first discharge. X-ray diffraction and 57 Fe Mössbauer spectroscopy provide us valuable information on both local and long range order. Hence, Li reaction with wüstite induces formation of highly divided metallic iron (-Fe and nano-Fe) and Li 2 O with a small amount of Fe 2 O 3 occurring in a diffusion layer at the surface of the primary particles. Based on the X-ray and Mössbauer spectroscopic analyses, a core-shell model is proposed in order to explain the irreversible capacity of about 1 Li observed at the first cycle. It involves cation diffusion induced by lithium acting as an "electronic pressure".

INTRODUCTION

Anodic materials having a discharge potential of 1V/Li are chosen among the metallic oxides [1,2]. Iron oxides are inexpensive and present low environmental impact for lithium battery applications [3,4]. Previous works have shown the interest of such iron oxides as electrochemically active material [5,6]. The mechanism is generally based upon the redox couple Fe 3+ /Fe 2+ working at approximately 0.5 V compared to lithium [7,8]. Wüstite is a nonstoichiometric compound with formulae Fe 1-x O, with 0.83 < x < 0.96 [9]. It has the rocksalt structure with a cell parameter of 4.307 Å [10]. In the structure some Schottky defects occur because of the presence of two oxidation states Fe II and Fe III , occupying the octahedral sites of the structure. Fe II is rather easy to oxidize to Fe III and can explain the wide range of composition. In order to get a better understanding of electrochemical Li reaction mechanism into Fe 1-x O, the present work deals with 57 Fe Mössbauer spectroscopy, X-ray diffraction and electrochemical tests. Coupling these experimental techniques allows us to get deeper insight into the mechanism ruling lithium reaction processes in the Fe 1-x O wüstite phase.

From X-ray diffraction, previous work has shown a correlation between amount of vacancies and cell parameters [11]. From 57 Fe Mössbauer spectroscopy the relative contribution of Fe II and Fe III to the spectrum can be determined [12,13]. Hence comparison between the amount of vacancies from X-ray diffraction and Mössbauer spectroscopy can be done. This nuclear technique gives local information at the atomic scale and is very sensitive of the neighborhood in terms of charge of ions and vacancy effects [14,15].

The electrochemical mechanism shows that a part of Li (~44 %) is involved in the reduction of FeO into metallic Fe 0 accompanied with small amount of nano-sized Fe 2 O 3 and a last part of Li (~56 %) is consumed to form a SEI in the first discharge. This last part seems to be responsible for the irreversible capacity in this conversion reaction. The reversible process is based on a reversible reaction: Fe II O + 2Li ↔ Fe 0 + Li 2 O.

EXPERIMENTAL

Synthesis conditions

Wüstite Fe 1-x O has been prepared from powdered hematite -Fe 2 O 3 by specific heat treatment [9]. Hematite has been ground in an agate mortar and then the fine powder has been put into a porcelain crucible. The crucible is then placed in the quartz tube of the oven at 800 °C under H 2 atmosphere used as reducing agent [16]. Since Fe 1-x O is metastable [17,18] and decomposes below 570 °C to magnetite Fe 3 O 4 and metallic Fe at ambient pressure and temperature, the crucible has been quenched outside by cooling with pulsed air.

X-ray diffraction

X-ray diffraction (XRD) patterns were collected on a conventional Philips -2 diffractometer with Cu-K  radiation (1.5418 Å) and a nickel filter in order to characterize the compounds before and after insertion of lithium. For electrochemically-inserted phase, the recording was made under vacuum in order to avoid undesirable reactions with air.

57

Fe Mössbauer spectroscopy 57 Fe Mössbauer spectra were recorded at room temperature (RT) using transmission geometry in the constant acceleration mode with a spectrometer based on electronic devices delivered by EG&G and WissEl. The absorbers contained 1-2 mg of 57 Fe per cm 2 were prepared inside the glove box, and sealed with parafilm to avoid contact with air. The velocity scale was calibrated using the 2 inner lines (see below) of the magnetic sextet of a high purity iron foil absorber as a standard, using 57 Co (Rh) as the source. The spectra were fitted using a Lorentzian approximation by least-squares method implemented in our GM5SIT program [19,20]. The quality of the fit was controlled by the conventional  2 test. Isomer shift (IS) values are given relative to the centre of the -Fe (10 µm foil) spectrum recorded at room temperature. The shape of a Mössbauer spectrum of a paramagnetic compound is generally determined by a doublet, characterized by the quadrupole splitting (QS) separating more or less the 2 Lorentzian lines and their mean position, the isomer shift (IS). The intensity of the absorption (A) is ruled by the number of sites (N) and the Lamb-Mössbauer factor (f) through A = N.f. In the case of magnetic contribution, the spectrum is split into 6 Lorentzian lines with a relative intensity as 3:2:1:1:2:3. Mössbauer spectra of bulk -Fe or Fe 2 O 3 present such a magnetic sextet. The sextet of -Fe used for velocity calibration is composed of two lines located at IS 1,6 = ±5.33 mm/s with relative intensities of 3, two other lines at IS 2,5 = ±3.02 mm/s with relative intensities of 2. The inner lines of the sextet have relative intensities of 1.

Since we present in the following Mössbauer spectra recorded in a small velocity range [±2.4 mm/s] to favor details in the Fe III /Fe II velocity scale, we can only detect the two inner lines IS 3,4 = ±0.84 mm/s. For convenience, we choose to fit these two lines with an equivalent doublet with IS ~0 mm/s and QS ~1.68 mm/s. After convergence of the fitting procedure, we have to correct the total absorption of our sample. We have multiplied by 6 the relative contribution of this doublet to get the real contribution of metallic -Fe in the mixture. The same procedure will be used with a magnetic contribution of Fe 2 O 3 . Then, all contributions are corrected to obtain the relative amount of the species.

Concerning the non-stoichiometric wüstite phase, Fe 1-x O will be written taking into account the amount of vacancies, x, as Fe II 1-3x Fe III 2x  x O. This quantity can be roughly estimated from the relative contributions A II and A III from the Mössbauer spectroscopic data by:

A A x III II / 2 3 1  
It is thoroughly admitted in the Mössbauer community that vacancies or "guest" ions can be detected in the vicinity of the probed ion or atom ( 57Fe) from the quadrupole splitting distribution as shown by Womes et al. [21]. The presence of both Fe III and vacancies in the vicinity of probed Fe II may change the electric field gradient (EFG) and therefore the measured quadrupole splitting (QS). The quadrupole splitting reflects the ferrous quadrupole interaction in the paramagnetic state and is related to the elements of the diagonalized electric field gradient tensor that can be estimated for ionic compounds from point charge model calculations [22].

Assuming a random distribution [23,24] of both vacancies and cations in the wüstite structure, the probability of a probed Fe II to have n Fe III ions, m vacancies at a concentration x and 12-m-n Fe II located in the 12 neighboring sites of the first coordination shell of Fe 1-x O is given by:

) 3 1 ( 2 12 , )! 12 ( ! ! ! 12 ) ( x x p m n m n m n m n m n x       
Since the solid solution domain is not very extended for Fe 1-x O (x < 0.1), we think that the main contributions are due to electric charge as often observed for ionic compounds [25]. The most probable configurations will be grouped according to the values of n and m. For instance a Fe II surrounded by n = 1 Fe III and m = 0, 1 or 2 vacancies will give an intensity proportional to P 1 = P 1,0 + P 1,1 + P 1,2 if we neglect the cases where m > 2 since x < 0.1. Hence 3 main contributions P n = P n,0 + P n,1 + P n,2 (n = 0, 1 or 2) will be the most probable and expected configurations. In this case we believe that the measured quadrupole splitting will be ndependant.

The intensity of the absorption of a given phase depends on its recoil-free factor f (named the Lamb-Mössbauer factor [26,27]). The factor f depends on the Debye temperature,  D , which decreases with decreasing particle size [28]. It is worth noticing that dispersed Fe nano-particles not bound to a rigid matrix show a large decrease in the apparent Debye temperature. Indeed the Debye temperature has been estimated to 388 ± 20 K for bulk -Fe.

This value decreases to 344 ± 16 K and 259 ± 18 K for particles of 2.5 nm and 1.5 nm in size, respectively [29]. For wüstite it has been estimated to  D = 417 K [30]. Knowing the Debye temperature of a given material, the Lamb-Mössbauer factor, which depends on the temperature and on the recoil energy (E R = 1.956 meV for 57 Fe), can be estimated through:

2 . 6 ) ( ln D R LM k T E T f   
In the discussion species fraction, n i , will be estimated assuming a given f i factor from the relative intensity A i of the Mössbauer absorption using n i = A i /f i .

Electrochemical tests

The cells consisted of lithium disks (anode) and pellets of wüstite Fe 1-x O samples (diameter = 7 mm, thickness ~ 0.3 mm) (cathode), and we used 2 Whatman separators wetted by LiPF 6 (1M) electrolyte in PC-EC-3DMC. Charge/discharge curves were carried out galvanostatically by means of a MacPile TM system operating at a current density of 10 A/kg (C/15 rate) between 3 and 0 V vs. Li. Some cells have been stopped at a given depth of discharge or charge, disassembled in a glove box to avoid air and/or moisture contamination.

RESULTS

Pristine material

Fe 1-x O has a rocksalt structure (B1, fcc, Fm-3m, a = 4.307 Å [START_REF] Bragg | Crystal structure of minerals[END_REF]). The X-ray diffraction pattern of the pristine material (Fig. 1) presents diffraction of  (111) = 18.03,  (200) = 20.92, (220) = 30.38,  (311) = 36.32 and  (222) = 38.14. The refined cell parameter is a = 4.313 (7) Å in agreement with the literature. From Aubry and Marion"s work [11], the estimated amount of vacancies from the cell parameter is x = 0.0500.013. Our sample presents some -Fe impurity (A2, bcc, Im-3m, a = 2.8665 Å [START_REF] Wyckoff | Crystal Structures[END_REF]) with diffraction peaks of which is antiferromagnetic (T N = 198 K [START_REF]Experimental Magnetochemistry[END_REF]) and shows a typical paramagnetic absorption at RT. A fourth doublet centered at IS ~ 0.55 mm/s, QS ~ 0.90 mm/s with a relative area of 11 % is usual of Fe III species as expected in the non-stoichiometric Fe 1-x O phase. The estimated amount of vacancies from the Fe II /Fe III ratio as explained in the experimental part is about 0.057±0.008. This value is rather close to 0.050 found from XRD. Finally, the doublet located at IS ~ 0 mm/s and QS ~ 1.68 mm/s, corresponding to -Fe as explained in the experimental part present a contribution of ~6 %. All these results are summarized in Table 1.

The calculated probabilities P 0 , P 1 and P 2 are 50.6, 37.0, 12.4 % for x = 0.054 while the experimental absorptions for Fe II species are 50.7, 31.3 and 18.0 % respectively. The weak discrepancy between calculated P 1 and P 2 and estimated absorptions may suggest that a situation of a Fe II surrounded by 2 Fe III would be most favorable situation than only 1 Fe III in the Fe II neighborhood. This is a small deviation to a random distribution and could indicate of tendency for defect ordering.

Combining XRD and 57 Fe Mössbauer spectroscopic results, we can concluded that our pristine sample is mainly (~94 %) composed of Fe II 0.838 Fe III 0.108 □ 0.054 O and -Fe (~6 %).

Lithiated samples

The electrochemical behavior of Fe 1-x O is shown in Figure 3. Starting from the pristine material (point (a) in Fig. 3), the first discharge with a capacity of ~800 Ah/kg is characterized by a pseudo-plateau starting at 0.64 V continuously decreasing to 0.5 V until an uptake of 1 Li (point (b) in Fig. 3), then becomes more sloppy to reach 0.01 V for 2.16 Li (point (c) in Fig. 3). The shape of this not-well defined plateau is rather characteristic for a reconstructive reaction within the pristine material inducing the formation of new phases. The curve of the first charge presents slight different shape with a mean potential of about 1.64 V.

At the end of charge (point (d) in Fig. 3), the potential was stopped at 3 V before starting a new cycle for which the Li uptake is 0.94. The difference between discharge and charge (2.16-0.94 = 1.22 Li) results in an irreversible capacity of about 350 Ah/kg. Then next discharge curve shows a very different shape, more sloppy, as compared to the first one. It suggests the formation of new phases or in a more divided state reacting at a mean potential of about 0.98 V. A high polarization of about 660 mV is observed and seems to confirm the nano-structuration of the primary particles during the first discharge. This situation is typical for conversion reactions as described by J.-M. Tarascon et al. [START_REF] Tarascon | [END_REF] for oxides, phosphides [35,36], stannides [37,38], and antimonides [39,40]. The following cycles present a rather stable and reversible capacity of about 450 Ah/kg (x ~ 1.20 Li).

Following the establishment of the electrochemical mechanism will only cover the first cycle studied by ex situ XRD and 57 Fe Mössbauer spectroscopy at points (a)-(d).

X-ray diffraction patterns for the ex situ lithiated samples are shown in Figure 4.

Depending on the discharge depth we clearly see a progressive consumption of pristine Fe 1-x O (Fig. 4a). Nevertheless a rather broadened pattern is observed at the end of the discharge (Fig. 4c), with residual peaks belonging to un-reacted Fe 1-x O. At this stage, diffraction peaks of -Fe are still visible, but with a large decrease in intensity compared to the pristine material.

Some rather broadened diffuse lines located at around 20-25, 30-33 and 37-42 are also visible and increase in intensity from (b) to (c). This result suggests the formation the new amorphous or nano-sized phases with smaller coherence lengths.

At the end of the first charge (Fig. 4d) the diffraction pattern presents no more crystallized phase. Since Fe 1-x O may be reduced to metallic Fe, we have plotted for comparison in Fig. 4d indexations of some known allotropic forms, i.e. -Fe (bcc), -Fe (fcc)

and -Fe (hcp).

Ex situ Mössbauer spectra of some discharge/charge depths are shown in Figure 5. We have chosen to show the spectra in a wide velocity scale including the total contribution of -Fe for comparison between the pristine (Fig. 5a) and the fully discharged sample (Fig. 5c).

Spectra are rather complex with various components but the overall trend in the shape of the spectra assumes formation of Fe III species upon discharge. The results of the fit have been reported in Table 1.

For an uptake of 1 Li (Fig. 5b), we observed in the spectrum a decrease for the Fe II contribution and an increase for Fe III (IS ~ 0.5 mm/s, QS ~ 0.66 mm/s). Hyperfine parameters (IS and QS) differ slightly from Fe III species in the pristine material. This Fe III contribution is not only due to Fe III in Fe 1-x O, but it could be nano-sized Fe III 2 O 3 which can be superparamagnetic if the particle size is small as we assume. It is worth to notice that in Fe 1-

x O some local defects are ordered giving Koch-Cohen clusters (V 13 T 4 type or V 16 T 5 ) with a composition close to magnetite Fe 3 O 4 [41,42].

The contribution of -Fe also increases in intensity with IS ~ 0.02 mm/s and QS ~ 1.675 mm/s. We have freed these two parameters in the fitting procedure to obtain convergence. Although this is within the error bars, it could be significant if we consider that new -Fe is formed with smaller particles, resulting in a slightly modified hyperfine magnetic field with a shifted spectrum. We also observe the formation of a rather broadened (= 0.6 mm/s) singlet located at negative velocity (IS = -0.054 mm/s). It cannot be associated to -Fe in micro-particles because in this case it must conserve its hyperfine magnetic field even at room temperature. It could be due to the presence of superparamagnetic Fe nano-particles with a size smaller than 2 nm [43]. This isomer shift is generally observed at room temperature in the case of paramagnetic spectra of austenitic steels. It could be associated to the formation of -Fe (fcc, Fm-3m, a = 3.587 Å). It is well established that -Fe is paramagnetic in a nonferromagnetic state at RT [44]. We know also a diffusive transition transforming -Fe into -Fe occurs at 912 °C. We don"t really believe that the signal observed can be reasonably attributed to -Fe. It is not excluded that -Fe (A3, hcp, P6 3 /mmc) is formed [45,46]. Indeed these two allotropic forms ( and ) differ only by the stacking sequences (ABCABC) and (ABABAB), respectively. On the other hand, at high pressure and room temperature -Fe becomes -Fe in a martensitic phase transition [47,48]. Anyway the broadened singlet could be attributed to either -Fe (bcc) or -Fe (hcp) in nano-particles. In the doubt, we prefer to call this metallic fingerprint nano-Fe 0 as mentioned in Table 1.

These results suggest that the active material Fe At the end of the discharge for an uptake of 2.16 Li (Fig. 5c), ~23 % of the absorption belongs to Fe II species. Hence, Fe All along the discharge by comparing XRD and Mössbauer spectroscopic data analysis we can propose the following: i) the active material Fe 1-x O reacts partially with Li forming a phase containing Fe III (Fe 2 O 3 ), ii) starting from Fe II , the production of Fe III must be necessarily accompanied by the production of Fe 0 (disproportionation), iii) a not well understood broadened singlet with negative velocity could be a nano-sizedFe, and iv) vacancies are progressively filled or used by Li during the transformation of FeO forcing Fe atoms to quit the structure.

Finally at the end of the charge (Fig. 5d), we observe a contribution of ~66 % for Fe II , ~4 % for Fe III and ~30 % Fe 0 . Metallic iron particles became less than 2 nm in size since no magnetic sextet of -Fe is observed, we can assume that it plays a role during the Li extraction mechanism. Form the relative area of Fe II species with x = 0.062, the proposed formulae for wüstite at this depth of discharge is Fe II 0.814 Fe III 0.124 □ 0.062 O.

A correlation between the quadrupole splitting of the Fe II sites and the number of Fe III n in the Fe II neighborhood has been found: QS(n) = 0.502 + 0.491 n for the pristine material and QS(n) = 0.569 + 0.467 n for a depth of discharge of 1 Li. These two tendencies are very close and the slight deviation between intercepts and slopes may be correlated to the amount of vacancies in both compositions. The correlation we found at the end of the first charge is QS(n) = 0.3 + 0.576 n. This tendency is very different to those previously estimated at the discharge. This result suggests that the conversion reaction deeply modifies the Fe II environment. It is worth to note that the isomer shift is also affected, decreasing from about 1 mm/s at the discharge to ~0.7±1 mm/s at the end of the charge.

DISCUSSION

A concept previously introduced by Connerade et al. [50] and described as an "electronic pressure" at atomic level (Hartree-Fock calculations) confirms that electrochemical lithium reaction acts as an increase of pressure [51,52]. It is worth noticing that lithium-based electrochemical reactions present some analogies with high-pressure studies: oxidation state change, phase transition, amorphization, volume collapse. In the case of oxides, the pressure may induce valence state transitions, structural transitions and in some cases amorphization of the compound [53,54]. For instance, Fe 2 O 3 plays a determining role in the understanding the physical properties of the Earth"s mantle [55]. At high-pressure Fe 2 O 3 undergoes an oxidation state transformation from Fe 3+ to a new valence state [56]. The same behavior has been observed for the disproportionation of SnO into -Sn and SnO 2 [57]. Recently, the disproportionation of wüstite (Fe 1-x O) has attracted increasing attention, as a mechanism for the formation of metallic iron in the origin of the chemical differentiation of planetary materials [58,59] At room temperature, metallic iron and tin present a well established martensitic phase transition from -Fe (bcc) to -Fe (hcp) and from -Sn (bct) to Sn-II (bcc) at about 10.0 and 9.2 GPa respectively [61,62]. Besides the structural phase transition, an important change in the magnetic properties of the iron atoms is found: namely iron which is ferromagnetic in the  phase becomes nonmagnetic in the  phase [63]. The interest in high-pressure (HP) behavior of elemental iron primarily stems from the fact that the hexagonal closed packed (hcp) phase of iron (-Fe) is considered as major constituent of the Earth"s inner core [64].

Results obtained under HP by 57 Fe Mössbauer spectroscopy clearly show a reduced atomic density in the boundary regions [65].

The effect of hydrostatic pressure isomer shift (IS) has been the subject of extensive research [66,[START_REF]Mössbauer Isomer Shifts[END_REF]. These changes of IS in Mössbauer effect measurements are proportional to those in "s"-electron density  v (0) at the nuclei in an absorber or a source. Variation of  (0) in compressed metals, alloys, and compounds is seen to involve several mechanisms, whose relative weight dictates the corresponding volume dependence. The main factors affecting the density of the ns electrons are: (i) reduction of the lattice parameters [START_REF] Pound | [END_REF], (ii) screening of the 5p ( 119 Sn, 121 Sb, 125 Te) or 4d ( 57 Fe) electrons [69].

In our case, the electrochemical lithium reaction induces an amorphization of the Fe 1-

x O active material or a formation of metallic nano-particles as shown by XRD. In our 57 Fe Mössbauer data we have a signature of some nanosized-Fe with an isomer shift IS = -0.054 mm/s which corresponds to metallic iron at a pressure of 6.7 GPa [65]. The line width is also pressure-sensitive, we found = 0.6mm/s which is consistent to -Fe at 14 GPa [45,70]. It seems that in our material, the metallic iron particles are formed at a scale around 2 to 10 nm since we have coexistence of some magnetic -Fe and superparamagnetic nano-particles.

These particles are more or less compressed by the Li 2 O matrix. 57 Fe Mössbauer spectra also give in our case more information in terms of vacancies. These vacancies play a role for cation diffusion in the FeO structure.

We propose the following electrochemical mechanism occurring upon Li reaction for the first discharge: From the data summarized in Table 1, we found = 0.256 and = 0.0215 between points (a) and (b) in Fig. 5 and = 0.549 and = 0.0315 between points (a) and (c) in Fig. 5.


These values mean that about 26 % of FeO is consumed during the first part. During the last part of the discharge curve, 29 % of FeO is consumed. Some active FeO material (~40 %) did not react. There is a concomitant reduction (reaction 1) and disproportionation (reaction 2).

Concerning the structural modification involved in such a mechanism, the Li applies some "electronic pressure" forcing diffusion of Fe ions. This increase of internal pressure induces Fe extrusion out from the FeO structure in which formation of vacancies forming small clusters of Fe 2 □O 3 . At the end of the discharge, it is more and more difficult to force Fe to quit the wüstite structure and these clusters are growing giving magnetic Fe 2 O 3 nanoparticles.

To summarize the electrochemical mechanism during the discharge we have: Compared to the electrochemical curve (Fig. 3) we clearly see that ~1.25 Li are missing although 2.16 Li were consumed for the global reaction. It suggests that ~58 % of Li is used for SEI formation at the surface of the electrode material and is not directly involved in the redox process. This SEI is at the origin of the irreversible capacity in the conversion reaction [71]. Apart from the charge consumed for the reduction of the solvent, this SEI is also believed to enable additional lithium storage on its surface in metallic form [72]. Any way, it has been shown that using nano-particle electrodes, electrolyte decomposition may occur simultaneously to the conversion reaction [73].

At the charge, starting from 2.16 Li, we extract about of the 3d transition-metal oxides such as CoO [74] and NiO [75], which can be reversibly reduced and oxidized, coupled with the formation and destruction of lithium oxide, respectively.

In Figure 6 we have plotted the relative amount n i deduced from Mössbauer data analysis of the species upon Li reaction during the first discharge up to 2.16 Li and the first charge stopped at 0.94 Li. After the middle of the first discharge, particles of -Fe 0 progressively decrease in size and transforms into nanosized Fe 0 . During the charge only a part of this nano-Fe 0 can reversibly react to form highly divided FeO. This is in agreement with the first derivative electrochemical curve shown in Figure 7. We have decomposed the observed shape into the sum of 2 or 3 Gaussian functions. The 1 st discharge is composed of 3 peaks with a position, a width and a surface of (0. to the pristine material. However, the formation of such nanocomposites should now be investigated using in situ and operando 57 Fe Mössbauer spectroscopy in order to follow global absorption of the sample to have a better estimation of the Lamb-Mössbauer factor of metallic nanoparticles [37]. In that case 2D correlation analyses of numerous recorded spectra would provide more precise information on the electrochemical mechanism as done on iron-based cathode materials [12]. TABLES Table 1. Isomer shift (IS), quadrupole splitting (QS), relative areas (A%), assignment and corrected areas from metallic iron magnetic sextet. The star (*) corresponds to inner lines 3 and 4 of the sextet of magnetic spectrum for -Fe. Since spectra have been recorded in a small velocity range (±2.4 mm/s), the estimated area for the lines 3 and 4 has to be multiplied by 6 respecting the expected intensity ratios 3:2:1:1:2:3. Estimated relative population of the species n i is given assuming Lamb-Mössbauer factors: f FeO ~f-Fe = 0.79, f FeIII = 0.81 and f nanoFe = 0.51. Proposed assignments are given. Relative contributions of most probable Fe II environments (see text) are given compared to the calculated probabilities P n = P n,0 + P n,1 + P n,2 for Fe II environments determined from 57 Fe Mössbauer spectra for the pristine material and after various discharge/charge depths. In P n,m , n stands for the number of Fe III first neighbors and m the number of vacancies. The vacancies amount is given by x in the formula Fe II



  (110) = 22.34,  (200) = 32.54, (112) = 41.18. The cell parameter has been roughly evaluated to a = 2.867(8) Å in agreement with the literature. The 57 Fe Mössbauer spectrum shown in Figure 2 presents 3 broadened doublets with an isomer shift IS ~ 1 mm/s and quadrupole splitting ranging from QS ~ 0.50 to QS ~ 1.50 mm/s and their relative areas were corrected from the -Fe magnetic contribution as described in the experimental part. The absorption intensities are 42, 26 and 15 %. The hyperfine parameters (IS and QS) are rather characteristic of Fe II species involved in wüstite Fe 1-x O

Fe II O + 2 (

 2  Li → ( Fe 0 + ( Li 2 O (reduction) 2) 3 Fe II O →  Fe 0 +  Fe 2 III O 3 (disproportionation)

0. 549

 549 Fe II O + 0.909 Li → 0.486 Fe 0 + 0.4545 Li 2 O + 0.0315 Fe 2 O 3,

( 2 .

 2 16-0.94) = 1.22 Li. The electrochemical mechanism deduced from Mössbauer data analysis is for the charge: 0.230 Fe 0 + 0.2285 Li 2 O + 0.0015 Fe 2 O 3 → 0.233 Fe II O + 0.457 Li Here again ~63 % of extracted Li are used for SEI consumption. It means that only ~50 % of the formed SEI during the first discharge is consumed during the first charge. Our observations are in agreement with previous works based on the lithium conversion reaction

Figure 1 :Figure 2 :Figure 3 :

 123 FIGURES

Figure 4 :

 4 Figure 4:Comparison of X-ray diffraction patterns of the pristine Fe 1-x O material (a) with small impurity of -Fe (*), after insertion of 1 Li (b) and the end for discharge at 2.2 Li (c). Label (d) corresponds to the end of the first charge at 1 Li. Indexation is given for FeO; -Fe is still present during the first discharge. Indexation for the 3 allotropic forms of metallic iron is given on the pattern (d): -Fe (bcc),-Fe (fcc),-Fe (hcp). For -Fe, indexation is represented above the pattern and positions are in the rather good agreement with the observed broadened peaks.

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: Comparison of57 Fe Mössbauer spectra at 300 K of the pristine Fe 1-x O material (a), after an uptake of 1 Li (b) and the end for discharge at 2.16 Li (c). Green contributions are attributed to Fe II . The thicker green line is sum of the thinner one corresponding to unreacted Fe 1-x O. Dashed blue line corresponds to the expected increasing contribution of -Fe from (a) to (c). In the case of (c), magnetic sextet has been slightly shifted as a guide of the eye. Spectrum (d) corresponds to the end of the first charge at 0.94 Li with no -Fe contribution, but the blue singlet nanosized metallic -Fe 0 is still present.

  1-x O reacts while new divided metallicFe is formed with a new species containing Fe III . Surprisingly, we can assume at this stage the disproportionation of 3Fe II into 2Fe III and 1Fe 0 .

	As previously explained, from the relative areas of Fe II species with found a vacancies
	amount of x = 0.036. The proposed formula for wüstite at this depth of discharge is
	Fe II 0.892 Fe III 0.072 □ 0.036 O.

  1-x O does not completely react. The Fe III species contribution decreases from ~12 % (1 Li) to ~6 %. At this stage the increase in size of Fe 2 O 3 particles induces a transition in the Mössbauer spectra from a superparamagnetic (doublet) to a magnetic (sextet) contribution. At RT this transition occurs for a particle size around 15 nm

	for hematite (K eff ~1 kJ.m -3 ) [49]. Metallic iron (micro and nano) contribution increases from
	~25 to ~31 %. From the relative area of the Fe II species with x = 0.032, the proposed formulae
	for wüstite at this depth of discharge is Fe II 0.904 Fe III	0.064 □ 0.032 O.

  Detailed analysis of Mössbauer spectra suggests that vacancies play a role in cation diffusion. These results reveal that in situ formation of nanometric metallic iron particles with a size smaller than 2 nm embedded in a Li 2 O matrix is accompanied with some traces of Fe 2 O 3 that growth to reach a size of about 15 nm. This suggests that Fe III defect in the FeO structure may be close to each other to permit small amounts of Fe 2 O 3 to reach rather high size compared to metallic iron nanoparticles. Therefore, a very high metal/Li 2 O interface induces an increase of pressure detected through isomer shift and quadrupole splitting variations. Upon reoxidation of these nanocomposites, very small clusters of oxides are formed back since hyperfine parameters of FeO are different compared

	CONCLUSION
	Electrochemical lithium conversion reaction of FeO wüstite has been investigated using X-ray
	diffraction and Mössbauer spectroscopy.
	57 V, 70 mV, 29 %), (0.48 V, 135 mV, 35 %) and (0.30
	V, 268 mV, 36 %) respectively. If we assume that these surfaces are proportional to the
	number of Li involved in a reaction, the very first one peak corresponds to about ~0.63 Li.
	The width of this peak is very thin and is a signature of a two-phase mechanism. It
	corresponds to the main reaction FeO + 2Li → Fe + Li 2 O. The end of the first discharge
	involves rather broadened peaks suggesting SEI growth. During the first charge a broadened
	peak located at about 1.31 V should correspond to SEI consumption while the thin peak at
	1.64 V may correspond to nanoFe 0 oxidation into FeO (~44 % compared to 37 % of Li). At
	the second discharge we found a thin peak (V ~ 47 mV) located at 0.98 V. Assuming the
	reversible reaction shows a rather high polarization of about 1.64-0.98 = 660 mV due to
	highly divided material.