Wenjie Fang

Hsien-Kuei Hwang

Mihyun Kang

Phase transitions from exp(n 1/2 ) to exp(n 2/3 ) in the asymptotics of banded plane partitions 1

We examine the asymptotics of a class of banded plane partitions under a varying bandwidth parameter m, and clarify the transitional behavior for large size n and increasing m = m(n) to be from c 1 n -1 exp c 2 n 1/2 to c 3 n -49/72 exp c 4 n 2/3 + c 5 n 1/3 for some explicit coefficients c 1 , . . . , c 5 . The method of proof, which is a unified saddle-point analysis for all phases, is general and can be extended to other classes of plane partitions.

Introduction

Asymptotics of partition-related generating functions with the unit circle as the natural boundary has been the subject of study since Hardy and Ramanujan's 1918 epoch-making paper [START_REF] Hardy | Asymptotic formulaein combinatory analysis[END_REF]. In particular, it is known that the number of partitions of n into positive integers is asymptotic to

p n := [z n ] k 1 1 1 -z k ∼ cn -1 e βn 1/2 , with (c, β) = 1 4 √ 3 , √ 2 π √ 3 , (1)
(see [START_REF] Andrews | The theory of partitions[END_REF][START_REF] Hardy | Asymptotic formulaein combinatory analysis[END_REF] or [18, A000041]), and that of plane partitions of n satisfies

p n = [z n ] k 1 1 1 -z k k ∼ cn -25/36 e βn 2/3 , with (c, β) = ζ(3) 7/36 e -ζ (-1) 2 11/36 √ 3π , 3ζ (3
) 1/3 2 2/3 , (2)
(see [START_REF] Andrews | The theory of partitions[END_REF][START_REF] Wright | Asymptotic partition formulae: I. Plane partitions[END_REF] or [18, A000219]). Here the symbol [z n ]f (z) denotes the coefficient of z n in the Taylor expansion of f and ζ(s) the Riemann zeta function [START_REF] Apostol | Introduction to analytic number theory[END_REF][START_REF] Whittaker | A course of modern analysis[END_REF]. Throughout this paper, the values of the generic (or local) symbols c, β or c j , β j may differ from one occurrence to the other, and will always be locally specified.

The increase of the sub-exponential (or stretched exponential) term from e βn 1/2 in the case of ordinary partitions to e βn 2/3 in the case of plane partitions is noticeable, and marks the essential difference in the respective asymptotic enumeration. As integer partitions are also encountered in statistical physics, astronomy, and other engineering applications, one naturally wonders if there is a tractable combinatorial model that interpolates between the two different orders e n 1/2 and e n 2/3 when some structural parameter varies. This paper aims to address this aspect of partition asymptotics and examines in detail a class of plane partitions with a natural notion of bandwidth m whose variation yields a model in which we can fully clarify the transitional behavior from being of order e βn 1/2 for bounded m to e βn 2/3 when m n 1/3 , providing more modeling flexibility of these partitions. Our study constitutes the first asymptotic realization of such phase transitions in the analytic theory of partitions. Readers are referred to [START_REF] Flajolet | Analytic combinatorics[END_REF]Section VII.10] for an introduction to phase transitions in combinatorial structures.

Intuitively, if we impose a constraint to one or two of the dimensions of plane partitions, then by suitably varying the constraint, we can generate families of objects whose asymptotic behaviors interpolate between e n 1/2 and e n 2/3 . An initial attempt can be found, e.g., in [START_REF] Gordon | Notes on plane partitions[END_REF], where Gordon and Houten computed the asymptotic counting formula for "k-rowed partitions" whose nonzero parts decrease strictly along each row of size n. However, they studied only the situations when k is bounded and when k → ∞, and do not consider how exactly the asymptotic behavior changes with respect to varying k (depending on n). See Section 6 for the phase transitions in plane partitions with a given number of rows.

The plane partitions of n 0 may be viewed as a matrix with nonincreasing entries along rows and columns and with the entry-sum equal to n. The class of plane partitions we work on in this paper is the double shifted plane partitions studied by Han and Xiong in [START_REF] Han | Skew doubled shifted plane partitions: calculus and asymptotics[END_REF] with an explicit notion of width, which for simplicity will be referred to as the banded plane partitions (or BPPs) in this paper. These are plane partitions arranged on the stair-shaped region T m = {(i, j) ∈ N 2 | j i j + m -1}, m ∈ Z + , where N = Z + ∪ {0}. Formally, a banded plane partition of width m is a function f : T m → N with finite support such that, for any (i, j) ∈ T m , we have f (i, j) f (i, j + 1) when (i, j + 1) ∈ T m , and f (i, j) f (i + 1, j) when (i + 1, j) ∈ T m . Figure 1 illustrates two instances of BPPs. The size of a BPP is the sum (i,j)∈Tm f (i, j). We denote by G n,m the number of BPPs of size n and width m, i.e., BPPs that can fit in T m . A closed-form expression for the generating function

G m (z) := n 0 G n,m z n is given in [11, Theorem 1.1] as G m (z) = P (z)Q m (z), where P (z) = k 1 1 1 -z k , and Q m (z) = k 0 1 h<j<m 1 1 -z 2mk+h+j . (3)
In particular,

Q 3 (z) = k 0 1 1 -z 6k+3 , Q 4 (z) = k 0 1 1 -z 8k+3 1 -z 8k+4 1 -z 8k+5 , Q 5 (z) = k 0 1 1 -z 10k+3 1 -z 10k+4 1 -z 10k+5 2 1 -z 10k+6 1 -z 10k+7 .
For a BPP f with m n, the function g on N 2 defined by g(i, j) = f (i + j, j) is a plane partition, and by replacing each row of g (which is an integer partition) by its conjugate partition, we obtain a column-strict plane partition (weakly decreasing in each row but strictly decreasing in each column). This transformation is clearly bijective. An example is given in Figure 2. The generating function of column-strict plane partitions is known to be of the form

k 1 1 (1 -z k ) (k+1)/2 ;
see [START_REF] Gordon | Notes on plane partitions. I, II[END_REF][START_REF] Stanley | Theory and application of plane partitions[END_REF] or [18, A003293].

Based on the generating function (3), Han and Xiong showed in [START_REF] Han | Skew doubled shifted plane partitions: calculus and asymptotics[END_REF], by an elementary convolution approach developed in [START_REF] Han | Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions[END_REF], that the number G n,m of BPPs of size n and width m satisfies

G n,m ∼ c(m)n -1 e β(m) √ n , (4)
for large n and bounded m 1, where

(c(m), β(m)) := √ m 2 + m + 2 2 (m 2 -3m+14)/4 √ 3m 3 j<m sin jπ 2m -(j-1)/2 , m 2 + m + 2 6m π . Thus log G n,m is still of asymptotic order √ n when m is bounded. Note that c(1) = c(2) = 1/(4 √ 3) and β(1) = β(2) = √ 2 π/ √ 3
, the same as c and β in (1), respectively. Now if we pretend that the formula (4) holds also for increasing m, then since β(m) ∼ m/6 π for large m, we see that β(m) √ n √ mn n 2/3 when m n 1/3 (where the Hardy symbol a n b n stands for equivalence of growth order for large n, equivalent to the Bachmann-Laudau notation a n = Θ(b n ); see [START_REF] Knuth | Big omicron and big omega and big theta[END_REF]). Furthermore, we will show in Proposition 3.1 that log c(m) ∼ -7ζ (3) 8π 2 m 2 for large m. Then equating m 2 √ mn also gives m n 1/3 . Thus we would expect that (4) remains valid for m = o n 1/3 and the "phase transition" occurs around m n 1/3 . However, while the latter is true by such a heuristic reasoning, the former is not as we will prove that (4) holds indeed only when m = o n 1/7 , although the weaker asymptotic estimate log G n,m ∼ β(m) √ n does hold uniformly for 1 m = o n 1/3 (see (78) and ( 82)). This implies particularly the estimate

log G n,m ∼ π √ 6 √ mn, (5)
which holds uniformly when m → ∞, m = o n 1/3 . On the other hand, Gordon and Houten [START_REF] Gordon | Notes on plane partitions[END_REF] showed that

G n,n = [z n ] k 1 1 (1 -z k ) (k+1)/2 ∼ cn -49/72 e β 1 n 2/3 +β 2 n 1/3 , (6)
where

(c, β 1 , β 2 ) = e ζ (-1)/2-π 4 /(3456ζ(3)) ζ(3) 13/72 2 3/4 (3π) 1/2 , 3ζ(3) 1/3 2 , π 2 24ζ(3) 1/3 . ( 7
)
This implies particularly the weak asymptotic estimate

log G n,n ∼ 3ζ(3) 1/3 2 n 2/3 . (8)
In Section 5 we will derive stronger asymptotic approximations to G n,m for all possible values of m, 1 m n, covering ( 4) and ( 6) as special cases. In particular, as far as log-asymptotics is concerned, we derive a uniform estimate, covering also the most interesting critical range when m n 1/3 ; see Proposition 5.5. Define

η d (z) := 1 e -z 2d-1 (1 + e -z ) (d ∈ N; Re(z) > 0). (9)
log G n,m n 2/3 ∼ G(α) := r + ζ(3) -2η 2 (αr) 2r 2 , ( 10
)
uniformly when α n -1/3 (or m → ∞), where r = r(α) > 0 solves the equation

r 3 -ζ(3) + 2η 2 (αr) -αrη 2 (αr) = 0. (11)
In particular,

G(α) ∼      π √ 6 √ α, if α → 0; 3 2 ζ(3) 1/3 , if α → ∞. (12)
We thus have a combinatorial model that interpolates nicely between integer partitions and columnstrict plane partitions, in the sense of asymptotic behavior. A very similar looking expression will be derived in Section 6 for m-rowed plane partitions, which bridge particularly ordinary partitions and plane partitions.

The BPPs we study here can be connected to ordinary plane partitions through the following decomposition. Given a plane partition g of size n, denote by t = i 0 g(i, i) its trace. We separate g by the diagonal i = j for (i, j) ∈ N 2 , obtaining two BPPs f 1 , f 2 of sizes n 1 , n 2 respectively, and an integer partition on the diagonal, such that n = n 1 + n 2 + t. The weak asymptotics of such a triple (n 1 , n 2 , t) is bounded above by

log G n 1 ,n 1 + log G n 2 ,n 2 + log p t β 1 (n 2/3 1 + n 2/3 2 ) + β 2 (n 1/3 1 + n 1/3 2 ) + O( √ t + log n) 2 1/3 β 1 n 2/3 -2 -2/3 β 1 n -1/3 t + 2 2/3 β 2 n 1/3 + O( √ t + log n),
with β 1 , β 2 defined in [START_REF] Flajolet | Analytic combinatorics[END_REF]. The last inequality uses the concavity of x → x 2/3 and the fact that (1x) 2/3 1 -x/2 for 0 x 1. Since t = O(n), the dominant term of the last upper bound matches that in [START_REF] Apostol | Introduction to analytic number theory[END_REF]. If t = ω(n 2/3 ), the subdominant term will be negative and of order Θ(n -1/3 t), making the bound exponentially smaller than [START_REF] Apostol | Introduction to analytic number theory[END_REF]. The main contribution thus comes from t = O(n 2/3 ). This is consistent with the results in [START_REF] Kamenov | The limiting distribution of the trace of a random plane partition[END_REF] on the asymptotic normality of t, with mean asymptotic to c 1 n 2/3 and variance to c 2 n 2/3 log n for some explicit constants c 1 and c 2 .

For the method of proofs, we will employ a more classical approach based on Mellin transforms (see [START_REF] Flajolet | Mellin transforms and asymptotics: harmonic sums. Theoret[END_REF]) and saddle-point method (see [START_REF] Andrews | The theory of partitions[END_REF][START_REF] Flajolet | Analytic combinatorics[END_REF][START_REF] Meinardus | Asymptotische Aussagen über Partitionen[END_REF]), instead of the elementary approach used in [START_REF] Han | Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions[END_REF][START_REF] Han | Skew doubled shifted plane partitions: calculus and asymptotics[END_REF], which becomes cumbersome when finer asymptotic expansions are required. The analytic approach we adopted, although standard as that presented in [START_REF] Andrews | The theory of partitions[END_REF][START_REF] Meinardus | Asymptotische Aussagen über Partitionen[END_REF], which applies for fixed m, becomes more delicate because we address the whole range 1 m n, and describing the transitional behaviors in different "phases" requires a finer analysis by maintaining particularly the uniformity of all error terms involved with varying m.

Of additional interest here is that, similar to the functional equation satisfied by the generating function of p n P (e -τ ) :=

n 0 p n e -nτ = τ 2π exp π 2 6τ - τ 24 P e -4π 2 /τ (Re(τ ) > 0), (13)
(see [START_REF] Ayoub | An introduction to the analytic theory of numbers[END_REF]), we also have the following (non-modular) relation satisfied by the generating function of G n,m .

Theorem 1.2. For Re(τ ) > 0, the function G m (e -τ ) satisfies the identity

G m (e -τ ) = g m √ τ exp m τ + φ m τ K m e -4π 2 /τ L m e -4π 2 /τ , (14)
where the constants depending on m are given by

         g m := (2π) -(m 2 -3m+4)/4 1 k<j<m Γ k + j 2m , m := π 2 24 m + 1 + 2 m , φ m := m 3 -7m + 2 96 , (15)
and the two functions K m and L m by

             K m (z) := P z 1/m P z 1/2 P z (m+2)/4 , L m (z) := exp - 1 2m 1 <m cos (2 -1)π m 1 -cos (2 -1)π m j 0 z j+ 2 -1 2m j + 2 -1 2m 1 -z j+ 2 -1 2m . ( 16
)
Both K m (z) and L m (z) are analytic in |z| < 1, z ∈ [-1, 0].
The expression ( 14) is complicated but exact, and is the basis of our saddle-point analysis for characterizing the asymptotic behaviors of G n,m . It is derived by Mellin transforms and the functional equation for the Hurwitz zeta function; see [2, §12.9]. Note that

Q 3 (z) = k 0 1 1 -z 6k+3 = P (z 3 ) P (z 6 ) = k 1 1 + z 3k ,
so we also have, by [START_REF] Hardy | Asymptotic formulaein combinatory analysis[END_REF], the functional equation

Q 3 (e -τ ) = e π 2 /(36τ )+τ /8 √ 2 Q 3 e -2π 2 /(9τ ) .
No such equation is available for higher Q m (z) with m 4. On the other hand, the sequence G n,3 coincides with A266648 in OEIS [START_REF]The on-line encyclopedia of integer sequences[END_REF].

The rest of this paper is structured as follows. The exact expression of G m in Theorem 1.2 is first proved in the next section. Then we turn to the asymptotics of G m in Section 3. A uniform asymptotic approximation to G n,m is then derived in Section 4, which is used in Section 5 to characterize the more precise behaviors of G n,m in each of the three phases: sub-critical, critical and super-critical. We then extend the same approach in Section 6 to m-rowed plane partitions, together with two other similar variants.

Notations. Since Q m (z) = 1 for m 2, we assume m 3 throughout this paper. The symbols c, c , β and c j , β j are generic whose values will always be locally specified. Other symbols are global except otherwise defined (e.g., in Section 6).

2 Exact expression for G m (e -τ ): proof of Theorem 1.2

In this section, we will prove Theorem 1.2 for the exact expression [START_REF] Kamenov | The limiting distribution of the trace of a random plane partition[END_REF] for G m (e -τ ) by Mellin transforms. We start with rewriting Q m (z) in (3) as

Q m (z) = k 0 1 j<2m 1 1 -z 2mk+j wm(j) , (17)
where

w m (j) := m -1 -|m -j| 2 (1 j < 2m). (18)
For convenience, the kth moment of w m is denoted by µ k (w m ):

µ k = µ k (w m ) := 1 j<2m j k w m (j) (k ∈ N).
By considering the parity of j and m, we deduce that

W m (z) := 1 j<2m w m (j)z j = z 3 (1 -z m-1 )(1 -z m-2 ) (1 + z)(1 -z) 2 (m 3). ( 19
)
From this expression, it is straightforward to compute the first few moments

µ k = k![s k ]W m (e s
), as given explicitly in Table 1.

µ 0 µ 1 µ 2 µ 3 (m -1)(m -2) 2 m(m -1)(m -2) 2 m(m -1)(m -2)(7m -3) 12 3m 2 (m -1) 2 (m -2) 4
Table 1: The exact expressions of µ k for 0 k 3.

Since all singularities of G m (z) lie on the unit circle, we consider the change of variables z = e -τ and examine the behavior of G m (e -τ ) in the half-plane Re(τ ) > 0. For that purpose, let

ζ(s, b) := k 0 (k + b) -s (Re(s) > 1, b > 0)
denote the Hurwitz zeta function. 13), we need only derive a similar expression for Q m (e -τ ) in order to prove [START_REF] Kamenov | The limiting distribution of the trace of a random plane partition[END_REF].

Proposition 2.1. For Re(τ ) > 0, q m (e -τ ) := log Q m (e -τ ) satisfies q m (e -τ ) = (m -1)(m -2)π 2 24mτ + 1 j<2m w m (j) log Γ j 2m - (m -1)(m -2) 4 log(2π) + (m -1)(m -2)(m + 3) 96 τ + E(τ ), (20)
where E(τ ) is given by

E(τ ) = E(m; τ ) := 1 2πi (-2) Γ(s)ζ(s + 1)M m (s)τ -s ds, (21)
with (c) representing c+i∞ c-i∞ and M m (s) := (2m) -s 1 j 2m w m (j)ζ s, j 2m . ( 22
)
Proof. Let M

[q] m (s) be the Mellin transform of q m (e -τ ). Then M

[q] m (s) = Γ(s)ζ(s + 1)M m (s) for Re(s) > 1,
where M m (s) is defined in [START_REF] Whittaker | A course of modern analysis[END_REF]. By the inverse Mellin transform, we have

q m (e -τ ) = 1 2πi (r) M [q] m (s)τ -s ds (r > 1). (23)
We will move the line of integration to the left, so as to include the leftmost pole at s = -1, and collect all the residues of the poles encountered. For that purpose, we need the growth properties of the integrand at c ± i∞ to ensure the absolute convergence of the integral. By the known estimate for Gamma function (see [4, §1.18])

|Γ(c + it)| = O |t| c-1/2 e -π|t|/2 , (c ∈ R, |t| > 1),
and that for Hurwitz zeta function (see [22, §13.51, p. 276])

|ζ(c + it, b)| = O |t| ν 0 (c) log |t| , with ν 0 (c) :=            1 2 -c, if c < 0; 1 2 , if c ∈ [0, 1 2 ]; 1 -c, if c ∈ [ 1 2 , 1]; 0, if c > 1, (24)
for |t| > 1, we have

|M [q] m (c + it)τ -s | = O m 2-c |t| ν(c) (log |t|) 2 e -π 2 |t|+t arg(τ ) , (25)
for c ∈ R, |t| > 1, where

ν(c) := 1 2 + |c|, if |c -1 2 | 1 2 ; min{ 1 2 + c, 3 2 -c}, if |c -1 2 | 1 2 . ν 0 (c) ν(c)
Thus the integral in ( 23) is absolutely convergent as long as | arg(τ )| π/2 -ε, and this justifies the analytic properties we need for summing the residues, which we now compute. Since w m (j) = w m (2m -j) (see ( 18)), we can rewrite [START_REF] Whittaker | A course of modern analysis[END_REF] as

M m (s) = 1 j<m w m (j) ζ s, j 2m + ζ s, 1 - j 2m + w m (m)ζ s, 1 2 . (26)
Observe that M m (-2j) = 0 for j ∈ Z + because ζ(-2j, x) = -B 2j+1 (x)/(2j + 1), where B 2j+1 (x) is the Bernoulli polynomial of order 2j + 1:

B j (x) := j![z j ] ze xz e z -1 , (27)
which satsfies B 2j+1 (x) = -B 2j+1 (1 -x); see [4, § 1.13]. On the other hand, ζ(s + 1) = 0 when s < -1 is odd. Thus the only poles of the integrand in ( 23) are s = 1 (simple), s = 0 (double) and s = -1 (simple); this similarity to that of log P (e -τ ) suggests the possibility of the identity [START_REF] Kamenov | The limiting distribution of the trace of a random plane partition[END_REF].

From these properties, it follows that

q m (e -τ ) = -1 j 1 Res s=j Γ(s)ζ(s + 1)M m (s)τ -s + E(τ ), (28)
where E(τ ) is as defined in [START_REF] Stanley | Theory and application of plane partitions[END_REF]. By the local expansions of Γ(s), ζ(s + 1) and ζ(s, b) for s ∼ 0 (see [START_REF] Erdélyi | Higher transcendental functions[END_REF]):

Γ(s) = 1 s -γ + O(|s|), ζ(s + 1) = 1 s + γ + O(|s|), ζ(s, b) = 1 2 -b + log Γ(b) - 1 2 log(2π) s + O(|s| 2 ),
where γ is the Euler-Mascheroni constant, we then have

q m (e -τ ) = π 2 µ 0 12mτ + 1 j<2m w m (j) log Γ j 2m - µ 0 2 log(2π) + - µ 2 8m + µ 1 4 - mµ 0 12 τ + E(τ ).
This, together with the expressions in Table 1, proves [START_REF] Wright | Asymptotic partition formulae: I. Plane partitions[END_REF].

We now evaluate E(τ ), beginning with a simple lemma.

Lemma 2.2. For integers m > 1, 1 2m and real number θ, we have

1 k<j<m sin θ + (k + j)π m = sin θ ×                      m -1 2 , for = 2m; - m -1 2 , for = m;
1, for 1 < 2m; = m and even;

- cos( π/m) 1 -cos( π/m)
, for 1 < 2m, = m and odd.

Proof. (Sketch) We consider the identity

1 k<m exp k πi m 2 = 2 1 k<j<m exp (k + j) πi m × 1 k<m exp 2k πi m ,
and perform straightforward simplifications in each case.

We now compute the error term E(τ ). Let p(z) := log P (z).

Proposition 2.3. The error term E(τ ) defined in [START_REF] Stanley | Theory and application of plane partitions[END_REF] satisfies

E(τ ) = κ m e -4π 2 /τ -p e -4π 2 /τ + λ m e -4π 2 /τ ,
for Re(τ ) > 0, where (K m , L m defined in [START_REF] Meinardus | Asymptotische Aussagen über Partitionen[END_REF])

κ m (z) := log K m (z) = m + 2 4 p(z) + 1 2 p z 1/m - 1 2 p z 1/2 , (29)
λ m (z) := log L m (z) = - 1 2m 1 <m cos (2 -1)π m 1 -cos (2 -1)π m k 0 z k+ 2 -1 2m k + 2 -1 2m 1 -z k+ 2 -1 2m . ( 30
)
Proof. We first rewrite the single-sum relation [START_REF] Whittaker | A course of modern analysis[END_REF] for M m (s) as a double sum:

M m (s) = (2m) -s 1 h<j<m ζ s, h + j 2m .
Combining this with the functional equation for the Hurwitz zeta function (see [2, §12.9])

ζ s, j d = 2Γ(1 -s) (2dπ) 1-s 1 d sin πs 2 + 2 jπ d ζ 1 -s, d (d = 1, 2, . . . ), (31)
we then have

M m (s) = Γ(1 -s) m(2π) 1-s 0 2m ζ 1 -s, 2m
1 k<j<m sin πs 2 + (k + j)π m .
Now, by Lemma 2.2, the sum above can be reduced to

M m (s) = Γ(1 -s) m(2π) 1-s sin πs 2 m -1 2 ζ(1 -s) - m -1 2 ζ 1 -s, 1 2 + 1 <m,2 =m ζ 1 -s, m - 1 <m,2 -1 =m cos (2 -1)π m 1 -cos (2 -1)π m ζ 1 -s, 2 -1 2m .
Then, by the relation

1 d ζ s, d = d s ζ(s) (d = 2, 3, . . . ), (32)
which implies, in particular, ζ(s, 1/2) = (2 s -1)ζ(s), we deduce that

M m (s) = Γ(1 -s) (2π) 1-s sin πs 2 c(m, s)ζ(1 -s) - 1 m 1 <m cos (2 -1)π m 1 -cos (2 -1)π m ζ 1 -s, 2 -1 2m , where c(m, s) := (m -2)/2 + m -s -2 -s .
By applying the change of variables s → -s in the integral representation in ( 21) of E(τ ), we obtain

E(τ ) = 1 2πi (2) Γ(-s)ζ(1 -s)M m (-s)τ s ds. ( 33
)
Note that the functional equation ( 31) with d = j = 1 implies for the Riemann zeta function that

ζ(s) = 2 s π s-1 Γ(1 -s)ζ(1 -s) sin πs 2 . ( 34
)
By this and Euler's reflection formula for the Gamma function

Γ(s)Γ(1 -s) = π sin(πs) , (35)
we then get

Γ(-s)ζ(1 -s) = - (2π) 1-s s sin(πs) ζ(s) cos πs 2 .
Consequently, the integrand in (33) can be written as

Γ(-s)ζ(1 -s)M m (-s)τ s = 1 2 4π 2 τ -s Γ(s)ζ(s) × c(m, -s)ζ(1 + s) -m -1 1 <m cos (2 -1)π m 1 -cos (2 -1)π m ζ 1 + s, 2 -1 2m .
The two expressions (29) (contributed by terms involving c(m, s)) and (30) (contributed by terms involving the partial sum with the cosine functions) then follow from inverting the Mellin transform using the relation

J(b, τ ) := 1 2πi (c) Γ(s)ζ(s)ζ(1 + s, b)τ s ds = k 0 e -(k+b)/τ (k + b) 1 -e -(k+b)/τ , (36)
for Re(τ ) > 0 and b > 0, where c > 1. In particular, the right-hand side equals p e -1/τ when b = 1. This completes the proof.

Proof of Theorem 1.2. Theorem 1.2 is a direct consequence of a combination of Proposition 2.1, Proposition 2.3 and (13).

3 Asymptotics of log G m (e -τ )

We derive the asymptotic behavior of log G m (e -τ ) as m → ∞ and |τ | → 0. From Theorem 1.2 and Proposition 2.3, we have

log G m (e -τ ) = m τ + 1 2 log τ + log g m + φ m τ + κ m e -4π 2 /τ + λ m e -4π 2 /τ , (37)
for Re(τ ) > 0. Since κ m (z) depends only on p(z) (see ( 29)), which, by [START_REF] Hardy | Asymptotic formulaein combinatory analysis[END_REF], satisfies

p(e -τ ) = π 2 6τ - τ 24 + 1 2 log τ - 1 2 log(2π) + p e -4π 2 /τ (Re(τ ) > 0), (38)
so we need only to examine more closely the asymptotics of log g m and λ m when m is large and |τ | → 0.

Complications arise when τ may depend also on m.

Asymptotics of log g m

We now derive an asymptotic expansion for log g m by the Euler-Maclaurin formula (see [12, Ch. VIII]).

Proposition 3.1. When m → ∞, log g m satisfies the asymptotic expansion

log g m ∼ - 7ζ(3) 8π 2 m 2 + 11 24 log m + c 1 - j 1 B 2j B 2j+2 (-π 2 ) j 8j(j + 1)(2j)! m -2j , (39)
where c 1 := w m (j) log Γ j 2m .

Since w m (j) = w m (2m -j), we have, by Euler's reflection formula (35), The last two sums are easily simplified by the elementary identity

S m = µ 0 2 log π - 1 j<m j -1 2 log sin jπ 2m = (m -1)(m -2) 4 log π - 1 j<m j -1 2 log sin jπ 2m + 1 2
1 j m/2 log sin jπ m = (m -1)(m -2) 4 log π - S m,1 2 + S m,2 2 + S m,3 2
1 j<k sin πj k = k 2 k-1 (k = 1, 2, . . . ), giving S m,2 = -(m -1) log 2 + log m 2 and S m,3 = - m -1 2 log 2 + log m 2 . ( 40
)
We now evaluate S m,1 . By the local expansion log(sin x) = log x+O(x 2 ) for x → 0, we decompose first the sum into two parts:

S m,1 = 1 j m j log sin jπ 2m -log jπ 2m + 1 j m j log jπ 2m ,
and then we apply Euler-Maclaurin formula (see [12, Ch. VIII]) to each sum, yielding

1 j m j log sin jπ 2m -log jπ 2m = c 2 m 2 - m 2 log π 2 - 1 12 1 + log π 2 + O(m -2 ),
where

c 2 := 1 m 2 m 0 x log sin xπ 2m -log xπ 2m dx = 7ζ(3) 4π 2 - log π 2 + 1 4 , and
1 j m j log jπ 2m = 1 2 log π 2 - 1 4 m 2 + m 2 log π 2 + log m 12 + 1 12 -ζ (-1) + O(m -2 ).
Summing up these two parts, we have

S m,1 = 7ζ(3) 4π 2 - log 2 2 m 2 + log m 12 -ζ (-1) + 1 12 log π 2 + O(m -2 ). (41)
By substituting (40) and (41) into

log g m = - 1 2 log π - m 2 -3m + 4 4 log 2 - S m,1 2 + S m,2 2 + S m,3 2 , ( 42
)
we obtain the expansion (39) up to an error of order m -2 . Further terms in (39) are computed by refining the expansion for S m,1 following the same procedure and using the relation

d k dx k log(sin(x)) x=π/2 = - d k-1 dx k-1 tan(x) x=π/2 = (2i) k k (2 k -1)B k (k 2);
see the OEIS sequence [18, A155585].

Asymptotics of E(τ )

We now consider the asymptotic behavior of the key "calibrating" term E(τ ) defined in [START_REF] Stanley | Theory and application of plane partitions[END_REF] as τ → 0. This term is asymptotically negligible when m = o(n 1/3 ), but plays a role for larger m, notably in the transitional zone when m n 1/3 . We then need finer asymptotic approximations for E(τ ), which, by Proposition 2.3, equals E(τ ) = κ m e -4π 2 /τ -p e -4π 2 /τ + λ m e -4π 2 /τ . We begin with the asymptotics of the first term, which is simpler.

Corollary 3.2. Assume Re(τ ) → 0 in the half-plane Re(τ ) > 0. Then the function κ m satisfies

κ m e -4π 2 /τ = 1 2 p e -4π 2 /(mτ ) + O e -Re(2π 2 /τ ) =    O e -Re(4π 2 /(mτ )) , if m|τ | 1, mτ 48 + 1 4 log 2π mτ - π 2 12mτ + 1 2 p(e -mτ ) + O e -Re(2π 2 /τ ) , if m|τ | 1. (43)
Proof. By (29), we obtain the first relation in (43). On the other hand, the series p e -4π 2 /τ = j 1 e -4jπ 2 /τ j 1 -e -4jπ 2 /τ is itself an asymptotic expansion when |τ | → 0. The other estimate in (43) when m|τ | 1 follows from the functional equation [START_REF] Hardy | Asymptotic formulaein combinatory analysis[END_REF].

We now examine the other term λ m e -4π 2 /τ , beginning with the asymptotics of the integral J(b, w) defined in (36).

Lemma 3.3. If b > 0, then J(b, τ ) =    b -1 e -b/τ 1 + O e -Re(b/τ ) + e -Re(1/τ ) , as |τ | → 0; ζ(2, b)τ - 1 2 log τ + 1 2 ψ(b) + O(1), as |τ | → ∞, (44)
uniformly in the half-plane Re(τ ) > 0, where ψ is the digamma function defined by ψ(x) = Γ(x)/Γ (x). These estimates hold also when b/|τ | → 0 and b/|τ | → ∞, respectively.

Proof. In the small |τ | case, the estimate follows from the series representation in (36), while in the large |τ | case it is from moving the line of integration in the integral representation in (36) to the left, adding the residues at s = 1 and s = 0.

Note that ζ(2, b) = b -2 + π 2 /6 + O(b) and ψ(b) → b -1 when b → 0. Define ϕ d (z) := 1 (2 -1) 1-2d e -2(2 -1)π 2 /z 1 -e -2(2 -1)π 2 /z (d ∈ Z; Re(z) > 0). (45
where

ξ 2 (z) := - 2 π 2 ϕ 2 (z), ξ 1 (z) := 5 6 ϕ 1 (z), ξ 0 (z) := ϕ 0 (z). (47)
Note that when m = O(1), the rightmost O-term is of the same order as ξ 1 (mτ ) e -Re(2π 2 /mτ ) .

Proof. In the defining series (30), we observe that the inner sum with z = e -4π 2 /τ is itself an asymptotic expansion when |τ | → 0, namely, the term with k = 0 is dominant and all others with k 1 are exponentially smaller. Thus

λ m e -4π 2 /τ = -1 + O e -Re(4π 2 /τ ) 1 <m cos (2 -1)π m 1 -cos (2 -1)π m • e -2π 2 (2 -1) mτ (2 -1) 1 -e -2π 2 (2 -1) mτ = -1 + O e -Re(2π 2 /τ ) 1 m/2 cos (2 -1)π m 1 -cos (2 -1)π m • e -2π 2 (2 -1) mτ (2 -1) 1 -e -2π 2 (2 -1) mτ , (48)
where in the second approximation we truncate terms with > m/2 whose total contribution is bounded above by O m 2 e -Re(2π 2 /τ ) . By expanding the ratio of cosines in (48) using the inequalities

-x 2 cos x 1 -cos x - 2 x 2 + 5 6 x 2 (0 x 1/2),
we then get (46) by summing the resulting terms and extending then the summation range to infinity. The error terms introduced are bounded above by

O > m/2 m 2 (2 -1) 3 + 1 2 -1 + 2 -1 m 2 e -Re(2(2 -1)π 2 /(mτ )) = O m -1 e -Re(2π 2 /τ ) .
This proves the proposition.

When z → 0, we see that ϕ d (z) is itself an asymptotic expansion. However, when z → ∞, the asymptotic behavior of ξ 2 , ξ 1 , ξ 0 cannot be read directly from their defining equations. We now consider this range of z. Recall the functions η d (z) defined in [START_REF] Gordon | Notes on plane partitions[END_REF], which are themselves asymptotic expansions for large |z| in the right half-plane.

Lemma 3.5. The functions ξ d (z) (d = 0, 1, 2) satisfy the identities:

ξ 0 (z) = z 2 48π 2 - 1 24 + z 2 2π 2 η 0 (z), (49)
ξ 1 (z) = 5z 96 + 5 24 log π 2z - 5 12 η 1 (z), (50)
ξ 2 (z) = - z 96 + 7ζ(3) 8π 2 - π 2 24z + ζ(3) 2z 2 - η 2 (z) z 2 , (51)
which are also asymptotic expansions for large |z| in Re(z) > 0.

Proof. We apply the same Mellin transform techniques, together with the functional equation (34) for the Riemann zeta function, as in the previous section. Consider first ξ 2 (z). By direct calculations using (32), we have

ξ 2 (z) = - 2 π 2 • 1 2πi (3/2) X 2 (s)z s ds, where X 2 (s) = Γ(s)ζ(s)(1 -2 -3-s )ζ(3 + s)(2π 2 ) -s .
By a similar analysis as in the proof of Proposition 20, we deduce that

ξ 2 (z) = - 2 π 2 -2 k 1 Res s=k (X 2 (s)z s ) + 1 2πi (-5/2) X 2 (s)z s ds .
The sum of the residues yields the first four terms on the right-hand side of (51). We then simplify the integral

(-5/2) X 2 (s)z s ds = (1/2) X 2 (-2 -s)z -s-2 ds.
By (34),

X 2 (-2 -s) = Γ(-2 -s)ζ(-2 -s)(1 -2 s-1 )ζ(1 -s) 2π 2 s+2 = π 2 2 (1 -2 1-s )ζ(s + 3)Γ(s)ζ(s),
which is nothing but the Mellin transform of π 2 2 η 2 (z). This proves (51). The proofs of the other two identities (49) and (50) are similar, and omitted. Corollary 3.6. Assume |τ | → 0 in the half-plane Re(τ ) > 0. Then the function λ m e -4π 2 /τ satisfies:

(i) if m|τ | 1, then λ m e -4π 2 /τ = m 2 ξ 2 (mτ ) + ξ 1 (mτ ) + O m -2 e -Re(2π 2 /(mτ )) ; ( 52
) (ii) if m|τ | 1, then λ m e -4π 2 /τ = m 2 ξ 2 (mτ ) + ξ 1 (mτ ) + O(|τ | 2 ) = ζ(3) -2η 2 (mτ ) 2τ 2 - π 2 m 24τ + 7ζ(3) 8π 2 m 2 - m 3 τ 96 + 5mτ 96 - 5 24 log 2mτ π - 5η 1 (mτ ) 12 + O(|τ | 2 ). ( 53
)
4 Asymptotics of G n,m

Our analytic approach to the asymptotics of G n,m relies on the Cauchy integral formula

G n,m = [z n ]G m (z) = 1 2πi |z|=e -ρ z -n-1 G m (z) dz (ρ > 0).
Since G m (e -τ ) grows very fast near the singularity τ = 0 (see ( 14)), we will apply the saddle-point method to the integral on the right-hand side. We derive first crude (but effective) approximations to G n,m and then sketch our approach to refining them, more details being given in the next sections.

Crude bounds

By the nonnegativity of the coefficients, we have the simple inequality

G n,m e nρ G m (e -ρ ) = exp (n + φ m )ρ + m ρ + κ m e -4π 2 /ρ + λ m e -4π 2 /ρ (n, m 1).
Here ρ = ρ(n, m) > 0 is taken to be the saddle-point, namely, it satisfies the equation

nG m (e -ρ ) = e -ρ G m (e -ρ ), or n + φ m = m ρ 2 -∂ ρ κ m e -4π 2 /ρ + λ m e -4π 2 /ρ .
Consider first the case when m is not too large. More precisely, if

κ m e -4π 2 /ρ + λ m e -4π 2 /ρ = O m 2 e -2π 2 /(mρ) = o m ρ m ρ ,
or, simply mρ → 0, then, by ( 43) and ( 52), the saddle-point satisfies

n + φ m ∼ m ρ 2 , or ρ ∼ m n + φ m .
Thus ρ is of order m/n, which in turn specifies the range of m: mρ m 3/2 /n 1/2 → 0, or m = o(n 1/3 ). In this range of m, we see that

log G n,m 2 (n + φ m ) m (1 + o(1)) ∼ π √ 6 √ mn,
which is tight when compared with the asymptotic estimate in [START_REF] Erdös | The distribution of the number of summands in the partitions of a positive integer[END_REF]. Note that κ m (e -4π 2 /ρ ) is not uniformly o(1) in this range, although it is of a smaller order than m/ρ; indeed, if

m 6π 2/3 n 1/3 (log n -2 log log n + log ω n ) 2/3 , (54)
for any sequence ω n tending to infinity, then

κ m (e -4π 2 /ρ ) m 2 e -2π 2 /(mρ) ω -2/3 n → 0.
For larger m with mρ ε > 0, we use (46) and Lemma 3.5, giving

log G m (e -ρ ) = ζ(3) 2ρ 2 + π 2 24ρ + log ρ 24 + O(1),
as ρ → 0 and mρ → ∞. Thus the saddle-point ρ satisfies

ρ ∼ ζ(3) 2 1/3 n -1/3 , implying that log G n,m 3ζ(3) 1/3 2 n 2/3 (1 + o(1)),
which is also tight in view of (8).

The uniform saddle-point approximation

The tightness of the crude bounds in the previous subsections is well-known. We now refine these bounds and derive a uniform asymptotic approximate for G n,m . For convenience, let Λ(z) := log G m (z) and write the Taylor expansion

Λ(e -ρ(1+it) ) = k 0 Λ k (ρ) k! (-it) k , with Λ k (ρ) := ρ k 0 j k k j e -jρ Λ (j) (e -ρ ), (55)
where k j denotes Stirling numbers of the second kind. In particular, Λ 1 (ρ) = ρe -ρ Λ (e -ρ ), and Λ 2 (ρ) = ρ 2 e -ρ Λ (e -ρ ) + e -2ρ Λ (e -ρ ) .

As we will see below, each Λ k (ρ) is of the same order as Λ(e -ρ ) = log G m (e -ρ ).

Theorem 4.1. Uniformly for m 1

G n,m = ρe nρ G m (e -ρ ) 2πΛ 2 (ρ) 1 + O Λ 2 (ρ) -1 , (56)
where ρ > 0 solves the equation

nρ -Λ 1 (ρ) = 0, or n = -∂ τ log G m (e -τ ) τ =ρ . (57)
The extra factor ρ in (56) is cancelled out with a factor ρ 2 in Λ 2 (ρ).

We will prove Theorem 4.1 in Section 4.5. The justification of the finer saddle-point approximation (55) consists of the following two propositions, which will be proved in Sections 4.3 and 4.4, respectively.

Another interesting use of (59) is the following very effective way of computing G n,m , with only weak dependence on m. Corollary 4.6. For m 1, G n,m satisfies G 0,m = 1 and for n 1

G n,m = 1 n 1 k n G n-k,m d | k [z d ]zU m (z),
where

[z d ]zU m (z) =            d 2 + dm 4 1 + (-1) d/m 2 d m -1 , if d is odd; dm 4 1 + (-1) d/m 2 d m -1 , if d is even, d 2m; d, if d | 2m. (64)
Proof. Since (1 -x)/(1 + x) = 1 -2x/(1 + x), we have, by a direct expansion,

V m (z) = m 4 d 1 1 + (-1) d/m 2 d m -1 z d . (65)
Now taking derivative with respect to z and then multiplying by z on both sides of (59) give

zG m (z) = G m (z) 1 z U m (z ),
or, taking coefficient of z n on both sides yields

G n,m = 1 n 1 k n G n-k,m [z k ] 1 z U m (z ) = 1 n 1 k n G n-k,m d | k [z d ]zU m (z).
By (63) and (65), we then deduce (64).

We now focus on uniform bounds for |V m (e -ρ-it )|.

Proposition 4.7. For any 3 m n and ρ → 0 + ,

|V m (e -ρ-it )| V m (e -ρ )    1 -cρ -2 t 2 , if |t| ρ; 7 8 , if ρ |t| π. (66)
Before the proof, we observe that V m (z) admits the partial fraction expansion,

V m (z) = m 4(1 -z) + 1 j m e 2 m,j m(1 -e m,j ) 2 (e m,j -z)
, with e m,j := e (2j+1)πi/m , which shows the subtlety of estimating

V m (e -ρ-it ) = e -ρ 2(1 -2e -ρ cos t + e -2ρ ) 1 -2e -mρ cos(mt) + e -2mρ 1 + 2e -mρ cos(mt) + e -2mρ . (67)
Proof. Our proof of (66) is long and divided into several parts.

we obtain

|υ(m(ρ + it))| υ(mρ) 1 + e -mρ (1 -cos(mt)) 4 3(1 + e -mρ ) 2 + 1 (1 -e -mρ ) 2 + 4e -2mρ (1 -cos(mt)) 2 3(1 -e -2mρ ) 2 ,
and then, by (68),

|V m (e -ρ-it )| V m (e -ρ ) 1 + Υρ -2 t 2 1 + ρ -2 t 2 (1 + O(t 2 )),
where Υ = Υ(ρ, t) is defined as

Υ(ρ, t) := ρ 2 t -2 e -mρ (1 -cos(mt)) 4 3(1 + e -mρ ) 2 + 1 (1 -e -mρ ) 2 = 1 -cos(mt) (mt) 2 /2 • e -mρ 2(mρ) 2 3(1 + e -mρ ) 2 + (mρ) 2 2(1 -e -mρ ) 2 .
Since (1 -cos t)/(t 2 /2) 1 for all t ∈ R and

max x 0 e -x 2x 2 3(1 + e -x ) 2 + x 2 2(1 -e -x ) 2 < 0.65, we have |V m (e -ρ-it )| V m (e -ρ ) 1 + 0.65ρ -2 t 2 1 + ρ -2 t 2 (1 + O(t 2 )) 1 -cρ -2 t 2 , (71)
for |t| ρ, where 0 < c < 0.35.

A uniform bound when ρ |t| π and mρ > π. In this case, we follow the same procedure as above, noting that 2e -mρ (1 + e -mρ ) 2 (1 -cos(mt)) 4e -mρ (1 + e -mρ ) 2 < 0.19 < 0.3, when mρ > π and |t| π. Then

|υ(m(ρ + it))| υ(mρ) 1 + 2e -mρ 4 3(1 + e -mρ ) 2 + 1 (1 -e -mρ ) 2 + 4e -2mρ (1 -cos(mt)) 2 3(1 -e -2mρ ) 2 < 1.25.
This, together with (69), gives

|V m (e -ρ-it )| V m (e -ρ ) < 1.25 2 = 5 8 , (72)
when mρ > π and ρ |t| π.

A uniform bound when ρ |t| π and mρ π. In this case, 1/(1 -z) 2 has a double pole at z = 1, while (1 -z m )/(1 + z m ) has simple poles at z = e t j i for -m/2 j m/2 , where t j := (2j -1)π/m. Since 1/|1 -e -ρ-it | 2 is monotonically decreasing in |t| when |t| π and |υ(m(ρ + it))| reaches the same maximum at t = t j for all j, we then deduce that

max ρ |t| π |V m (e -ρ-it )| max{|V m (e -ρ-iρ )|, |V m (e -ρ-it 1 )|},
where t 1 = π/m ρ when mρ π. By (71), we have

|V m (e -ρ-iρ )| V m (e -ρ ) 1.65 2 (1 + O(ρ 2 )) < 7 8 .
On the other hand, when t = t 1 ,

|υ(m(ρ + it 1 ))| υ(mρ) = (1 + e -mρ ) 2 (1 -e -mρ ) 2 .
It follows, by (68), that

|V m (e -ρ-it 1 )| V m (e -ρ ) = (1 + e -mρ ) 2 (1 + π 2 (mρ) -2 )(1 -e -mρ ) 2 (1 + O(t 2 1 )) < 7 8 ,
when mρ π, since the value of the monotonic function

x → (1 + e -x ) 2 (1 + π 2 x -2 )(1 -e -x ) 2 ,
lies between 4/π 2 and 0.6 when x ∈ [0, π]. Summarizing, we proved that, for ρ |t| π,

|V m (e -ρ-it 1 )| V m (e -ρ ) 7 8 , (73)
whether mρ π or mρ > π.

By collecting the estimates (71), (72), and (73), we obtain (66) and complete the proof of the uniform bounds. By (71), for some constants c, c > 0, we have

J 1 = O ρ δρ e -cVm(e -ρ )t 2 /ρ 2 dt = O ρe -cVm(e -ρ )δ 2 = O ρe -c (nρ) 1/5 .
On the other hand, by (73), J 2 is bounded above by

J 2 = O e -cVm(e -ρ ) = O e -c nρ . ( 74
)
This completes the proof of Proposition 4.2.

4.4 Asymptotic nature of the expansion (55): proof of Proposition 4.3

We now prove Proposition 4.3 from which the asymptotic approximation (56) will then follow. We begin with the following uniform estimates for log G m (e -τ ). On the other hand, if mρ 1, then, by (37) using the expressions in ( 15), ( 39), ( 43) and (53), we deduce that

log G m (e -τ ) = ζ(3) -2η 2 (mτ ) 2τ 2 + π 2 24τ + log τ 24 + ζ (-1) 2 - log 2 4 + τ 48 - 5η 1 (mτ ) 12 + 1 2 p(e -mτ ) + O |τ | 2 + m -2 , (76)
where many terms in m /τ + log g m + φ m τ are cancelled with the corresponding ones in (53). Thus, by [START_REF] Gordon | Notes on plane partitions[END_REF], we have log G m (e -τ ) = O(|τ | -2 ).

Lemma 4.9. For k 0, we have, uniformly for |t| = O(ρ),

|Λ (k) (e -ρ-it )| = O ρ -k Λ(e -ρ ) .
Proof. We apply a standard argument (or Ritt's Lemma; see [19, § 4.3]) for the asymptotics of the derivatives of an analytic function in a compact domain, starting from the integral representation

Λ (k) (e -ρ-it ) = k! 2πi |w-e -ρ-it |=cρe -ρ Λ(w) (w -e -ρ-it ) k+1 dw,
where c > 0 is a suitably chosen small number. Then, since ρ → 0, we see that

Λ (k) (e -ρ ) = O ρ -k max |θ| π |Λ(e -ρ-it (1 + cρe iθ ))| = O ρ -k max |θ| π |Λ(e -ρ-it+cρe iθ )| .
By choosing c sufficiently small, the circular range specified by ρ + it -cρe iθ for |θ| π is covered in the cone |t| = O(ρ), and we can then apply the bounds for Λ given in (75).

Proof. (Proposition 4.3) Lemma 4.9 implies, by the definition (55), that

Λ k (ρ) Λ(e -ρ ) = log G m (e -ρ ), (k = 1, 2, . . . ).
Thus the Taylor expansion (55) is also an asymptotic expansion when |t| → 0.

4.5

The saddle-point approximation. Then by the expansion (55), Proposition 4.3 and the estimate in Lemma 4.9, we have

1 2π δρ -δρ e n(ρ+it) G m (e -ρ-it ) dt = ρe nρ G m (e -ρ ) 2π δ -δ exp it(nρ -Λ 1 (ρ)) - Λ 2 (ρ) 2 t 2 + Λ 3 (ρ) 6 (-it) 3 + O(Λ(e -ρ )t 4 ) dt.
Choose ρ > 0 to be the solution of the equation ( 57), which exists by the estimates in (75). Then take δ as we described above, namely, Λ 2 (ρ)δ 2 → ∞ and Λ 2 (ρ)δ 3 → 0. The evaluation of the integral is then straightforward, and omitted.

Remark 1. The same calculations lead indeed to an asymptotic expansion of the form

G n,m ∼ ρe nρ G m (e -ρ ) 2πΛ 2 (ρ) 1 + j 1 γ j (ρ)Λ 2 (ρ) -j ,
for some (messy) coefficients γ j (ρ) depending on ρ. In particular (for simplicity, Λ j = Λ j (ρ)),

γ 1 (ρ) = 3Λ 2 Λ 4 -5Λ 2 3 24Λ 2 2
, and

γ 2 (ρ) = -24Λ 3 2 Λ 6 + 168Λ 2 2 Λ 3 Λ 5 + 105Λ 2 2 Λ 2 4 -630Λ 2 Λ 2 3 Λ 4 + 385Λ 4 3 1152Λ 4 2
.

Phase transitions

Based on the less explicit saddle-point approximation (56), we now derive more precise asymptotic estimates according to the relative growth rate of m with n 1/3 , which prove Theorem 1.1.

Subcritical phase

: m = o(n 1/3 (log n) -2/3 )
We consider here m in the range

3 m m -, with m -:= 6π 2/3 n 1/3 (log n -1 2 log log n + log ω n ) 2/3 , (77)
for any sequence ω n tending to infinity; compare (54).

Proposition 5.1. If m lies in (77), then

G n,m ∼ g m √ m 2 √ π n e 2 √ m(n+φm) ∼ g m √ πm 4 √ 6 n e 2 √ m(n+m 3 /96) , (78)
where g m , m and φ m are defined in [START_REF] Knuth | Big omicron and big omega and big theta[END_REF]. If m → ∞ and still lies in the interval (77), then

G n,m ∼ c 1 n -1 m 23/24 e -c 2 m 2 +2 √ m(n+m 3 /96) , with (c 1 , c 2 ) := e ζ (-1)/2 π 1/24 2 67/24 √ 3 , - 7ζ(3) 8π 2 .
Proof. When 3 m m -, log G m (e -ρ ) satisfies, by (37) together with the expressions in ( 15), ( 43) and ( 52),

log G m (e -ρ ) = m ρ + 1 2 log ρ + log g m + φ m ρ + O m 2 ξ 2 (mρ) , (79)
where m 2 ξ 2 (mρ) m 2 e -2π 2 /(mρ) , and the saddle-point equation has the form (by an argument similar to the proof of Lemma 4.9 using (37))

n + φ m = m ρ 2 - 1 2ρ + O m 3 ξ 2 (mρ) . (80)
Asymptotically, we have, by a direct bootstrapping argument,

ρ = m n + φ m + O n -1 + m 1/2 n -1/2 e -4 √ 6πn 1/2 /m 3/2 . (81)
Then the upper limit m -of m in (77) implies that the O-terms in the above three equations are all of order o(1); in particular,

m 3 ρξ 2 (mρ) mρ -1 e -2π 2 /(mρ) = Θ(ω -2/3 n ) → 0, m 2 ξ 2 (mρ) m 2 e -2π 2 /(mρ) = o mρ -1 e -2π 2 /(mρ) = o ω -2/3 n .
[This range is slightly smaller than (54) because we need an expansion for nρ up to o(1) error, or

(n + φ m )ρ = m /ρ -1/2 + o (1)
.] Substituting this choice of ρ and using (80) into (79), we have

nρ + log G m (e -ρ ) = m ρ + 1 2 log ρ + log g m + (n + φ m )ρ + o(1) = 2 m (n + φ m ) + 1 2 log ρ + log g m + o(1).
On the other hand, we also have ρ n+φm) , proving (78) by ( 81). The values of c 1 , c 2 are computed using (39).

2πΛ 2 (ρ) ∼ ρ 3/2 2 √ π m ; thus G n,m ∼ g m ρ 2 2 √ π m e 2 √ m (
From this estimate, it is straightforward to show that (4) holds only when m = o n 1/7 :

where the constants (c, β 1 , β 2 ) are defined in [START_REF] Flajolet | Analytic combinatorics[END_REF].

In particular, the growth of the number of BPPs when their widths get close to the typical length behaves asymptotically like a Gumbel distribution. Along with [START_REF] Flajolet | Mellin transforms and asymptotics: harmonic sums. Theoret[END_REF], the ratio between G n,m and G n,n thus has the form (90).

Similar to Theorem 1.1 in [START_REF] Pittel | On dimensions of a random solid diagram[END_REF], we may conclude that there is an exponential decay of the number of BPPs of size n and width m when m is close to the typical width, which is of order Θ(n 1/3 log n). See [START_REF] Erdös | The distribution of the number of summands in the partitions of a positive integer[END_REF] for a similar Gumbel limiting distribution of the largest part in random integer partitions, which is one of the first results of this type, and also [START_REF] Mutafchiev | The size of the largest part of random plane partitions of large integers[END_REF] for the same phenomenon in random ordinary plane partitions.

Phase transitions in m-rowed plane partitions

Our method of proof extends to some other classes of plane partitions. For simplicity, we only consider briefly in this section plane partitions with m rows, which has the known generating function (see [START_REF] Andrews | The theory of partitions[END_REF]) For 2 m 9, these partitions appear in OEIS with the following identities.

Figure 1 :

 1 Figure 1: Two instances of banded plane partition of size 40 and width 4 (with and without the outer banded staircase).

Figure 2 :

 2 Figure 2: Example of the bijection between BPPs with m n and column-strict plane partitions: (a) a BBP f with m n, (b) the associated plane partition g, (c) the column-strict plane partition obtained by taking the conjugate partition of each row of g.

Figure 3 :Theorem 1 . 1 .

 311 Figure 3: A plot of the increasing function G(α).

1 2 ζ

 2 (-1) -11 24 log π -7 24 log 2 and B j = B j (0) denote the Bernoulli numbers. Proof. Starting from the definition of g m in (15), we write log g m as log g m = -m 2 -3m + 4 4 log(2π) + S m , where S m := 1 j<2m

) Proposition 3 . 4 .

 34 Uniformly for |τ | → 0 in the half-plane Re(τ ) > 0, λ m e -4π 2 /τ = 1 + O e -Re(2π 2 /τ ) m 2 ξ 2 (mτ ) + ξ 1 (mτ ) + O m -2 |ξ 0 (mτ )| ,

  Proof. (Proposition 4.2: smallness of the integral over δρ |t| π) By (60), we obtain δρ |t| π e n(ρ+it) G m (e -ρ-it ) dt = O e nρ G m (e -ρ ) m (e -ρ ) + V m (e -ρ-it ) dt =: O e nρ G m (e -ρ )(J 1 + J 2 ) .

Lemma 4 . 8 .

 48 Let τ = ρ + it. Then, uniformly for ρ → 0 and |t| = O(ρ) in the half-plane ρ > 0, log G m (e -τ ) = O(m/|τ |), if mρ 1, O(|τ | -2 ), if mρ 1. (75) Proof. If mρ 1, then, by (43) and (52), we obtain κ m e -4π 2 /τ + λ m e -4π 2 /τ = O m 2 e -Re(2π 2 /(mτ )) = O m 2 e -c/(mρ) , which is obviously O(m/|τ |). Now, by (37) and the asymptotic expansion (39), we have log G m (e -τ ) = O(m/|τ | + m 2 + m 3 |τ |) = O(m/|τ |), since m|τ | = O(1).

Theorem 4 .

 4 1 is a direct consequence of Propositions 4.2 and 4.3. Proof. (Theorem 4.1) By (58), we obtain G n,m = 1 2π δρ -δρ e n(ρ+it) G m (e -ρ-it ) dt + O e nρ G m (e -ρ )e -c (nρ) 1/5 .

m m 3 n√ π 2 m 2 √m(n+φm) - 2 mProposition 5 . 2 .

 322252 -1/2 /192+O(m 13/2 n -3/2 ) .A connection to the modified Bessel functions. By the same analysis used in the proof of Proposition 4.2 (see (74)), we haveG n,m = 1 2πi ρ+iρ ρ-iρ e nτ G m (e -τ ) dτ + O e nρ G m (e -ρ )e -c nρ .The integral on the right-hand side is indeed well-approximated by the modified Bessel function when 3 m m -(see (77)). By (14) and (52),e nτ G m (e -τ ) dτ = g m 2πi ρ+iρ ρ-iρ √ τ e (n+φm)τ + m/τ 1 + O me -Re(2π 2 /(mτ )) dτ = g m 2πi H √ τ e (n+φm)τ + m/τ dτ + O me -Re(2π 2 /(mτ )) + e -cnρ ,where H denotes a Hankel contour, which starts from -∞, encircles around the origin counterclockwise, and then returns to -∞ (the exact shape being immaterial). The last integral over H is nothing but the modified Bessel function:G n,m ∼ g m 2πi H √ τ e (n+φm)τ + m/τ dτ = g m j 0 j m (n + φ m ) j+3/2 j!Γ(j -1/2) = g m (n + φ m ) -3/2 4 (n + φ m ) -1 e (n + φ m ) + 1 e -2 √ m(n+φm) ,which holds as long as 3 m m -. (Numerical fit of the last expression is very satisfactory.)5.2 Supercritical phase: m n 1/3 log nWe now consider m in the following stationary rangem m + , with m + := n ζ(3) 1/3 2 3 log n + log log n + ω n ,(83)for any sequence ω n tending to infinity with n. If m + m n, then G n,m ∼ G n,n ∼ cn -49/72 e β 1 n 2/3 +β 2 n 1/3 ,

Proof. 2 ,

 2 For this range of m, we have, by (76) and the definition of η d in[START_REF] Gordon | Notes on plane partitions[END_REF],log G m (e -ρ ) =ζ-2 η 2 (mρ) + e -mρ + ρ ρ (η 2 (mρ)/ρ 2 ) + me -mρ + ρ .

Corollary 5 . 6 .

 56 Assume that m satisfies m G n,n = exp -e -x 1 + O n -1/3 log n ,(90)uniformly for x = o(log n).Proof. By a standard bootstrapping argument applied to[START_REF] Han | Skew doubled shifted plane partitions: calculus and asymptotics[END_REF], we have, for large α,r = ζ(3) 1/3 1 -ζ(3) 1/3 α + 2 3ζ(3) e -ζ(3) 1/3 α 1 + O (1 + α 2 )e -ζ(3) 1/3 α .

n 0 H 1 1k<m 1 -

 011 n,m z n = k -z k -min{k,m} = P (z) m Qm (z) = exp 1 Ũm (z ) ,where H n,m denotes the number of m-rowed plane partitions of n, P is given in (3), andQm (z) := 1 z k m-k , and Ũm (z) := z(1 -z m ) (1 -z) 2 .

1 1

 1 simplicity, we only describe the transitional behavior of log H n,m . Define η(t) := j -e -jt j 3 . (91)

  In addition to Mellin transforms, we need some properties of ζ(s, b) and the Gamma function Γ(s); see, for example, [2, Ch. 12], [4, Ch. 1] or [22, Chs. XII & XIII]. Since P (e -τ ) satisfies (

Acknowledgements

We thank the two referees for their valuable comments and suggestions that substantially improved the paper.

 [START_REF] Andrews | The theory of partitions[END_REF]This work was partially supported by a joint FWF-MOST project under the Grants I2309 (FWF) and 104-2923-M-009-006-MY3 (MOST).

2 Also partially supported by an Investigator Award from Academia Sinica under the Grant AS-IA-104-M03 and a joint ANR-MOST project under the Grants MOST 105-2923-E-001-MY4. 1

Proposition 4.2. Let δ := (nρ) -2/5 > 0. Then for a certain constant c > 0, δρ |t| π e n(ρ+it) G m (e -ρ-it ) dt = O e nρ G m (e -ρ )e -c (nρ) 1/5 .

(58) Proposition 4.3. Let δ := (nρ) -2/5 > 0. Then, uniformly for |t| δ, the Taylor expansion (55) is itself an asymptotic expansion as |t| → 0.

Note that δ = (nρ) -2/5 > 0 is a specially tuned parameter, chosen in the standard way such that (nρ)δ 2 → ∞ and (nρ)δ 3 → 0.

Proof. By [START_REF] Mutafchiev | The size of the largest part of random plane partitions of large integers[END_REF], we have, for |z| < 1,

w m (j)

w m (j)

.

w m (j)z j .

Then (59) follows from [START_REF] Olver | Asymptotics and special functions[END_REF]. Lemma 4.5. For ρ > 0

where

Proof. Since each U m (z ) contains only nonnegative Taylor coefficients, we have, by (59),

From (59), we have the decomposition

where each term contains only nonnegative Taylor coefficients; this implies that we also have

from which (60) follows.

Growth order of V m (e -ρ ). By the definition (61) of V m (z), we easily obtain the estimates

In all cases, we have V m (e -ρ ) nρ.

Uniform bounds for |z/(1-z) 2 |. We consider first the modulus of |z/(1-z) 2 |, which is independent of m and simpler. Observe that

for -π t π. Now if |t| = O(ρ), then we have the uniform expansion

while if ρ |t| π, then, by monotonicity,

A uniform bound when |t| ρ. The other factor in (67) is more complicated. For convenience, write υ(w) := 1 -e -w 2(1 + e -w ) .

Consider first the range |t| ρ, beginning with the expression

.

When |t| ρ, we have the inequality

Then, by the inequalities

Solving asymptotically the saddle-point equation (85) gives, with

Then we obtain

).

Thus we have expansions for nρ + log G m (e -ρ ) and ρ to within an error of order o(1), which, together with the relation Λ 2 (ρ) ∼ 3ζ(3)ρ -2 , gives the same asymptotic approximation as in (6).

Critical phase: log m ∼ 1 3 log n

In this range, we begin with the expansion (76) and the approximate saddle-point equation

We recall that, in this regime, α = mn -1/3 . Define

where the η d (x) are defined in [START_REF] Gordon | Notes on plane partitions[END_REF]. We begin with two simple lemmas establishing the positivity of σ and the existence of a positive solution r of the equation R(α, r) = 0, respectively.

Lemma 5.3. The function σ(x) is positive for x > 0.

Proof. Note that σ(x) ∼ 3ζ(3) as x → ∞, and σ(x) ∼ ζ(2)x/2 as x → 0. So the monotonicity of σ(x) for x 0 follows from the identity: 4 ,

Once m is given, α = m/n 1/3 is fixed and then r can be solved from the equation R(α, r) = 0, which is nothing but [START_REF] Han | Skew doubled shifted plane partitions: calculus and asymptotics[END_REF].

Lemma 5.4. For any α > 0, the equation R(α, r) = 0 has a unique solution r > 0. Moreover, r = r(α) is increasing as a function in α.

Proof. Consider the function R(x) := ζ(3) -2η 2 (x) + xη 2 (x), which has the explicit series form

for x > 0. Thus for each fixed α > 0, the equation r 3 = R(αr) has a unique positive solution.

We now state the transitional behavior of G n,m for m n 1/3 . Proposition 5.5. Let α = mn -1/3 , where log m = 1 3 (1 + o(1)) log n. Then we have the asymptotic approximation

uniformly in m, where r is the unique positive solution of R(α, r) = 0, β 1 (α, r) = G(α) in (10):

.

The error term in (87) suggests that (87) remains valid as long as m n 1/5+ε , but outside the range m = 1 3 (1 + o( 1)) log n it is simpler to use other simpler approximations such as (78) and (84).

Proof. Write first m = αn 1/3 and

where the coefficients r j = r j (ρ, η 1 , η 2 ) can be computed as follows. Substitute first this expansion into (86), expand in decreasing powers of n, equate the coefficient of each negative power of n on both sides, and then solve for r 1 , r 2 , . . . , one after another. In this way, we obtain, for example,

The determination of further terms r j with j 4 requires a longer expansion in (86). The asymptotic estimate (87) then follows from substituting the expansion (88) into the uniform saddle-point approximation (56) and expand terms up to an error of O n -2/3 , together with the relation

The more precise error term in (87) results from computing more terms in the expansion and examining the asymptotic behaviors when αr is large and small; we omit the less interesting details. In particular,

Proof. (Sketch) We consider τ with Re(τ ) > 0. By the Euler-Maclaurin summation formula (see [START_REF] Flajolet | Analytic combinatorics[END_REF]Chapter A.7]), we obtain

which holds uniformly as long as τ → 0 and m → ∞. Then in this range

In particular, when m/n 1/3 → ∞, then η(mτ ) ∼ ζ(3) and η (mτ ) = o(1). Thus r ∼ (2ζ(3)) 1/3 , and log [z n ]P (z) m Qm (z) ∼ 3ζ(3) 1/3 (n/2) 2/3 , consistent with (2). On the other hand, when m = o(n 1/3 ), we use the asymptotic expansion

the series being convergent when |z| < 2π. Thus in this case, using the saddle point method,

The theorem is proved by examining the error terms in each case. We omit the details.

When mρ = o(1), we can write down more precise expansions, similar to (82), beginning with log Qm (e -τ ) ∼

while in the case of BPPs the corresponding expansion is a finite one (with exponentially smaller error in 1/τ ). The infinite series is divergent when m|τ | 2π. Here

is a polynomial in m of degree j + 2 and divisible by m(m -1), the B j (x) being Bernoulli polynomials; see (27). In particular,

The saddle-point equation is now of the form

Then, writing ς j (m) = mς j (m),

where r := π m/(6n) and

Thus, by the Lagrange Inversion Formula, ρ ∼ j 1

Since each d j = d j (m) is a polynomial in m of degree m -1, we see that the general term in the expansion of ρ is of the form m (3j-2)/2 /n j/2 , which, after substituting such ρ into the corresponding saddle-point approximation gives an expansion in terms of r as follows:

, where e j (m) is a polynomial of degree (3j + 4)/2. In general, if n j 0 /(3j 0 +4) m = o(n (j 0 +1)/(3j 0 +7) ), we have the asymptotic approximation

In particular, if m = o(N 1/7 ), then j 0 = 0, while if m = o(N 1/5 ), then retaining the term e 1 (m)/ √ N and dropping the remaining terms yields an error of order o(1).

Remark 2. (m-rowed plane partitions whose non-zero parts decrease strictly along each row) The generating function now has the form (see [START_REF] Gordon | Notes on plane partitions[END_REF])

where P (z) is as in (3) and Qm (z

where 1 m odd is the indicator function for m being odd. We then deduce the same type of transitional behavior as that of m-rowed plane partitions: log [z n ]P (z) m Qm (z) ∼ r + η(αr) 2r 2 n 2/3 , where η is defined in (91) and r > 0 solves the equation 2r 3 -2η(αr) + αrη (αr) = 0.

Remark 3. In a very similar manner, we can derive the phase transitions in the asymptotics of

the difference here being that for small m = O(1) the saddle-point method fails and one needs instead the singularity analysis [START_REF] Flajolet | Analytic combinatorics[END_REF] for the corresponding asymptotic approximation. Indeed, singularity analysis applies when 1 m = o(n 1/3 ):

1 k m k k ∼ n m(m+1)/2-1 Γ(m(m + 1)/2) 1 k m k k , while our saddle-point analysis applies when m → ∞. Furthermore, similar to (92), the transitional behavior is described by the function 1 -e -jαr j 3 -αre -jαr j 2 ,

where Li 2 (z) denotes the dilogarithm function, and r > 0 solves the equation 2η(αr) -2αrLi 2 (e -αr ) + (αr) 2 log(1 -e -αr ) = 0.


B La Cassir

Scola

Manon Boxberger
email: manon.boxberger@hotmail.fr

Angéline Antezack
email: angeline.a@wanadoo.fr

Sibylle Magnien
email: sibylle.magnien@gmail.com

Nadim Cassir
email: cassirnadim@gmail.com

Bernard La Scola
email: bernard.la-scola@univ-amu.fr

Draft genome and description of Corynebacterium haemomassiliense strain Marseille-Q3615 T sp. nov., a new bacterium isolated from a 59-year-old man suffering from chronic obstructive pulmonary disease symptoms

Keywords: Bacteria, Corynebacterium, human, genome, species, sp. nov., taxonogenomics

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

Introduction

The genus Corynebacterium counts 173 species [START_REF] Parte | LPSN -List of Prokaryotic names with Standing in Nomenclature (bacterio.net), 20 years on[END_REF], some of them are of medical, veterinary or biotechnological interest [START_REF] Oliveira | Insight of Genus Corynebacterium: Ascertaining the Role of Pathogenic and Non-pathogenic Species[END_REF]. Corynebacterium haemomassiliense strain Marseille-Q3615 T was isolated as part of the diagnosis in IHU-Méditerrannée Infection Institute in Marseille (France). A taxono-genomics approach-including matrix-assisted laser desorption-ionization time-of-flight mass spectrometry (MALDI-TOF MS), phylogenetic analysis, main phenotypic description and genome sequencing-was used to describe this species [START_REF] Lagier | Culturing the human microbiota and culturomics[END_REF][START_REF] Ramasamy | A polyphasic strategy incorporating genomic data for the taxonomic description of novel bacterial species[END_REF]. The genome of Corynebacterium haemomassiliense strain Marseille-3615 T is 2.578.128 bp long with 65.28% G+C content. This new bacterium is most closely related to Corynebacterium pilbarense strain DSM 45350 with a 16S rRNA sequence identity value of 99.12 %. Furthermore, genomic comparison using the OrthoANI parameter provided a value of 73.65 % with Corynebacterium striatum strain KC-NA01 (NZ_CP014634.1) and a dDDH value of 43.3 % with Corynebacterium afermentans strain DSM 44280, closest species standing in nomenclature.

Material and method

Strain isolation and phenotypic tests

Corynebacterium haemomassiliense strain Marseille-Q3615T was initially isolated from a liquid aerobic hemoculture bottle (BACT/ALERT®, bioMe ŕieux, Marcy l'Etoile, France) incubated 24h at 37°C, and is routinely cultivated on Columbia Agar with 5% Sheep Blood media (BioMérieux, Marcy L'Etoile, France) incubated in aerobiosis at 37°C. MALDI-TOF mass spectrometry (MS) protein analysis was carried out using a Microflex spectrometer (Bruker Daltonics, Bremen, Germany) [START_REF] Lee | OrthoANI: An improved algorithm and software for calculating average nucleotide identity[END_REF]. Spectra from strain Marseille-Q3615 T (Fig. 1) were imported into the MALDI BioTyper software (version 3.0, Bruker) and analyzed by standard pattern matching (with default parameter settings). The study was validated by the ethics committee of the Institut Fédératif de Recherche IFR48. Different growth temperatures (20; 31.5; 37; 45 and 56°C), atmosphere conditions, anaerobic, aerobic and microaerophilic (CampyGEN, Oxoid, USA) and pH (5,5; 6.5; 7.5; 8.5) were tested. API ZYM, API Coryne and API 50 CH strips (BioMérieux, Marcy L'Etoile, France) were used to evaluate the biochemical properties of the strain according to the manufacturer's instructions. For scanning electronic microscopy, a colony was collected from agar and immersed into a 2.5 % glutaraldehyde fixative solution. The slide was gently washed in water; air dried and examined with approximately 60 centimeters in height and 33 cm in width to evaluate bacterial structure on a TM4000 microscope (Hitachi, Tokyo, Japan). Motility test was performed using the semi solid TCC media as described by Tittsler et al. [START_REF] Tittsler | The Use of Semi-solid Agar for the Detection of Bacterial Motility[END_REF]

Genome sequencing

Genomic DNA (gDNA) of Corynebacterium haemomassiliensis strain Marseille-Q3615 T was extracted the EZ1 biorobot (Qiagen) with EZ1 DNA tissues kit.after a mechanical and enzymaic pretreatment -respectively by glass beads acid washed (G4649-500g Sigma) using a FastPrep-24™ 5G Grinder (mpBio) and lysozyme incubation at 37°C . gDNA was next sequenced on the MiSeq Technology (Illumina Inc, San Diego, CA, USA) with the paired end strategy using the Nextera XT DNA sample prep kit (Illumina). Purification step was performed using AMPure XP beads (Beckman Coulter Inc, Fullerton, CA, USA) and libraries were normalized according the Nextera XT protocol (Illumina). They were pooled into a single library for sequencing on the MiSeqAutomated cluster generation and paired end sequencing with dual index reads were performed in a single 39-hours run in 2x250-bp. Total information of 4.8 Gb was obtained from a 511/mm 2 cluster density with a cluster passing quality control filters of 90.7%. Within this run, the index representation for Corynebacterium haemomassiliensis Marseille-Q3615 T was determined to index 4.8%. The 9 843 335 paired end reads were filtered according to the read qualities.

Phylogeny, Genome annotation and genome comparison

Assembly was performed using SPAdes software v3.10 using default parameters. [START_REF] Bankevich | SPAdes: A New Genome Assembly Algorithm and Its Applications to Single-Cell Sequencing[END_REF] Genome annotation was obtain through the NCBI Prokaryotic Genome Annotation Pipeline [START_REF] Tatusova | NCBI prokaryotic genome annotation pipeline[END_REF]. By extracting the sequence,a 16S rRNA based phylogenetic tree was obtained using the Maximum Likelihood method parameter within the MEGA 7 software [START_REF] Abdallah | The impact of culturomics on taxonomy in clinical microbiology[END_REF]. The Genome-to Genome Distance Calculator (GGDC) web server (http://ggdc.dsmz.de ) was used to estimate the overall identity among compared genomes and to replace the wet-lab DNA-DNA hybridization (DDH) by a digital DDH (dDDH) [15,16]. The degree of genomic identity of Corynebacterium haemomassiliense strain Marseille-Q3015 T with closely related species was estimated using the OrthoANI software [START_REF] Lee | OrthoANI: An improved algorithm and software for calculating average nucleotide identity[END_REF]. Antibiotic resistance genes and presence of pathogenesis-related proteins was investigated using the ABRicate tool and CARD, Resfinder, VFDB and PlasmidFinder databases of the Online Galaxy platform. [START_REF] Afgan | The Galaxy platform for accessible, reproducible and collaborative biomedical analyses: 2016 update[END_REF]

Results

Strain identification and classification

Corynebacterium haemomassiliense strain Marseille-Q3615 T was isolated from a blood specimen of a 59-year-old man suffering from chronic obstructive pulmonary disease symptoms. This strain failed to be identified by our systematic MALDI-TOF MS screening, suggesting that the corresponding species was not in the database -https://www.mediterraneeinfection.com/acces-ressources/base-de-donnees/urms-data-base/. By analyzing its conserved sequences, Corynebacterium haemomassiliense strain Marseille-Q3615 T exhibited a 99.12% 16S gene sequence identity with Corynebacterium pilbarense strain DSM 45350 (NR_116953.1), the phylogenetically closest bacterium standing in nomenclature and a 95.60% rpoB gene -shown to be more discriminant for Corynebacterium species, validating the <96.6% identity cuttoff described by Khamis et al. [START_REF] Khamis | rpoB Gene Sequencing for Identification of Corynebacterium Species[END_REF]-sequence identity with Corynebacterium ureicelerivorans strain DSM 45051 (CP_009215.1) (Fig. 2). Digital DNA-DNA hybridization analysis between the novel organism with Corynebacterium afermentans strain DSM 44280 type strain revealed an identity of only 43.3 % and OrthoANI parameter provided a value of 73.65 % with Corynebacterium aurismucosum strain ATCC 700975. These both values are below the species delineation cut off. [START_REF] Abdallah | The impact of culturomics on taxonomy in clinical microbiology[END_REF]

Phenotypic characteristics

Colonies from strain Marseille-Q3615 T showed a white pigmentation and no hemolysis. Bacterial cells were Gram-positive, non-motile, rod-shaped bacilli with a size of 1.8 x 0.2 µm determined by electronic scanning microscopy (Fig. 3). Strain Marseille-Q3615 T is a facultative aerobe. Optimal growth medium pH and NaCl concentration is comprised between 5,5-8,5 and 10-15 g/L respectively. The sporulation test (20 minutes at 80°C) was negative.

Using an API strips (BioMérieux, Marcy L'Etoile, France) positive reactions were shown for Pyrazinecarboxamide, 2-naphthyl-phosphate, D-glucose, D-ribose, D-saccharose (sucrose), Alkaline phosphatase, Esterase (C4), Esterase Lipase (C8), Acid phosphatase, Naphthol-AS-BI-phosphohydrolase, N-acetyl--glucosaminidase. All other reactions tested were negative.

In addition, this bacterium shows catalase positive and oxidase negative. Table .1 synthetized the main characteristic of the strain compared to its relative species standing in nomenclature.

Genome properties

The genome size of strain Marseille-Q3615 T is 2,578,128-bp long with a 65.28% G+C content. The genome de novo assembly of this strain was achieved on 4 contigs. (Fig. 4) Of the 2.431 predicted genes, 2,365 were protein-coding genes and 66 were RNAs (4 16S rRNA, 4 additional 5S rRNAs, 4 additional 23S rRNAs 3 ncRNA and 51 tRNAs). The genome properties and distribution of genes into COGs functional categories are detailed in Table 2.

The in silico resistome of the strain Marseille-Q3615 T shows 3 genes erm(X)_4, tet(W)_4 and cmx_1with high identity percentage (94.85%, 99.01% and 99.83% respectively) that could be involved in tetracycline and chloramphenicol resistance. Neither associated plasmid nor virulence factor was found. OrthoANI parameter provided a value of 73.65 % with Corynebacterium striatum strain KC-NA01 (Fig. 5) and a dDDH value of 43.3 % with Corynebacterium afermentans strain DSM 44280.

Discussion and conclusion

Using the taxono-genomics concept i.e. the combination of the genomic and phenotypic properties of a putative new taxon, we have characterized a new bacterial species representing a new species within the family Corynebacteriaceae found in human.(Table .3) It was named as Corynebacterium haemomassiliense strain Marseille-3615T. Gr. fem. n. korynê, a club; L. neut. n. bacterium, a rod, and in biology a bacterium (so called because the first ones observed were rod-shaped); N.L. neut. n. Corynebacterium, a club bacterium.

Haemomassiliense, blood (Lat. transliteration haema) referring to the nature of the specimen. massiliense, 'to Massilia,' the antic name of Marseille, France, where the strain was isolated.

Deposit in culture collections and Sequences Database

Corynebacterium haemomassiliense strain Marseille-Q3615 T , was deposited in CSUR collection under accession CSUR-Q3615. This Whole Genome Shotgun project has been deposited at GenBank under the accession JACDTZ000000000. 16S gene sequence has been deposited under the accession MT772001.

List of abbreviation

API -Analytical

  Profile Index CARD -Comprehensive Antibiotic Resistance Database CSUR -Collection de Souches de l'Unité de Rickettsies dDDH -digital DDH DDH -DNA-DNA hybridization DNA -Deoxyribonucleic acid DSM-Z -Deutsche Sammlung von Mikroorganismen und Zellkulturen GGDC -Genome-to-Genome Distance Calculator MALDI-TOF MS -Matrix Assisted Laser Desorption Ionization -Time of Flight -Mass Spectrometry NCBI -National Center for Biotechnology Information OrthoANI -Ortho -Average Nucleotide Identity rRNAs -Ribosomal ribonucleic acid VFDB -Virulence Factors DataBase Transparency declaration None to declare

Figure. 1 Figure 2 :

 12 Figure.1 MALDI-TOF MS reference mass spectrum. Spectra from 12 individual colonies of Corynebacterium haemomassiliense strain Marseille-Q3615 T were compared and a reference spectrum was generated

Figure. 3 :

 3 Figure.3: Scanning electron microscopy of Corynebacterium haemomassiliense. nov., strain Marseille-Q3615 T using a Tabletop microscope TM 4000plus (Hitachi, Tokyo, Japan). The scale bar represents 5 µm.

Figure. 4 :Figure. 5 :

 45 Figure.4:Graphical circular map of the genome from strain Marseille-Q3615 T obtained by CG view tool.[START_REF] Grant | The CGView Server: a comparative genomics tool for circular genomes[END_REF]

Table . 1

 . : Differential characteristics of Corynebacterium haemomassiliense strain Marseille-Q3615 T and closest species standing in 232

		D-maltose	-	-	-	-	-	+
	233	nomenclature D-lactose	-	-	na	-	-	+
		G+C content (mol%) Habitat	C. haemonassili ense 65,28% Human healthy skin	C. pilbarense na ankle aspirate from a male patient	na C. ureicelerivor ans blood culture	64 mol% C. mucifaciens Human clinical material	62 mol% C. coyleae Human clinical specimens	65.1 mol% C. ihumii Fecal flora of a 62 yo male patient

Table 3 :

 3 Description of Corynebacterium haemomassiliense sp. nov. strain Marseille-Q3615 T 238 Corynebacterium haemomassiliense strain Marseille-3615T. Gr. fem. n. korynê, a club; L. neut. n. bacterium, a rod, and in biology a bacterium (so called because the first ones observed were rod-shaped); N.L. neut. n. Corynebacterium, a club bacterium. Haemomassiliense, blood (Lat. transliteration haema) referring to the nature of the specimen. massiliense, 'to Massilia,' the antic name of Marseille, France, where the strain was isolated.

	Type of description	New description
	Species name	haemonassiliense
	Genus name	Corynebacterium
	Specific epithet	Corynebacterium
	Species Status	sp. nov
	Species etymology

Acknowledgements

The authors are indebted to Ludivine Brechard for sequencing the genome and the platform of electron microscopy of IHU -MI for the electron micrographs.

Funding sources MB PhD grant is supported by the collaboration between M&L Laboratories and Aix Marseille University referenced PVM: 2018-200. This study was supported by the French State managed by the National Research Agency under the "Investissements d'avenir (Investments for the Future)" program under the reference ANR-10-IAHU-03 (Méditerranée Infection) and by the Région Provence-Alpes-Côte-d'Azur and the European funding FEDER PRIMI.

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