Thierry E Huillet

OCCUPANCY PROBLEMS RELATED TO THE GENERALIZED STIRLING NUMBERS

Keywords: Combinatorial probability, Stirling numbers, Ewens-Pitman sampling formula, random allocation of particles, integer partitions, Markov chains, forests of increasing trees

.

Introduction

We report on the importance of Stirling numbers at the heart of two processes from Combinatorics.

In Section 1, we first recall the definition of the generalized Stirling (GSN) triangle, together with examples. It depends on 3 real parameters (w 2 , α 1 , α 2 ) . It can be defined through a generating function with two indeterminate variables. Conditions under which its entries are positive are given, to be found mostly when (w 2 ≥ 0, α 1 , α 2 ≥ 0) and α := α 1 /α 2 ≤ 1.

In Section 2, in the range α ∈ [0, 1], we revisit their occurrence in the Ewens-Pitman sampling problem from a three-parameters Poisson-Dirichlet partition, say PD(w 1 , w 2 , α) , of the unit interval in the presence of missing samples, controlled by w 2 . The additional parameter w 1 > 0 is the one of Dirichlet-Ewens (often called θ in the literature). For such a model, the interval is broken into an uncountable number of pieces representing species abundance and after n hits of the interval (some of which can be missing), the problem is to estimate the number of distinct species S n that were sampled, together with the associated species counts and intervals between successive records (newly sampled species). It includes the case α = 0 which corresponds to a two-parameters Ewens' sampling formula from the Poisson-Dirichlet partition PD(w 1 , w 2 , 0) . The number S n , as a Markov chain, grows like n α when α < 1 and like log n when α = 0. The parameter α is a diversity parameter: the larger α, the larger the abundance of rare species, making them more likely to be sampled.

In Section 3, we show that: -In the range (A) when α ∈ [0, 1], the GSN triangle also occurs in the enumeration problem of the number of unordered forests of k increasing weighted ordered trees with n labelled nodes. The physical image rather is the one of a nucleation/aggregation process for which a new incoming atom may nucleate while forming the germ of a new tree or aggregate while joining one of the preexisting trees. A positive value of w 2 translates the presence in this process of a reservoir when incoming atoms get stored whenever they fail to aggregate or nucleate. We show that the number K n of the trees constituting the size-n forest goes to a finite Poisson limit when α ∈ (0, 1] and grows like log n when α = 0; the joint statistics of the connected components' sizes are of Maxwell-Boltzmann type. The random numbers (K n ) n≥0 constitute a triangular Markov sequence.

-In the range (B) when α ∈ [-1, 0], the GSN triangle occurs in the enumeration problem of the number of unordered forests of sequenced k increasing weighted ordered trees with n labelled nodes. The number K n of the trees constituting such forests grows like n -α/ (1-α) .

-In the range (C) when α < -1, the local exponential generating function (EGF) has no combinatorial interpretation in terms of a labelled weighted tree, be it increasing or not. However, the GSN triangle occurs in the set partition problem of n labelled atom into k subsets with atoms' random weight W being Gamma(-α, 1/α 2 )-distributed. Some randomness of the atoms' weight is required to interpret this case.

In an Appendix, some salient facts concerning ordered increasing trees and forests are recalled, together with a reminder on weighted versions of these.

On generalized Stirling numbers S n,k

Let (α 1 , α 2 , w 2 ) be three real numbers different from (0, 0, 0). Let If w := w 1 + w 2 , this is alternatively

[w : α 2 ] n = n k=0 S n,k [w 1 : α 1 ] k .
The S n,k obey the (Stirling's triangle) recurrence

(1)

S n+1,k = S n,k-1 + (nα 2 -kα 1 + w 2 ) S n,k
with boundary conditions S n,0 = [w 2 : α 2 ] n and S n,n = 1 for all n ≥ 0. Note the special cases: -S n,k ≡ S n,k (-1, 0, 0) = |s n,k |, the unsigned first kind Stirling numbers (counting the number of permutations of n elements with exactly k disjoint cycles, including fixed points).

-S n,k ≡ S n,k (1, 0, 0) = s n,k = (-1) n-k |s n,k |, the first kind Stirling numbers. -S n,k ≡ S n,k (0, 1, 0) = S n,k , the second kind Stirling numbers (counting the number of partitions of a set of size n into k non-empty classes).

-S n,k ≡ S n,k (0, -1, 0) = (-1) n-k S n,k .

-S n,k ≡ S n,k (-1, 1, 0) = n-1 k-1 n! k! , the Lah numbers. -S n,k ≡ S n,k (1, -1, 0) = (-1)

n-k n-1 k-1 n!
k! , the signed Lah numbers.

-S n,k ≡ S n,k (0, 0, 1) = n k , the binomial coefficients. -S n,k ≡ S n,k (-2, -1, 0) = (2 (nk) -1)!! 2n-k-1 2(n-k) = 1 2 n-k (2n-k-1)! (n-k)!(k-1)! , the square-root coefficients (see (34) below).

We shall let S 0 n,k ≡ S n,k (-α 2 , -α 1 , 0) obtained from the general S n,k when w 2 = 0.

2.1. Generalized Stirling numbers generating functions. We need to distinguish various cases, [START_REF] Hsu | A unified approach to generalized Stirling numbers[END_REF]:

• If α 1 α 2 = 0 and α 1 = α 2 , with σ n (θ) = n k=0 S n,k θ k , and τ k (z) = n≥k S n,k z n n! , the joint generating function of the S n,k 's is

Φ (z, θ) : = n,k≥0 S n,k z n n! θ k = n≥0 z n n! σ n (θ) = k≥0 θ k τ k (z) = (1 -α 2 z) -w2/α2 e θ α 1 (1-(1-α2z) α 1 /α 2 ) =: r (z) Φ 0 (z, θ) .
Note that r (z) = (1α 2 z) -w2/α2 and Φ 0 (z, θ) is the restriction of Φ (z, θ) when w 2 = 0. We shall let

(2)

φ (z) = 1 α 1 1 -(1 -α 2 z) α1/α2
be the generating function appearing in the exponential argument of Φ 0 (z, θ) =: e θφ(z) . The power-series Φ (z, θ) has horizontal generating function

(3) σ n (θ) = n! [z n ] Φ (z, θ)
and vertical generating function

τ k (z) = θ k Φ (z, θ) = r (z) 1 k! φ (z) k =: r (z) τ 0 k (z) = 1 k! (1 -α 2 z) -w2/α2 1 -(1 -α 2 z) α1/α2 α 1 k .
It holds that, with S n,0 = [w 2 : α 2 ] n = 0 if and only if w 2 = 0,

S 0 n,k = n! [z n ] τ 0 k (z) = n! k! [z n ] φ (z) k θ k σ 0 n (θ) . and S n,k = n! [z n ] τ k (z) = n! k! [z n ] r (z) φ (z) k θ k σ n (θ) .
The generating function φ (z) appearing in the exponential argument of Φ 0 (z, θ) =: e θφ(z) , to be interpreted later (in most cases) as the one of rooted weighted ordered increasing trees (defined in [START_REF] Bergeron | Varieties of increasing trees[END_REF]). Conditions under which φ (z) is a power-series with positive coefficients will be important to us (in particular when α 2 ≥ α 1 ≥ 0) in which case, Φ 0 (z, θ) will then be the generating function of a forest of such trees with θ marking their numbers.

We observe that, with β := -α 1 /α 2 , φ (z) may be written as

φ (z) = - 1 α 1 1 1 -b (z) -1 , b (z) = 1 -(1 -α 2 z) β
where b (z) is a power-series with positive coefficients if and only if α 2 > 0 and β ∈ [0, 1] . In that case, φ (z) is a power-series with positive coefficients as well, together with Φ 0 (z, θ), r (z) and Φ (z, θ), assuming θ ≥ 0. The double-generating function Φ (z, θ) of the S n,k 's is defined for θ ≥ 0 and z < z c = 1/α 2 .

By Dobiński formula, the horizontal generating function takes the form (Theorem 4: [START_REF] Hsu | A unified approach to generalized Stirling numbers[END_REF]):

σ n (θ) = e θ/α1 k≥0 (-1) k (θ/α 1 ) k k! [-kα 1 + w 2 : α 2 ] n σ n (1) = e 1/α1 k≥0 (-1) k α -k 1 k! [-kα 1 + w 2 : α 2 ] n ,
where σ n (1) is the generalized Bell number. As a result, with

-a k = α -k 1 k! and -b n,k = (-1) k α -k 1 k! [-kα 1 + w 2 : α 2 ] n = a k (-1) k [-kα 1 + w 2 : α 2 ] n =: a k c n,k , observing a l a k-l = a k k l , θ k σ n (θ) = S n,k = (a . * b n,. ) k = a k k l=0 k l c n,l ,
and we recover the following alternating sum expression of the S n,k 's:

Proposition 1. [3] If α 1 = 0, (4) S n,k = α -k 1 k! k l=0 (-1) l k l [-lα 1 + w 2 : α 2 ] n = α -k 1 α n 2 k! k l=0 (-1) l k l -l α 1 α 2 + w 2 α 2 n = (-1) n α n 2 (-1) k α k 1 1 k! k l=0 (-1) k-l k l l α 1 α 2 - w 2 α 2 n
where the last two lines require α 2 = 0 and (z

) n := z (z -1) ... (z -n + 1) = Γ (z + 1) /Γ (z -n + 1) (the falling factorials of z).
In the special case w 2 = 0, [ [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF], (1.22), page 21 and [START_REF] Charalambides | Review of the Stirling numbers, their generalizations and statistical applications[END_REF]], with S 0 n,0 = δ n,0 , for k = 1, ..., n,

S n,k = : S 0 n,k = n! k! [z n ] φ (z) k = n l=k (-1) n-k s n,l S l,k α n-l 2 α l-k 1 = (-1) n α n 2 (-1) k α k 1 n l=k s n,l S l,k (α 1 /α 2 ) l .
is an alternative developed expression to the one

(5) S 0 n,k = α -k 1 k! k l=0 (-1) l k l [-lα 1 : α 2 ] n
obtained from (4) while plugging w 2 = 0. Another (non-alternating) combinatorial expression of S 0 n,k in terms of Bell polynomials also holds (see (32) below). We can compare the expression of S n,k in (4) with the more complex convolution expression (r

n := n! [z n ] r (z) = [w 2 : α 2 ] n ) (6) S n,k = n! 1 .! r . * 1 .! S 0 .,k n = n n0=0 n n 0 r n0 S 0 n-n0,k , resulting from Φ (z, θ) = r (z) Φ 0 (z, θ). It involves a nested sum. • If α 1 α 2 = 0 and α 2 = α 1 , Φ (z, θ) = (1 -α 2 z) -w2/α2 e θz with Φ 0 (z, θ) = e θz and τ k (z) = θ k Φ (z, θ) = z k (1 -α 2 z) -w2/α2 /k! S n,k = n! [z n ] τ k (z) = n k α n-k 2 [w 2 /α 2 ] n-k S 0 n,k = δ n,k=n .
We have φ (z) = z and the trees constituting the forest are reduced to singletons.

• If α 1 α 2 = 0 and α 2 = -α 1 , Φ (z, θ) = (1 -α 2 z) -w2/α2 e θ z 1-α 2 z leading to S n,k = (-α 2 ) -k k! k l=0 (-1) l k l [lα 2 + w 2 : α 2 ] n .
If w 2 = 0 and α 2 = 1, this is an alternative alternating sum representation of the Lah numbers.

• If α 1 = 0 and w 2 , α 2 = 0 Φ (z, θ) = (1 -α 2 z) -(w2+θ)/α2 σ n (θ) = n! [z n ] Φ (z, θ) = α n 2 [(w 2 + θ) /α 2 ] n so that (7) S n,k = α n 2 θ k [θ/α 2 + w 2 /α 2 ] n = α n 2 θ k n l=0 |s n,l |(θ/α 2 + w 2 /α 2 ) l = α n-k 2 n l=k l k w 2 α 2 l-k |s n,l | -If α 1 , w 2 = 0 and α 2 = 0 Φ (z, θ) = (1 -α 2 z) -θ/α2 σ n (θ) = α n 2 [θ/α 2 ] n so that (8) S n,k = α n 2 θ k [θ/α 2 ] n = α n-k 2 |s n,k | • If α 2 = 0 and w 2 , α 1 = 0 Φ (z, θ) = e w2z e θ α 1 (1-e -α 1 z ) σ n (θ) = n! [z n ] Φ (z, θ) = e θ/α1 k≥0 (-1) k (θ/α 1 ) k k! (-kα 1 + w 2 ) n (9) S n,k = (-1) k α -k 1 k! k l=0 (-1) l k l (-lα 1 + w 2 ) n -If α 2 , w 2 = 0 and α 1 = 0 (10) S n,k = (-1) k α -k 1 k! k l=0 (-1) k-l k l (-lα 1 ) n = (-α 1 ) n-k k! k l=0 (-1) l k l (k -l) n = (-α 1 ) n-k S n,k ,
where S n,k are the second kind Stirling numbers.

2.2.

Positive generalized Stirling numbers. We will be interested in the conditions under which the generalized Stirling numbers are positive for their relation to probability theory.

Proposition 2. The generalized Stirling numbers S n,k are positive if:

(A) α 2 ≥ α 1 ≥ 0 and w 2 ≥ 0, in which case, α := α 1 /α 2 ∈ [0, 1] . (B) α 2 ≥ 0 ≥ α 1 , with β := -α 1 /α 2 = -α ∈ (0, 1] and w 2 ≥ 0. (C) α 2 ≥ 0 ≥ α 1 , with β := -α 1 /α 2 = -α > 1 and w 2 ≥ 0. (d) α 2 < 0, α 1 = mα 2 < 0 and w 2 = 0, m ∈ N.
This statement can be verified by considering the conditions under which the z m -coefficients of φ (z) themselves are non-negative, together with the ones of r (z). It is likely that the above conditions are also sufficient but we were unable to prove it.

We have split the regimes (B) and (C) for reasons to appear later. In the regimes (A) and (B) (C), the boundary regime α 1 = 0 is obtained when

α 1 → 0 in which case φ (z) = -1 α2 log (1 -α 2 z), with φ (z) = z if in addition α 2 → 0 (a degenerate limiting condition under which α 1 = α 2 = 0). If α 1 = α 2 in the regime (A), φ (z) = z. If 0 > α 1 = -α 2 in the regimes (B) (C), φ (z) = (1/ (1 -α 2 z) -1) /α 2 with φ m := m! [z m ] φ (z) = m!α m-1 2 , m ≥ 1.
In the non-degenerate condition of (A) with α 1 α 2 = 0 :

φ m := m! [z m ] φ (z) = α m-1 2 [1 -α] m-1 , m ≥ 1.
In the non-degenerate condition of (B) (C) with α 1 α 2 = 0 :

φ m := m! [z m ] φ (z) = α m-1 2 [1 + β] m-1 , m ≥ 1. Case (B) with β = 1/m, m ∈ N (α 1 < 0 and α 2 = -mα 1 > 0) is a special case for which φ (z) = m α2 (1 -α 2 z) -1/m -1 .
Case (d) is a very special case for which φ (z) = -1 mα2 ((1α 2 z) m -1), a polynomial of degree m. It is anecdotal for our purposes and will not be considered further.

For some special cases with (w 2 , α 2 , α 1 ) integers, the

S n,k = n! k! [z n ] r (z) φ (z) k are positive integers.
From the above identities, the simplest examples are: -S n,k ≡ S n,k (-1, 0, 0) = |s n,k |, the unsigned first kind Stirling numbers (counting the number of permutations of n elements with exactly k disjoint cycles, including fixed points).

-S n,k ≡ S n,k (0, 1, 0) = S n,k , the second kind Stirling numbers (counting the number of partitions of a set of size n into k non-empty classes).

-

S n,k ≡ S n,k (-1, 1, 0) = n-1 k-1 n! k! , the Lah numbers. -S n,k ≡ S n,k (0, 0, 1) = n
k , the binomial coefficients. In the sequel, it will be tacitly assumed that we work under the conditions on the parameters that make S n,k positive.

Generalized Stirling numbers and Möhle's random walk (regime (A))

If α 2 ≥ α 1 ≥ 0, w 1 > 0, w 2 ≥ 0 (condition (A)) and w = w 1 + w 2 , define, as in [START_REF] Möhle | A restaurant process with cocktail bar and relations to the three-parameter Mittag-Leffler distribution[END_REF], the random walk S n for which S 0 = 0 and ( 11)

P (S n+1 = k + 1 | S n = k) = 1 -P (S n+1 = k | S n = k) = α 1 k + w 1 α 2 n + w ,
defining the (space-time inhomogeneous) transition probabilities of a Markov chain on the nonnegative integers. Then,

P (S n = k) = S n,k [w 1 : α 1 ] k [w : α 2 ] n , k = 0, ..., n (12)
are the integrated running probabilities of the chain.

3.1. Möhle's random walk. -If α 1 = 0, with

S n,k = S n,k (-α 2 , 0, w 2 ) = α n-k 2 n l=k l k w 2 α 2 l-k |s n,l |, then P (S n = k) = w k 1 S n,k [w : α 2 ] n = w k 1 S n,k σ n (w 1 )
, k = 0, ..., n.

-If α 1 = 0, w 1 = 1,

P (S n = k) = S n,k [1 + w 2 : α 2 ] n = S n,k σ n (1) , k = 0, ..., n.
corresponding to a random walk S n as in [START_REF] Yamato | Ordered sample from two-parameter GEM distribution[END_REF].

-If α 1 = 0, w 1 = 1, α 2 = 1, w 2 = 0,

P (S n = k) = S n,k n!
showing that S n,k = S n,k (-1, 0, 0) = |s n,k |, the unsigned first kind Stirling numbers. In this case S n interprets as the distribution of the number of permutations of size n with k cycles chosen at random. If α 1 = -1, α 2 = 0, w 1 = 1, w 2 = 0,

P (S n = k) ! = S n,k /σ n (1)
showing that S n,k = S n,k (0, 1, 0) = S n,k where S n,k > 0 are the second kind Stirling numbers (with σ n (1) the Bell numbers). A probability mass function but not admissible as a model of S n because α 1 < 0 violates condition (A).

For the ratios α1k+w1 α2n+w to be transition probabilities for all n ≥ 0 and k = 0, ..., n, necessarily indeed w 1 > 0 and (α 2 ≥ α 1 ≥ 0 and w 2 > 0) or w 1 > -α 1 and (α 2 > α 1 ≥ 0 and w 2 = 0), so for a small sub-class of the positive S n,k . Note that, when α 2 > 0, we do not lose any generality in assuming α 2 = 1 since the transition probabilities may be written as ( 13)

P (S n+1 = k + 1 | S n = k) = 1 -P (S n+1 = k | S n = k) = (α 1 /α 2 ) k + (w 1 /α 2 ) n + (w/α 2 )
which is the same model with the changes

α := α 1 /α 2 , θ 1 := w 1 /α 2 , θ := (w 1 + w 2 ) /α 2 and θ 2 = θ -θ 1 = w 2 /α 2 . We note α := α 1 /α 2 ∈ [0, 1] . So in (13)
, we can take α 2 = 1 and α 1 = α ∈ [0, 1], avoiding to introduce the new variables θ 1 and θ 2 .

If w 2 = 0, w 1 = w and S n may be seen as the number of new species in a n-sample from the two-parameters stick-breaking Poisson-Dirichlet PD(α, w) partition of the unit interval, as defined in the Chinese restaurant problem [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF]. Necessarily then, P (S n = 0) = 0 for each n ≥ 1 because the first sample necessarily leads to a new sampled species (S 1 = 1 with probability one and S n+1 ≥ S n almost surely). Note the new feature emphasizing the role of w 2 > 0: if w 2 > 0

P (S n = 0) = S n,0 [w : 1] n = [w 2 : 1] n [w : 1] n = [w 2 ] n [w] n > 0!
When w 2 > 0, S n may no longer be seen, stricto sensu, as the number of new species in a n-sample from a partition of the unit interval. However ([13], Theorem 2), S n is the number of new species (excluding a fictitious 'species' 0 with 'abundance'

B 0 d =Beta(w 2 , w 1 )) in a n-sample from the 3-parameters Poisson-Drichlet partition PD(α, w 1 , w 2 ) := (B 0 , (1 -B 0 ) PD (α, w 1 )) . Note that if T 1 = inf (n ≥ 1 : S n = 1) , then (14) P (T 1 = n) = P (S n-1 = 0) P (S n = 1 | S n-1 = 0) = w 1 [w 2 ] n-1 [w] n , n ≥ 1
so that L n := S n+T1-1 (with L 1 = 1 almost surely) coincides (in law) with the number of new species from the PD(α, w 1 ) partition of the unit interval. Whenever a sample hits the 'fictitious species' 0, sampling simply fails to sample any new species: this event thus represents the possibility of a failure of the sampling process (missing sampling event). Note that P (T 1 > n) = P (S n = 0) ∼ Γ(w) Γ(w2) n -w1 as n → ∞, translating that T 1 is heavy-tailed with tail index w 1 (see [START_REF] Huillet | On Bernoulli trials with unequal harmonic success probabilities[END_REF]).

Concerning S n , observing Γ(n+w) Γ(n+w+α) (S n + w 1 /α) was a martingale, it was shown in (Theorem 3 of [START_REF] Möhle | A restaurant process with cocktail bar and relations to the three-parameter Mittag-Leffler distribution[END_REF]) that, for α ∈ (0, 1]

(15) n -α S n a.s. → Z d ∼ GM L (α, w 1 , w 2 ) as n → ∞
where GM L (α, w 1 , w 2 ) > 0 (generalized Mittag-Leffler distribution) reduces, when α = 1, to a [0, 1] -valued Beta(w 1 , w 2 ) random variable (rv) [as a balanced Pólya urn with initial balls having weights w 1 , w 2 with returned balls having weight

α 1 = α 2 = α = 1, [ 19
E (Z q 0 ) = Γ (w/α + q) Γ (w) Γ (w/α) Γ (w + αq) , q > max (-1/α, -w/α) .
The resulting moments of Z are given by

E (Z q ) = Γ (w 1 /α + q) Γ (w 1 /α) Γ (w) Γ (w + αq)
, q > max (-1/α, -w 1 /α) .

Time to

k-th success. If T k = inf (n ≥ 1 : S n = k) , then (16)
P (T k = n) = P (S n-1 = k -1) P (S n = k | S n-1 = k -1) = P (S n-1 = k -1) w 1 + α (k -1) w + n -1 , n ≥ k.
It is thus important to have an expression of

P (S n = k) = [w1:α] k [w] n S n,k and thus of S n,k ≡ S n,k (-1, -α, w 2 ).
Recalling S 0 = 0 and

P (S n+1 = k + 1 | S n = k) = 1 -P (S n+1 = k | S n = k) = w 1 + αk w + n ,
for k ∈ {0, .., n}, P (S n = k) is the k-th entry of (1, 0, ..., 0, ..)

n-1 m=0 P m where P m is the upper-diagonal transition matrix of S n at step m. For k ∈ {0, ..., n} , therefore,

P (S n = k) = (17) = * * m k k l=1 P (S m l = l | S m l -1 = l -1) k+1 l=1 m l-1 <m<m l P (S m = l -1 | S m-1 = l -1) = * * m k k l=1 w 1 + α (l -1) w + m l -1 k+1 l=1 m l-1 <m<m l w 2 -α (l -1) + m -1 w + m -1 = [w 1 : α] k * * m k k+1 l=1 m l-1 <m<m l (w 2 -α (l -1) + m -1) k l=1 (w + m l -1) k+1 l=1 m l-1 <m<m l (w + m -1) =: [w 1 : α 1 ] k * * m k N (m k ) D (m k )
where the double star-sum runs over the integers m k = (m l ; l = 1, ..., k) obeying 0 =: m 0 < m 1 < ... < m k < m k+1 := n + 1, and

N (m k ) : = k+1 l=1 m l-1 <m<m l (w 2 -α (l -1) + m -1) D (m k ) : = k+1 l=1 m l-1 <m≤m l (w + m -1) = k l=1 [w] m l [w] m l-1 [w] n [w] m k = [w] n
The latter expression of P (S n = k) translates the fact that there are k unit moves up at m k with no other moves but for this sequence. There are n k terms in this sum, due to the Fermi integer partition of k into n admissible summands each in {0, 1}:

z k n m=1 (1 + zx m ) = 0=:m0<m1<...<m k <m k+1 :=n+1 k l=1 x m l = im∈{0,1}: n m=1 im=k n m=1
x im m Note (an empty product being 1)

P (S n = n) = n-1 m=0 P (S m+1 = m + 1 | S m = m) = n-1 m=0 w 1 + αm w + m = [w 1 : α] n [w] n P (S n = 0) = n-1 m=0 P (S m+1 = 0 | S m = 0) = n-1 m=0 w 2 + m w + m = [w 2 ] n [w] n
We finally get

P (S n = k) = [w 1 : α] k [w] n * * m k N (m k ) = [w 1 : α] k [w] n S n,k
where S n,k ≡ S n,k (-1, -α, w 2 ). Therefore,

Proposition 3. With the double star-sum running over the integers

m k = (m l ; l = 1, ..., k) obeying 0 =: m 0 < m 1 < ... < m k < m k+1 := n + 1 (18) S n,k = * * m k N (m k ) = * * m k k+1 l=1 m l-1 <m<m l (w 2 -α (l -1) + m -1) is another representation of S n,k = S n,k (-1, -α, w 2 ), α ∈ (0, 1). Coming back to the full S n,k = S n,k (-α 2 , -α 1 , w 2 ) , S n,k = α n 2 * * m k k+1 l=1 m l-1 <m<m l w 2 α 2 - α 1 α 2 (l -1) + m -1 = * * m k k+1 l=1 m l-1 <m<m l (w 2 -α 1 (l -1) + α 2 (m -1)) Consistently, S n,0 = [w 2 : α 2 ] n , S n,n = 1. For instance, S 2,1 = 2w 2 + α 2 -α 1
, consistently with the first-step recurrence characterizing these S n,k 's. Recall from (4) that S n,k has an alternative expression as an alternating sum.

Corollary 1. (i) With S n,k (-1, -α, w 2 ) = * * m k k+1 l=1 m l-1 <m<m l (w 2 -α (l -1) + m -1) , for n ≥ k, (19)
P (T k = n) = P (S n-1 = k -1) P (S n = k | S n-1 = k -1) = P (S n-1 = k -1) w 1 + α (k -1) w + n -1 = [w 1 : α] k [w] n S n-1,k-1 (-1, -α, w 2 ) is a closed-form expression of the law of T k . (ii) When α ∈ (0, 1), it holds that (20) k -1/α T k → W := Z -1/α a.s., as k → ∞.
Statement (ii) follows from the almost sure (a.s.) convergence of n -α S n to the generalized Mittag-Leffler rv Z. We have

E (W q ) = Γ ((w 1 -q) /α) Γ (w 1 /α) Γ (w) Γ (w -q) , q < w 1 ,
showing that W is heavy-tailed with tail index w 1 .

Occupancy statistics (Ewens-Pitman sampling formulae).

If w 2 ≥ 0, with n 0 ∈ {0, ..., n}, n l > 0, l = 1, ..., k, and [START_REF] Kolchin | Random Mappings. Translated from the Russian[END_REF] of [START_REF] Möhle | A restaurant process with cocktail bar and relations to the three-parameter Mittag-Leffler distribution[END_REF])

k l=0 n l = n, with r n0 = [w 2 : 1] n0 = [w 2 ] n0 , (see
P (N n (0) = n 0 , N n (1) = n 1 , ..., N n (k) = n k ; S n = k) = n! k! [w 1 : α] k [w] n r n0 n 0 ! k l=1 [1 -α] n l -1 n l !
gives the joint distribution that, given the sample size is N = n, there was n 0 visits to the reservoir (accounting for early failure events of the sampling process, or missing samples) and S n = k distinct visited species with positive sample sizes (n l ; l = 1, .., k) not in the reservoir ( k l=1 n l = nn 0 ) and observed in an arbitrary way (independently of the sampling mechanism). When w 2 = 0 (w = w 1 ), r n0 = δ n0,0 and the latter expression is the Ewens-Pitman sampling formula, [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF], [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF], [START_REF] Yamato | Moments of some statistics of Pitman sampling formula[END_REF] and [START_REF] Yamato | Ordered sample from two-parameter GEM distribution[END_REF]. If in addition α = 0, the latter expression reduces to the famous Ewens sampling formula, see [START_REF] Ewens | The sampling theory of selectively neutral alleles[END_REF], [START_REF] Tavaré | Lectures on Probability Theory and Statistics[END_REF] and [START_REF] Tavaré | Multivariate Ewens distribution[END_REF]. In a PD(α, w 1 ) partition model with α > 0, the ordered abundances decay algebraically fast with their rank k. Species with small abundances are long-tailed (there are many small size groups or rare species in the model with α > 0, compared to the Ewens model when α = 0 where the ordered abundances decay geometrically fast with k). The latter sampling formula yields an occupancy statistics when uniform sampling is from a PD(α, w 1 ) partition abundance model of the unit interval. Specifically,

P (S n = k) = n! k! [w 1 : α] k [w] n r n0 n 0 ! * k l=1 [1 -α] n l -1 n l ! S n-n0,k = S n-n0,k (-1, -α, w 2 ) = n! k! r n0 n 0 ! * k l=1 [1 -α] n l -1 n l ! ,
where the star sum runs over the positive sample sizes (n l ; l = 1, .., k) obeying

k l=1 n l = n -n 0 . Note (w 2 = 0) (21) S 0 n,k = S n,k (-1, -α, 0) = n! k! * k l=1 [1 -α] n l -1 n l !
Summing over the n l > 0, l = 1, ..., k, with k l=1 n l = n-n 0 , the joint probability that, in a sample of size n, there are S n = k new sampled species and n 0 = n visits to the fictitious species is thus

P (N n (0) = n 0 ; S n = k) = n n 0 [w 2 ] n0 [w 1 : α] k [w] n S 0 n-n0,k where S 0 n,k = S n,k (-1, -α, 0) . Recalling S n,k = S n,k (-1, -α, w 2 ) is given from (6) by S n,k = n n0=0 n n 0 [w 2 ] n0 S 0 n-n0,k ,
this gives, consistently,

P (S n = k) = [w 1 : α] k [w] n S n,k , with n k=0 S n,k [w 1 : α] k = [w] n . Observing now, n-n0 k=0 S n-n0,k (-1, -α, 0) [w 1 : α] k = [w 1 ] n-n0
the probability that in a sample of size n there are n 0 failure events is thus the Beta-binomial probability mass function

P (N n (0) = n 0 ) = n n 0 [w 2 ] n0 [w 1 ] n-n0 [w] n , n 0 = 0, ..., n.
Note the conditional probabilities

P (S n = k | N n (0) = n 0 ) = [w 1 : α] k [w] n-n0 S 0 n-n0,k P (N n (0) = n 0 | S n = k) = n n 0 [w 2 ] n0 S 0 n-n0,k S n,k . If w 2 ≥ 0, with n 0 ≥ 0, n-n0 m=1 ma m = n -n 0 and n-n0
m=1 a m = k, the second alternative Ewens-Pitman Sampling formula, accounting for repetitions of occupancies, reads ( 22)

P (N n (0) = n 0 , A n (1) = a 1 , ..., A n (n -n 0 ) = a n-n0 ; S n = k) = n! [w : α] k [w] n r n0 n 0 ! n-n0 m=1 [1 -α] am m-1 m! am a m ! = n! k! [w : α] k [w] n r n0 n 0 ! k a 1 ...a n-n0 n-n0 m=1 [1 -α] am m-1 m! am .
Here, A n (m) is the number of boxes (species) visited m times by the n-sample and S n = n -N n (0) the number of visited boxes.

In particular, the distribution involving the number of singleton species is

P (N n (0) = n 0 , A n (1) = k; S n = k) = n! k! [w : α] k [w] n r n0 n 0 ! .
With i := n-n0 m=1 i m and j := n-n0 m=1 mi m , integers, given N n (0) = n 0 , the joint falling factorials of the A n (m)'s are also given by (from (4.1) in [START_REF] Yamato | Moments of some statistics of Pitman sampling formula[END_REF])

(23) E n-n0 m=1 (A n (m)) im | N n (0) = n 0 = (n -n 0 ) j (n) n0 [w] n [w] n-n0 [w : α] i [w + iα] n-n0-j [w 1 ] n-n0 n-n0 m=1 [1 -α] m-1 m! im ,
giving access to the marginals when all integers i m s are null but one and also to the two-point correlations

Corr(A n (m) , A n (m ′ )), m ′ = m.
We refer to [START_REF] Huillet | On Bernoulli trials with unequal harmonic success probabilities[END_REF] for additional aspects of this model. 4.1. The horizontal case. Suppose S n,k > 0 and let K n be the rv for which K 0 = 0 and ( 24)

P (K n = k) = S n,k n k=0 S n,k =: S n,k σ n (1) , k = 0, ..., n where σ n (θ) = n k=0 θ k S n,k . A tilted version of this distribution is (θ > 0) P θ (K n = k) = θ k S n,k n k=0 θ k S n,k =: θ k S n,k σ n (θ) , k = 0, ..., n E θ u Kn = σ n (θu) σ n (θ) , E θ (K n ) = σ ′ n (θ) /σ n (θ) ≤ ∞.
We used P 1 (K n = k) = P (K n = k). We will give below a combinatorial interpretation of this random variable as the uniform probability to observe an unordered forest of k weighted increasing ordered trees given a population of n atoms. It follows from the identity for the S n,k 's that, if

α 1 = 0, with θ := w 1 /α 1 , α := α 1 /α 2 and θ 2 := w 2 /α 2 E [w 1 : α 1 ] Kn = [w : α 2 ] n σ n (1) else E [θ] Kn = α -n α 1 [θα + θ 2 ] n σ n (1)
In the special case α 1 = 0 and w 1 ≥ 0,

E w Kn 1 = [w : α 2 ] n σ n (1) = α n 2 [w 1 /α 2 + θ 2 ] n σ n (1) = [w 1 /α 2 + θ 2 ] n [1/α 2 + θ 2 ] n yields the probability generating function (pgf) for K n , observing σ n (1) = α n 2 [1/α 2 + θ 2 ] n (see (41)). Note that with k = 0, ..., n, (25) S n,k = α n 2 w k 1 [w 1 /α 2 + θ 2 ] n = α n 2 w k 1 n l=0 |s n,l |(w 1 /α 2 + θ 2 ) l = α n-k 2 n l=k l k θ l-k 2 |s n,l | = α n-k 2 n l=k l k w 2 α 2 l-k |s n,l |
where |s n,l | are the absolute (unsigned) first kind Stirling numbers. In this case

σ n (1) σ n+1 (1) = 1 α 2 (1/α 2 + θ 2 + n) = 1 1 + w 2 + α 2 n P (K n+1 = k + 1 | K n = k) = 1 -P (K n+1 = k | K n = k) = σ n (1) σ n+1 (1)
independent of k: K n has independent but 'time-inhomogeneous' increments. As

n → ∞ ∼ Γ (1/α 2 + θ 2 ) Γ (w 1 /α 2 + θ 2 ) e -(1-w1) log n/α2 ,
suggesting that K n has a Poisson component with mean log n/α 2 but with an additive independent factor with pgf Γ(1/α2+θ2) Γ(w1/α2+θ2) , (is the reciprocal of the Gamma function a pgf??) independent of n. From (26), E w Kn

1 1/ log n → e -1/α2(1-w1) a Poisson pgf with mean 1/α 2 , suggesting (27) K n log n → 1 α 2 a.s., as n → ∞.
From (26),

K n = n m=1 I m
where (I m ; m ≥ 1) is an independent sequence of Bernoulli rv's with success probability inversely proportional to the trial's number:

P (I m = 1) = 1/α2 (1+w2)/α2+m-1 = 1 1+w2+α2(m-1) . 4.2. The transversal case. If α 1 α 2 = 0, with σ n (θ) = n k=0 S n,k θ k , and τ k (z) = n≥k S n,k z n n! , recall Φ (z, θ) = n,k≥0 S n,k z n n! θ k = n≥0 z n n! σ n (θ) = k≥0 θ k τ k (z) = (1 -α 2 z) -w2/α2 e θ α 1 (1-(1-α2z) α 1 /α 2 ) =: r (z) Φ 0 (z, θ) with σ n (θ) = n! [z n ] Φ (z, θ) and τ k (z) = θ k Φ (z, θ) = 1 k! r (z) φ (z) k .
The tilted probability to observe a population with n unlabelled (indistinguishable) atoms given it has k connected components is

P z (N k = n) = z n S n,k /n! τ k (z) , n ≥ k, E z u N k = n≥k u n P z (N k = n) = τ k (uz) τ k (z) , z < z c = 1/α 2 .
We have

E z u N k = τ k (uz) τ k (z) = r (uz) r (z) φ (uz) φ (z) k
showing that, under P z ,

N k = X 0 + k l=1 X l ,
the sum of mutually independent rv's with X 0 , X 1 , ..., X k with X 0 ≥ 0 and X 1 , ..., X k ≥ 1 independent and identically distributed (iid), having pgf's

E z u X0 = r (uz) r (z) and E z u X1 = φ (uz) φ (z) ,
respectively. Note X 0 has a negative-binomial distribution. As a result, with

E z (X 1 ) = φ ′ (z) φ(z) = 1 1-α2z 1-α1φ(z) φ(z)
and σ 2 z (X 1 ) the variance,

1 k N k → E z (X 1 ) , P z -a.s. and N k -kE z (X 1 ) σ z (X 1 ) √ k → N (0, 1) as k → ∞.
If 0 < α 2 < 1, both φ (1) < ∞ and r (1) < ∞ and the result holds under P = P 1 (the uniform case). In that case, with

E z X0 = r (z) /r (1) E z X1 = φ (z) /φ (1) two pgf's, τ k (z) = θ k Φ (z, θ) = 1 k! r (1) φ (1) k E z X0 E z X1 k τ k (1) = θ k Φ (z, θ) = 1 k! r (1) φ (1)
k Then, with N k = X 0 + X 1 + ... + X k the sum of mutually independent rv's with X 1 , ..., X k ≥ 1 iid, having pgf E z X0 and E z X1 respectively,

τ k (z) τ k (1) = E z N k and [z n ] τ k (z) τ k (1) = P (N k = n) = 1 r (1) φ (1) k k! n! S n,k .
X 0 > 0 has negative-binomial distribution. In other words, we obtained

Proposition 4. If 0 < α 2 < 1, a new representation of S n,k is (28) S n,k = r (1) φ (1) k n! k! P (N k = n) , with τ k (1) = n≥k S n,k /n! = r (1) φ (1) k /k!.

Combinatorial interpretation of S n,k

Aggregation phenomena manifest themselves on a wide variety of physical scales, ranging from the large structures of the Universe to the elementary particles and are still poorly understood. A natural way of modeling nucleation/aggregation phenomena of elementary particles into clusters is at the statistical physics level, regardless of the nature of their constitutive elements or of the properties of their binding force.

We can formulate the problem as follows: nucleation/aggregation phenomena consists of phenomena where elementary particles (atoms), possibly with a mass or weight, are given the opportunity of forming assemblies (groups or clusters). Inspecting more closely this problem amounts to asking for the connection politics of a new atom when it sees a certain previously-formed group-pattern of n similar atoms. In this sequential approach, nucleation of a cluster occurs when the inserted atom does not connect at all, whereas aggregation takes place when it joins any existing cluster [or more?]. Depending on this local logic of pattern formation, the asymptotic structure of groups (in the thermodynamic limit n → ∞) can be quite different; also the group sizes distributions may vary widely. These asymptotic studies deserve interest. We will illustrate such ideas on a monomer addition model. In this simplistic model, the additional atom, if it connects, connects to a previouslyformed single group or cluster: clusters can only grow by the iterative addition of single particles and the resulting assembly of atoms is the one of a recursive tree. We could think of a more complicate model where clusters themselves are allowed to aggregate (when the added atom is allowed to join two or more clusters thereby merging them), which should match with more realistic situations. This is ruled out in the nucleation/aggregation model described by the generalized Stirling numbers, as a result of the recurrence (1) which the S n,k satisfy: when adding an additional atom n → n + 1, the only way to get k clusters is that there were initially k of them (the new atom aggregates to one of the pre-existing clusters) or that they were k -1 (the new atom nucleates). We shall make use of the notion of generating (partition) functions from combinatorics and more precisely of the one of forests (ordered or not) of increasing ordered trees. Clusters are indeed essentially trees or sequences of trees (as a collection of trees attached to a linear open string (or chain) of labelled (distinguishable) atoms with free dangling endpoints) which are increasing due to their sequential formation in which a new atom, if it connects, connects to a randomly chosen node of a pre-existing tree. See the Appendix.

5.1. The regime (A) : α 2 > α 1 > 0, α ∈ (0, 1). The generating function Φ 0 (z, θ) is the one of a forest of weighted increasing trees with labelled nodes and clusters (connected components); see the Appendix. The parameter θ 'marks' the number of trees (connected components) of the forest. An increasing tree is a rooted simply connected tree in which the labels of the nodes along any branch from the root to any leaf are forced to go in increasing order, [START_REF] Bergeron | Varieties of increasing trees[END_REF]. The exponential generating function (EGF) φ (z) of such weighted ordered trees solve a non-linear ODE having an admissible branching generating mechanism g (z) in the sense that

φ (z) = 1 -(1 -α 2 z) α 1 /α2 α 1 , with φ ′ (z) = (1 -α 2 z) α1/α2-1 = g (φ (z)) and g (z) = (1 -α 1 z) -(α2/α1-1) and, b! z b g (z) = α b 1 [(α 2 -α 1 ) /α 1 ] b > 0 if α 2 > α 1 > 0.
The function g, as a weighted generating function with non-negative Taylor coefficients, is called the branching mechanism of the increasing g-tree. It describes locally the variability of the edges pointing outwards any node of the tree, assigning a weight b! z b g (z) to a b-branching node (with b = 0 corresponding to the leaves of the tree). Similarly, n! [z n ] φ (z) is the resulting weight of all g-trees with n constitutive nodes and S 0 n,k :

= n! k! [z n ] φ (z)
k is the resulting weight of all possible unordered forests made of k increasing ordered g-trees (connected components) having n constitutive labelled nodes. If α 1 , α 2 are integers with α 2 ≥ 0, α 2 ≥ α 1 , the weights n! [z n ] φ (z) are themselves integers and so reduce to counts. Note that

n! [z n ] φ (z) = α n-1 2 [1 -α] n-1 > 0 if α 2 > α 1 > 0, else 0 < α = α 1 /α 2 < 1.
Remark: When α 1 → 0 + , φ (z) boils down to φ (z) =log (1α 2 z), the weighted cycle-EGF, so that Φ 0 (z, θ) = e θφ(z) is the weighted EGF of a forest (or set) of cycles (the usual Ewens interpretation; see the Appendix). However, with

g (z) = α 2 e z , φ (z) = -log (1 -α 2 z) also solves φ ′ (z) = g (φ (z)) , φ (0) = 0,
showing that φ (z) (respectively Φ 0 (z, θ)) is also the EGF of an (a forest of) increasing tree(s) generated by g.

The regime (B) :

α 2 > 0 > α 1 , β := -α = (-α 1 ) /α 2 ∈ (0, 1]
. Here, we have

φ (z) = 1 -(1 -α 2 z) α1/α2 α 1 , with n! [z n ] φ (z) = - α n 2 α 1 [-α] n = α n-1 2 [1 -α] n-1 > 0.
This is a 'negative-binomial' assembly of atoms (nodes), not in the increasing tree class [see the following sub-section]. In this case indeed, with (a) b = a (a -1) ... (ab + 1) the falling factorial of order b of a, φ (z) solves φ

′ (z) = g (φ (z)) with b! z b g (z) = (-α 1 ) b ((α 2 -α 1 ) / (-α 1 )) b = (-α 1 ) b (1 + β) b , not all positive.
When α 1 < 0 with β ∈ (0, 1), φ (z) may however be written as [START_REF] Flajolet | Analytic Combinatorics[END_REF]. The EGF φ (z) is then the weighted EGf of a sequence of b (z), where b (z) is the EGF of a weighted increasing tree with admissible branching mechanism g :

(29) φ (z) = 1 -α 1 b (z) 1 -b (z) = 1 -α 1 1 1 -b (z) -1 for b (z) = 1 -(1 -α 2 z) β obeying b (0) = 0 and n! [z n ] b (z) = α n 2 β [1 -β] n-1 > 0 for all n ≥ 1
(30) b ′ (z) = g (b (z)) , g (z) = α 2 β (1 -z) -( 1-β β ) .
This shows that, if in general α 2 > 0 > α 1 with β = -α = (-α 1 ) /α 2 ∈ (0, 1), the interpretation of Φ 0 (z, θ) is now the one of a weighted forest of sequenced increasing weighted ordered trees; see the Appendix.

Remarks:

There are two special cases where type (B) models can also be relevant to the class (A) :

(i) If α 1 < 0, whenever (α 2 -α 1 ) / (-α 1 ) = d (else -(α 2 /α 1 -1) = d)
, a positive integer ≥ 2, then φ (z) also obeys φ ′ (z) = g (φ (z)) where g (z) = (1α 1 z) d is a degree-d polynomial with positive Taylor coefficients (the admissible branching mechanism generating d-ary increasing plane trees). In that polymer case where atoms represent monomers, α 2 = (d -1) (-α 1 ) > 0 so that 1 -1/ (1α) = 1/d. This shows that the interpretation of Φ 0 (z, θ) as an EGF of a forest of increasing weighted trees remains valid if α 1 < 0 but only when α 2 = (d -1) (-α 1 ) > 0. Second kind Stirling numbers S 0 n,k are obtained when α 2 = 0, α 1 = -1 and d = 1 (a boundary linear polymer case).

(ii) If α 1 < 0 and α 2 = -α 1 (d = 2), the boundary (Lah) case with

β = -α 1 /α 2 = 1, φ (z) = α 2 z/ (1 -α 2 z) with φ ′ (z) = g (φ (z)) for g (z) = α 2 (1 + z) 2
which is an admissible binary EGF: the interpretation of Φ 0 (z, θ) as an EGF of a weighted forest of increasing weighted binary trees remains valid in this boundary case as well.

The regime (C) :

α 2 > 0 > α 1 , β := -α = (-α 1 ) /α 2 > 1.
In the range β > 1 (α < -1), the EGF φ (z) has no combinatorial interpretation in terms of a labelled weighted tree, be it sequenced, increasing or simple. Rather, it can be interpreted from a set partition (number of ways to partition a set of atoms) with atoms' random weight W being Gamma(β, 1/α 2 )-distributed, namely Φ 0 (z, θ) = e θφ(z) = e θ(E(e W z )-1) , with

φ (z) = E(e W z ) -1 = (1 -α 2 z) -β -1.
Some randomness on the atoms' weight needs to be incorporated. Recall α 2 is the scale parameter of the Gamma distribution, while β is its' shape. As a result, in this parameter range, the splitting of atoms is of a completely different nature.

This point is the combinatorial version of the fact that a negative-binomial distribution is a Gamma-Poisson mixture. 5.4. The reservoir. Finally, the EGF r (z) may be viewed as the EGF of a weighted(w 2 /α 2 ) forest of weighted(α 2 ) cycles resulting from: r (z) = e

r n+1 = r n (w 2 + nα 2 ) , r 0 = 1,
for an increasing number of its constitutive labelled atoms. When dealing with Φ (z, θ) = r (z) Φ 0 (z, θ), there is thus an additional reservoir of labelled atoms with weights r n and S n,k := n! k! [z n ] r (z) φ (z) k is the weight of all possible unordered forests made of k increasing ordered g-trees (connected components) having n constitutive labelled nodes in total, including the ones of the background reservoir (the solute containing the branched polymer soup).

Normalizing S n,k by σ n (1) amounts to choosing one of them at random, proportionally to its weight. 5.5. Occupancy statistics: Maxwell-Boltzmann sampling formulae. The weight of those configurations having n atoms as a whole, n 0 of which are from the reservoir forest, is

n! z n0 1 z n-n0 0 r (z 1 ) Φ 0 (z 0 , 1) = n n 0 r n0 σ 0 n-n0 (1)
where

σ 0 n (θ) = n! [z n ] Φ 0 (z, θ) = n k=0 S 0 n,k θ k yielding (if α 1 α 2 = 0), by Dobiński formula, σ 0 n (1) = e 1/α1 k≥0 (-1) k α -k 1 k! [-kα 1 : α 2 ] n . Normalizing the above weight by w 2 σ n (1) = n! [z n ] Φ (z, 1) = e 1/α1 k≥0 (-1) k α -k 1 k! [-kα 1 + w 2 : α 2 ] n
gives the probability to observe such a configuration. Similarly,

n n 0 r n0 S 0 n-n0,k S n,k
is the probability to observe a configuration having n atoms as a whole, n 0 of which are from the reservoir, the remaining part being a forest made of nn 0 atoms with k increasing trees. Consequently,

n n 0 r n0 σ 0 n-n0 (1) σ n (1)
, n 0 ∈ {0, ..., n} is the probability to observe a configuration having n atoms as a whole, n 0 of which are from the reservoir. Given n 0 atoms are from the reservoir, the probability to observe k new species not in the reservoir is

S 0 n-n0,k σ 0 n-n0 (1)
σ n (1) S n,k .

Remark: We have terms in such sums],

n! [z n ] Φ 0 (z, θ) = n! [z n ] e θφ(z) = n k=0 B n,k (φ • ) θ k where B n,k (φ • ) = n! k! [z n ] φ (z) k are the exponential Bell polynomials in the vari- ables φ • = [α 2 -α 1 : α 2 ] •-1 = α •-1 2 [1 -α 1 /α 2 ] •-1 with φ (z) = m≥1 φ m z m m! . So,
S 0 n,k = B n,k (φ • ) = n! k! [z n ] φ (z) k = n! k! m k : |m k |=n k l=1 φ m l m l ! (32) = n! k! α n-k 2 m k : |m k |=n k l=1 [1 -α] m l -1 m l ! > 0 (33)
are Bell polynomials in the variables φ • .

Remark:

when α 2 = 2, α 1 = 1 (α = 1/2), φ • = (2 • -3)!! with φ (z) = 1 - √ 1 -2z. Then, with (-1)!! = 1 (see (1.10) of [20]) (34) S 0 n,k = S n,k (-2, -1, 0) = B n,k (φ • ) = (2 (n -k) -1)!! 2n -k -1 2 (n -k)
If w 2 = 0, with n l > 0 and k l=1 n l = n, the sampling formula arising from a random allocation scheme of particles [see [START_REF] Kolchin | Random Mappings. Translated from the Russian[END_REF]] to potentially infinitely many species (boxes) yields, see [START_REF] Huillet | Occupancy distributions arising in sampling from Gibbs-Poisson abundance models[END_REF],

P θ (N n (1) = n 1 , ..., N n (k) = n k ; K n = k) = n! k! θ k σ 0 n (θ) k l=1 φ n l n l ! .
This first Maxwell-Boltzmann sampling formula (MBSF) gives the joint distribution that, given the population is made of N = n atoms, it has k connected components with positive sizes (n l ; l = 1, .., k) . Alternatively, in the latter Ewens-Pitman spirit [following Kolchin's balls in boxes occupancy model], this is the joint distribution that, given the sample size is N = n, there was k distinct visited species with positive sample hits (n l ; l = 1, .., k) , observed in an arbitrary way. The tilt parameter θ is related to 'temperature' in that θ = e -β , β = 1/T. Note

P θ (K n = k) = θ k σ 0 n (θ) n l >0: k l=1 n l =n n! k! k l=1 φ n l n l ! = θ k S 0 n,k σ 0 n (θ)
.

Proposition 5. If w 2 > 0, with n 0 ≥ 0, n l > 0, l = 1, ..., k, and

k l=0 n l = n -n 0 (35) P θ (N n (0) = n 0 , N n (1) = n 1 , ..., N n-n0 (k) = n k ; K n = k) = n! k! θ k σ n (θ) r n0 n 0 ! k l=1 φ n l n l ! = n! k! (θ/α 2 ) k σ n (θ) r n0 n 0 ! α n-n0 2 k l=1 [1 -α] n l -1 n l !
gives the joint distribution that, given the sample size is N = n, there was n 0 visits to the reservoir (accounting for early failure events of the sampling process, or missed samples) and k distinct visited species with positive sample sizes (n l ; l = 1, .., k) not in the reservoir.

Note that (36)

P θ (N n (0) = n 0 ; K n = k) = n! (n -n 0 )! θ k r n0 σ n (θ) n 0 ! n l >0: k l=1 n l =n-n0 (n -n 0 )! k! k l=1 φ n l n l ! = n n 0 θ k r n0 S 0 n-n0,k σ n (θ) .
and, as required,

P θ (K n = k) = n n0=0 P θ (N n (0) = n 0 ; K n = k) = θ k S n,k σ n (θ) ,
equivalently,

E θ u Kn = n k=0 u k P θ (K n = k) = σ n (θu) σ n (θ) , x ≤ 1, with mean: E θ (K n ) = θσ ′ n (θ) /σ n (θ)
. The probability to observe a configuration with N = n atoms as a whole is (z < z c = 1/α 2 )

P z (N = n) = n!z n [z n ] Φ (z, 1) n!Φ (z, 1) = z n σ n (1) n!Φ (z, 1) , equivalently E z u N = Φ (zu, 1) Φ (z, 1) = 1 -α 2 z 1 -α 2 zu w2/α2 e -(1-α 2 z) α 1 /α 2 α 1 1- 1-α 2 zu 1-α 2 z α 1 /α 2
is the pgf of N . Note E z (N ) = zΦ ′ (z, 1) /Φ (z, 1) . In the Ewens sampling problem spirit, the rv N is also the random number of particles to be assigned to the different boxes.

Second Maxwell-Boltzmann sampling formula: over the a m for short), if w 2 = 0, (37)

P θ (A n (1) = a 1 , ..., A n (n) = a n ; K n = k) = n!θ k σ 0 n (θ) n m=1 (φ m /m!) am a m ! with * am:(n,k) P θ (A n (1) = a 1 , ..., A n (n) = a n ; K n = k) = θ k S 0 n,k σ 0 n (θ) .
Proposition 7. For each sequence of non-negative integers (i m ) obeying n m=1 i m = i ≤ k and n m=1 mi m = j ≤ n, we also have the joint falling factorial moments of the A n (m)'s as (see [START_REF] Huillet | Occupancy distributions arising in sampling from Gibbs-Poisson abundance models[END_REF]), ( 38)

E θ n m=1 (A n (m)) im = θ i (n) j σ 0 n-j (θ) σ 0 n (θ) n m=1 φ m m! im = α j-i 2 θ i (n) j σ 0 n-j (θ) σ 0 n (θ) n m=1 [1 -α] m-1 m! im . Taking i m ′ = 0 for all m ′ = m, E θ (A n (m)) im = θ im (n) mim σ 0 n-mim (θ) σ 0 n (θ) φ m m! im . If in addition i m = 1, E θ (A n (m)) = n m θ σ 0 n-m (θ) σ 0 n (θ) φ m = α m-1 2 n m θ σ 0 n-m (θ) σ 0 n (θ) [1 -α] m-1 .
giving the mean number of boxes (species) visited m times by the n-sample.

If w 2 > 0, with n 0 ≥ 0, m≥1 ma m = nn 0 and m≥1 a m = k.

(39)

P θ (N n (0) = n 0 , A n (1) = a 1 , ..., A n (n -n 0 ) = a n-n0 ; K n = k) = n!θ k σ n (θ) r n0 n 0 ! n-n0 m=1 (φ m /m!) am a m ! . Note that, in view of * am:(n-n0,k) (n -n 0 )! n-n0 m=1 (φ m /m!) am am! = S 0 n-n0,k , as re- quired, P θ (N n (0) = n 0 ; K n = k) = * am:(n-n0,k) P θ (N n (0) = n 0 , A n (1) = a 1 , ..., A n-n0 (n -n 0 ) = a n-n0 ; K n = k) = n n 0 θ k r n0 S 0 n-n0,k σ n (θ) . Furthermore, Proposition 8. If w 2 > 0, observing P θ (N n (0) = n 0 ) = n n0 rn 0 n-n 0 k=0 θ k S 0 n-n 0 ,k σn(θ) = n n0 [w2] n 0 σ 0 n-n 0 (θ) σn(θ) , (40) E θ n m=1 (A n (m)) im | N n (0) = n 0 = r n0 α j-i 2 θ i (n -n 0 ) j n 0 !P θ (N n (0) = n 0 ) σ n-n0-j (θ) σ n-n0 (θ) n m=1 [1 -α] m-1 m! im = α j-i 2 θ i (n -n 0 ) j (n) n0 σ n (θ) σ 0 n-n0 (θ) σ n-n0-j (θ) σ n-n0 (θ) n m=1 [1 -α] m-1 m!
im yields the joint conditional falling factorial moments of the A n (m)'s.

• If α 2 = 0 (requiring α 1 < 0) Φ (z, θ) = e w2z e θ α 1 (1-e -α 1 z ) = r (z) e θφ(z) φ (z) = 1 α 1 1 -e -α1z and g (z) = 1 -α 1 z φ ′ (z) = e -α1z = g (φ (z))
(linear trees with branching number b = 0 or 1).

• If α 1 = 0 Φ (z, θ) = (1 -α 2 z) -(w2+θ)/α2 (41) σ n (θ) = α n 2 [(w 2 + θ) /α 2 ] n φ (z) = - 1 α 2 log (1 -α 2 z) and g (z) = e α 2 z φ ′ (z) = 1 1 -α 2 z = g (φ (z)) n! [z n ] φ (z) = (n -1)!α n-1 2 If w 2 = 0 and α 2 = 1, σ 0 n (θ) = [θ]
n the second MBSF boils down to the classical 'critical' Ewens sampling formula

P θ (A n (1) = a 1 , ..., A n (n) = a n ; K n = k) = n!θ k σ 0 n (θ) n m=1 (φ m /m!) am a m ! = θ k [θ] n n! n m=1 m am a m ! .
• If α 1 = 0 and α 2 = 0 Φ (z, θ) = e w2z e θz = e z(w2+θ) φ (z) = z = g (φ (z)) with g (z) = 1 (singleton trees as isolated atoms with branching number b = 0).

σ n (θ) = (w 2 + θ) n .

Singularity and saddle point analysis

It is shown, making use of combinatorics, that two regimes are to be distinguished: the one of strongly connected aggregates where aggregation dominates nucleation (creation of a new group, appearance of a new species are rare) and the one of weakly connected aggregates where nucleation dominates aggregation.

Nucleation occurs when the new incoming particle starts a new group or cluster (an event which may be interpreted as the appearance of a new species). Aggregation occurs when the new incoming particle joins any already pre-existing group or cluster (tree).

In the first (condensed) regime, the number of groups K n and size-m groups N n (m) may converge in the thermodynamic limit, with a special role played by the Poisson distribution. The asymptotic variables become observable macroscopically. Such situations with few large clusters are favoured under condition (A).

In the second (diluted) regime, the number of clusters K n always diverges as n → ∞, the asymptotic equivalent of which being under control. Large deviation results are shown to be available. Such situations with many 'small' clusters are observed under condition (B). The size-m groups N n (m) are estimated by the Maxwell-Boltzmann sampling formula.

There is a transition between the condensed and diluted phases occurring when α 1 → 0, either from below or from above. At the transition point α 1 = 0, the occupancy statistics of clusters is the one akin to the Ewens' sampling formula.

There is a clear connection with species sampling problems. Tree sizes of some forest at some intermediate step may represent the number of visits to different species. If a newly introduced atom nucleates, a new species is discovered; if it aggregates to a preexisting tree, the number of visits to the corresponding species is incremented by one unit. This relation between species sampling problems and nucleation/aggregation of particles in Physics was outlined in [START_REF] Mekjian | Cluster distributions in physics and genetic diversity[END_REF] in the special Ewens case.

6.1. The regime (A) : α 1 , α 2 , w 2 > 0 and α := α 1 /α 2 ∈ (0, 1) . Exp-algebraic singularity of Φ (z, θ). We first describe the singularity analysis of σ n (θ) . We have:

Φ (z, θ) = e θ/α1 k≥0 (-θ/α 1 ) k k! (1 -α 2 z) -(w2/α2-kα) ,
with singularity at z = 1/α 2 . Singularity analysis of the singular expansion (valid if w 2 = kα 1 for some integer k)

σ n (θ) = n! [z n ] Φ (z, θ) = n!e θ/α1 k≥0 (-θ/α 1 ) k k! α n 2 n w2/α2-kα n 1 Γ (w 2 /α 2 -kα) = (n -1)!α n 2 n w2/α2 Γ (w 2 /α 2 ) e θ/α1 1 - θn -α α 1 Γ (w 2 /α 2 ) Γ (w 2 /α 2 -α) + o n -α ∼ (n -1)!α n 2 n w2/α2 Γ (w 2 /α 2 ) e θ/α1 1-n -α Γ(w 2 /α 2 ) Γ(w 2 /α 2 -α) Therefore σ n (1) ∼ (n -1)!α n 2 n w2/α2 Γ (w 2 /α 2 ) e 1 α 1 1-n -α Γ(w 2 /α 2 ) Γ(w 2 /α 2 -α) σ n (1) σ n+1 (1) ∼ 1 nα 2 1 - w 2 /α 2 n
Moreover:

E θ Kn = σ n (θ) σ n (1) ∼ e -(1-θ)/α1 1-n -α Γ(w 2 /α 2 ) Γ(w 2 /α 2 -α) .
This shows that: Proposition 9. (condensed phase). Under conditions (A) with α 1 > 0, K n has a Poisson limit-law with mean 1/α 1 as n → ∞. It corresponds to the distribution

P (K n = k) = S n,k σ n (1) , k = 0, ..., n
of the normalized weight of unordered forests with n labelled atoms and k increasing ordered trees, resulting from a random uniform choice of a configuration proportional to its weight.

One can track the size of the clusters as follows: with z := (z 1 , z 2 , ...) marking their sizes, define

φ z (z) = m≥1 z m φ m z m /m! = φ (z) + m≥1 (z m -1) φ m z m /m! and σ 0 n (θz) = n! [z n ] Φ 0 (z, θz) ; σ n (θ, z) = n! [z n ] Φ (z, θz) where Φ 0 (z, θz) = e φ θz (z) and Φ (z, θz) = r (z) Φ 0 (z, θz). Clearly, E n m=1 z An(m) m θ Kn = σ n (θz) σ n (1) ∼ e -[(1-θ)/α1+θ n m=1 (1-zm)φ m /m!](1-O(n -α ))
showing that, asymptotically, the number A n (m) of size-m clusters in a population with n atoms are jointly Poisson with mean φ m /m!.

• If α 1 → 0 + , 1 -n -α Γ(w2/α2) Γ(w2/α2-α) /α 1 ∼ 1 -e -α log n /α 1 ∼ log n/α 2 and σ n (θ) σ n (1) ≃ e -(1-θ) log n/α2
consistently with (26) and the fact that Φ (z, θ) has an Exp-log singularity. There is a phase transition from the condensed phase to the diluted one when α 1 → 0 + .

6.2.

The regimes (B) (C) : α 2 > 0, α 2 > 0 > α 1 with α := α 1 /α 2 < 0. Essential singularity of Φ (z, θ). We now perform a saddle point analysis of σ n (θ) . We have

σ n (θ) = n! [z n ] Φ (z, θ) = n! 2iπ e σ θ (z) z n+1 dz,
where

σ θ (z) = log Φ (z, θ) = -w 2 /α 2 log (1 -α 2 z) + θ α 1 (1 -(1 -α 2 z) α ) , with σ ′ θ (z) = w 2 1 -α 2 z + θ (1 -α 2 z) α-1 , σ ′′ θ (z) = w 2 α 2 (1 -α 2 z) 2 + θα 2 (1 -α) (1 -α 2 z) α-2 .
The saddle point solving zσ ′ θ (z) = n is given by z

n (θ) = 1 α2 1 -(nα 2 /θ) -1/(1-α) [else 1 -α 2 z n (θ) = (nα 2 /θ) -1/(1-α) ],
for which to the dominant order

σ θ (z n (θ)) ∼ - 1 α 1 (nα 2 ) -α/(1-α) θ 1/(1-α) σ ′′ θ (z n (θ)) ∼ θα 2 (1 -α) (nα 2 /θ) (2-α)/(1-α) = α (3-2α)/(1-α) 2 (1 -α) θ -1/(1-α) n (2-α)/(1-α) With b θ (z) : = z 2 σ ′′ θ (z) + zσ ′ θ (z) b θ (z n (θ)) ∼ α -2 2 σ ′′ θ (z) , observing z n (θ) n ∼ α -n 2 e -n(nα2/θ) -1/(1-α) = α -n 2 e cαθ 1/(1-α) n -α/(1-α) σ θ (z n (θ)) ∼ cn -α/(1-α) θ 1/(1-α) = c (θ) n -α/(1-α)
where c := -α

-1/(1-α) 2 /α > 0, c (θ) := cθ 1/(1-α) > 0, we get the saddle point estimate σ n (θ) ∼ n! e σ θ (zn(θ)) z n (θ) n 2πb θ (z n (θ)) ∼ n!α n-1 2 e c(1-α)n -α/(1-α) θ 1/(1-α) 2π (-cα) -1 (1 -α) θ -1/(1-α) n (2-α)/(1-α) = (n -1)!α n-1 2 e c(θ)(1-α)n -α/(1-α) 2π (-c (θ) α) -1 (1 -α) n α/(1-α)
In particular,

σ n (1) ∼ (n -1)!α n-1 2 e c(1-α)n -α/(1-α) 2π (-cα) -1 (1 -α) n α/(1-α) σ n (1) σ n+1 (1) ∼ 1 α 2 n e c(1-α)n -α/(1-α) 1-(1+ 1 n ) -α/(1-α) 1 + 1 n α/[2(1-α)] ∼ 1 α 2 n e -cαn -1/(1-α) 1 + α 2 (1 -α) 1 n ∼ 1 α 2 n 1 -cαn -1/(1-α) Eθ Kn = σ n (θ) σ n (1) = n! [z n ] Φ (z, θ) n! [z n ] Φ (z, 1) ≃ e σ θ (zn(θ))-σ1(zn(1)) σ ′′ θ (z n (θ)) σ ′′ 1 (z n (1)) -1/2 ≃ e c(1-α)n -α/(1-α) [θ 1/(1-α) -1] θ -1/(1-α) -1/2
We obtained the following result characterizing the diluted phase with many clusters:

Proposition 10. Suppose α 2 > 0 > α 1 with α := α 1 /α 2 < 0 and w 2 ≥ 0. With c := -α -1/(1-α) 2 /α > 0, a := -α/ (1 -α) = 1 -1/ (1 -α) ∈ (0, 1) and c n := cn a → ∞ (slower than n), as n → ∞, it holds that Ee -λKn 1/cn → α (λ) := e -(1-α)[1-e -λ/(1-α) ] = e -F (λ) , λ ≥ 0,
where α (λ) is the Laplace transform of a mean 1 Poisson rv with intensity 1α.

From [4], (42) lim n→∞ 1 c n log P K n c n ≷ x = f (x) , x ≷ 1 where f (x) = inf λ≥0 (λx -F (λ)) = -(1 -α) [x log x + 1 -x] ≤ 0, x ≥ 0
is the Legendre transform of the log-Laplace F (λ). In particular, as n → ∞,

K n c n → 1 P-a.s. and in law.

The same holds true under P θ , simply substituting c (θ) to c in c n . These results are valid both under the regimes (B) and (C) . In the regime (B), c n := cn a → ∞ (slower than n 1/2 ).

We recall from (26) that, as α 1 → 0 -and n → ∞,

(44) K n log n → 1 α 2 a.s., K n -1 α2 log n 1 α2 log n → N (0, 1)
There is a phase transition from the diluted phase to the condensed one when α 1 → 0 -.

7.

Relations with probability theory 7.1. Grand canonical approach.

(i) Φ z (θ) := Φ (z, θ) Φ (z, 1) = e -φ(z) (1-θ) pgf of a Poisson rv with rate φ (z

) = 1-(1-α2z) α 1 /α 2 α1 > 0 if α 2 z ≤ 1. P (X z = k) = 1 Φ (z, 1) θ k Φ (z, θ) = 1 Φ (z, 1) τ k (z) = e -φ(z) φ (z) k k! If z > 1/α 2 , the rv X z is ill-defined. (ii) Φ θ (z) := Φ (z, θ) Φ (1, θ) = 1 -α 2 1 -α 2 z w2/α2 e θ α 1 ((1-α2) α 1 /α 2 -(1-α2z) α 1 /α 2 ) P (X θ = n) = 1 Φ (1, θ) [z n ] Φ (z, θ) = 1 Φ (1, θ) σ n (θ) n! If 0 < α 2 < 1, the term 1-α2 1-α2z w2/α2
is the pgf of a negative-binomial distribution. For the second term,

e θ α 1 ((1-α2) α 1 /α 2 -(1-α2z) α 1 /α 2 ) = e -θφ(1)(1-φ(z)/φ(1))
the pgf of a compound-Poisson rv with compounding rv ∆ ρ having pgf E z ∆ρ = φ (z) /φ (1) if α 2 ≤ 1. Both contributions summing to X θ are mutually independent. If α 2 > 1, the rv X θ is ill-defined.

If α 2 = 1 and 0 < α 1 ≤ α 2 = 1, the 'negative-binomial' term is degenerate and φ (z) = (1 -(1z) α1 ) /α 1 . The pgf of the second compound Poisson term is e -θ α 1

(1-z) α 1 , the one of a discrete-stable distribution with scale parameter (θ/α 1 ) 1/α1 and tail parameter α 1 ∈ (0, 1]. The pgf of the compounding distribution is 1 -(1z)

α1 , the one of a Sibuya rv, [START_REF] Sibuya | Generalized hypergeometric, digamma and trigamma distributions[END_REF].

7.2. Canonical approach. Vertical gf:

n≥k S n,k z n n! = 1 k! (1 -α 2 z) -w2/α2 1 -(1 -α 2 z) α1/α2 α 1 k E θ Kn = σ n (θ) /σ n (1) pgf E (K n ) = σ ′ n (1) /σ n (1) ≤ ∞. P (K n = k) = 1 σ n (1) θ k σ n (θ) = S n,k σ n (1)
, k = 0, ..., n.

The rv K n is well-defined only if the S n,k are non-negative.

Let us come back to our case study. We first recall that for φ

• = [α 2 -α 1 : α 2 ] •-1 = α •-1 2 [1 -α] •-1 Bell polynomials obey S 0 n+1,k (φ • ) = S 0 n,k-1 (φ • ) + (nα 2 -kα 1 ) S 0 n,k (φ • ) . When w 2 > 0 S n+1,k = S n,k-1 + (nα 2 -kα 1 + w 2 ) S n,k
Define K n with tilted P θ -law:

P θ (K n = k) = θ k σ n (θ)
S n,k , k = 0, ..., n.

We get the recursion (45)

P θ (K n+1 = k) = θ k σ n+1 (θ) (S n,k-1 (φ • ) + (nα 2 -kα 1 + w 2 ) S n,k (φ • )) = σ n (θ) σ n+1 (θ) (θP θ (K n = k -1) + (nα 2 -kα 1 + w 2 ) P θ (K n = k))
showing that (46)

P θ (K n+1 = k + 1 | K n = k) = θσ n (θ) σ n+1 (θ) =: q (n) k,k+1 ,
independent of k but dependent on n and,

q (n) k,k := σ n (θ) σ n+1 (θ) (nα 2 -kα 1 + w 2 ) ,
depending on both k and n. The transition coefficients are q

(n) k,k and q (n) k,k+1
. The latter one is the transition weight at which a new species is being discovered given k of them were previously discovered in a size-n sample.

This shows that the event K n+1 = k only depends on the event K n = k -1 (respectively K n = k), when a new species (respectively no new species) is being discovered as the sample size is increased by one unit. And not on further past events such as K m = k -1 for k -1 ≤ m < n. The sequence K n is Markovian.

The three-term ('space-time' inhomogeneous) recurrence ( 45) is not the one of a standard Markov chain with a usual probability transition matrix. It rather is the one of a triangular Markov probability sequence. Write (45) as

P (K n+1 = k) = q (n) k,k P (K n = k) + q (n) k-1,k P (K n = k -1) ,
defining the (positive) transition coefficients q (n) k,k and q (n) k-1,k , not transition probabilities.

Introduce the (n + 1) × (n + 1) superdiagonal transition matrix

Q n+1,n+1 :=        q (n) 0,0 q (n) 0,1 0 • • • 0 . . . . . . 0 0 0 q (n) n-1,n-1 q (n) n-1,n . . . . . . 0 q (n) n,n       
, and let Q n,n+1 be its n × (n + 1) truncated version. With π K 0 := 1, taking into account the boundary conditions, the distributions π K n := (P (K n = k) , k ∈ {0, ..., n}), n ≥ 0, satisfy the recursion

π K n = π K n-1 , 0 Q n+1,n+1 = π K n-1 Q n,n+1 , n ≥ 1. Thus, π K n = ← - m=1,...,n Q m,m+1
, n ≥ 1 is an integrated form solution of the recursion, as a left product of nested rectangular matrices. Because for each n, π L n is a probability vector, we get that π L n is orthogonal to the 1-shifted column sum vector q n -1 of Q n,n+1 with k-th entry q

(n) k,k + q (n)
k,k+1 -1, k = 0, ..., n. This can easily be seen while post-multiplying π L n+1 by the column vector 1 and observing

π L n+1 1 = π L n 1 = 1.
Assuming θ = 1 and summing the above recursion over k = 0, ..., n + 1 yields (if α 1 = 0) an expression of the mean value

E (K n ) = E θ=1 (K n ) = 1 |α 1 | 1 + nα 2 + w 2 - σ n+1 (1) σ n (1)
If α 1 > 0, recalling σn(1) σn+1(1) ∼ 1 nα2 1 -w2 nα2 in that case, as n → ∞

E (K n ) ∼ 1 α 1 1 + nα 2 + w 2 -nα 2 1 + w 2 α 2 n = 1 α 1 ,
consistently with the singularity analysis in this regime. If α 1 < 0, recalling σn(1) σn+1(1) ∼ 1 nα2 1cαn -1/(1-α) in that case, where α = α 1 /α 2 < 0, as n → ∞

E (K n ) ∼ - 1 α 1 1 + nα 2 + w 2 -nα 2 1 + cαn -1/(1-α) ∼ cn 1-1/(1-α) = cn -α/(1-α) ,
with algebraic divergence, consistently with the saddle point analysis in this regime.

If

α 1 = 0, recalling σ n (1) = α n 2 [(w 2 + 1) /α 2 ] n , P (K n+1 = k) = σ n (1) σ n+1 (1) (P (K n = k -1) + (nα 2 + w 2 ) P (K n = k)) = 1 w 2 + nα 2 + 1 (P (K n = k -1) + (nα 2 + w 2 ) P (K n = k))
which is a time-inhomogeneous Markov chain with independent increments of the type S n with w 1 = 1, so with Kn log n → 1 α2 a.s., as n → ∞. Note that, summing the above recursion over k = 0, ..., n + 1,

E (K n+1 ) = E (K n ) + 1 w 2 + nα 2 + 1 ⇒ E (K n ) ∼ log n/α 2 .
Finally, Proposition 11. Time T 1 of first appearance of a new species (cluster): with S n,0 = [w 2 : α 2 ] n and for n ≥ 1,

P (T 1 = n) = P (K n-1 = 0) P (K n = 1 | K n-1 = 0) = S n-1,0 σ n-1 (1) σ n-1 (1) σ n (1) = S n-1,0 σ n (1) .
More generally, the law of the time T k of first appearance of k-th new species (cluster): ( 47)

P (T k = n) = P (K n-1 = k -1) P (K n = k | K n-1 = k -1) = S n-1,k-1 σ n (1)
, n ≥ k.

Appendix: from counting to weighted EGF's

We refer to ( [START_REF] Flajolet | Analytic Combinatorics[END_REF], Chapter II) and to [START_REF] Drmota | Random trees. An interplay between combinatorics and probability[END_REF] for more details on the trees' aspects concerning this point.

We first recall some elementary facts from combinatorics of labelled structures.

-Sequence EGF: S (z) = 1/ (1z) ; (linear chain of labelled atoms with free endpoints). Here, C n = n! [z n ] S (z) = n! : there are n! linear sequences of n labelled atoms (the number of its permutations).

-Set EGF: E (z) = e z ; (set of labelled and disconnected atoms). Here, C n = n! [z n ] E (z) = 1 : there is one single set of n labelled disconnected atoms.

-Cycle EGF:

C (z) = 1 -log (1 -z) ; C n = n! [z n ] C (z) = (n -1)
! counts the number of unrooted cycles with n labelled atoms. There are n! = nC n of them if rooted.

-Set of cycles (identity): E (C (z) -1) = S (z) counting the number of sets of unrooted cycles with n labelled atoms is also the number of linear sequences.

-Set of sequences: E (S (z) -1) = exp (z/ (1z)) .

-Set of sets (Bell): E (E (z) -1) = exp (e z -1) . Number of partitions of a set.

-All rooted labelled trees EGF's are solutions to the functional equations: T (z) = zE T (z) (non-plane or unordered) or T (z) = zS T (z) (plane), T (0) = 0. When the empty tree is included: T (z) = 1 + T (z) . Here, C n = n! [z n ] T (z) counts the number of rooted labelled trees with n nodes. And C n /n counts the number of all such unrooted trees with n nodes.

In the unordered Cayley case, C n = n n-1 and C n /n = n n-2 (Cayley's formulae).

In the plane case,

T (z) = 1- √ 1-4z 2 , with C n = (2n-2)! (n-1)! and C n /n = (2n-2)! n!
. These numbers can be obtained by Lagrange inversion formula:

C n = n! 1 n z n-1 g (z)
n respectively with g (z) = E (z) or g (z) = S (z) .

The number of rooted unlabelled plane trees is the Catalan number

K n-1 = C n /n! = 1 n 2n-2 n--1 .
Plane (or ordered) means that there is an ordering of the offspring (giving an embedding of the tree in the plane), with the root at the top and the offspring of each node lower than that node. If one gives a direction of the offspring, say left to right, then the embedding orders them. Rotated configurations around the root are considered different (cyclic permutations of the offspring are not allowed). Plane (ordered) trees are clearly 'labellisable'. For plane trees, both C n and c n := C n /n! are integers, the latter one counting the number of rooted unlabelled trees.

-The EGF's of rooted labelled increasing trees solve the ODE's: I ′ (z) = S I (z) or I ′ (z) = E I (z) , I (0) = 0. With empty tree included I (z) = 1+I (z) .

The first EGF is explicitly ; it is the one of a set of sequences (or threadlike trees). Indeed, the EGF exp(z/(1z)) corresponds to unordered collections of permutations with fragments obtained by breaking a permutation into pieces. The EGF exp θ 1/ 1 -I (z) -1 is then the one of a set of sequenced increasing trees I (z) appearing in regime (B) .

I (z) = -log (1 -z), with C n = n! [z n ] I (z) = (n -1)!, while the second is I (z) = 1 - √ 1 -2z, with C n = n! [z n ] I (z) = (2n -
Weighted versions of any of these simple or composed EGF's, say G (z), are obtained while substituting wz to z (w > 0) to form G (wz) [the w's can be different for each of them in a composite EGF, so that w could be a vector of weights in a composition of EGF's]. In that case, n! [z n ] G (wz) = w n are the positive weights of the configurations, i.e. w n = w n C n (the weight times the number). Note that, for z ∈ [0, 1], G w (z) := G (wz) /G (w) is a probability generating function indexed by w, with

[z n ] G w (z) = w n /n! G (w) = w n C n /n! G (w) ,
translating that a configuration is chosen at random with probability proportional to its weight. The admissible values of w > 0 are those for which G (w) < ∞.

Examples:

-Considering the weighted EGF E (C (z) -1) with C (z) =log (1w 1 z) and E (z) = e w2z , assigning a weight w 1 (respectively w 2 ) to each of the cycles (respectively forest) nodes then

E (C (z) -1) = (1 -w 1 z) -w2
which is of the form of the reservoir term r (z) appearing in Φ (z, θ) [with the proper identifications of the parameters w = (w 1 , w 2 ) in terms of the ones (α 2 , w 2 /α 2 ) of r (z)]. The weight of such a configuration with n labelled atoms is

n! [z n ] E (C (z) -1) = w n 1 [w 2
] n , summing the weights over each possible cyclic connected components.

-Considering the weighted EGF E (S (z) -1) with S (z) = 1/ (1w 1 z) and E (z) = e w2z , assigning a weight w 1 (respectively w 2 ) to each of the sequenced (respectively forest) nodes, then E (S (z) -1) = e w2(1/(1-w1z)-1) .

-Consider the weighted EGF E I (z) with I ′ (z) = g I (z) for g (z) = (1w 1 z)

-(w2/w1-1)

(arising from an above weighted version of E (C (z) -1)). This leads (if w 1 , w 2 > 0 and w 1 /w 2 ∈ (0, 1)) to

I (z) = 1-(1-w2z) w 1 /w 2 w1 . If in addition, E (z) = e w3z , E I (z) = e w 3 w 1 (1-(1-w2z) w 1 /w 2 )
which is of the form of Φ 0 (z, θ) appearing in Φ (z, θ) [with the proper identifications of the parameters w = (w 1 , w 2 , w 3 ) in terms of the ones (α 1 , α 2 , θ) of Φ 0 (z, θ)].

-If w 2 > 0 > w 1 and -w 1 /w 2 ∈ (0, 1), a well-defined EGF model for Φ 0 (z, θ) is E S I (z) -1 representing a set (unordered forest) of sequenced increasing trees (see the body of the text). It is of the form of Φ 0 (z, θ) when w 2 = α 2 > 0 > w 1 = α 1 .

The key-point to respect to build admissible weighted composite EGF's is that their Taylor coefficients should be non-negative, possibly leading to weight parameters constraints as illustrated in the examples. From the first row of (48), the mean number of leaves is readily obtained to be respectively (growing as a fraction of n): The variance grows similarly proportionally to n and a Central Limit theorem can be shown to hold for n 0 (τ n ). Such increasing trees may serve as models for phylogenetic trees in which nodes represent species and labels encode their order of appearance in the tree, and thus the chronology of evolution. The leaves of the tree are the currently living species; the different trees consist of genera.

E [n 0 (τ n )] = n -α 2 -α , n 2
-

  [z : α] n := z (z + α) ... (z + (n -1) α) = α n [z/α] n = α n Γ(z/α+n) Γ(z/α) , where [z] n := [z : 1] n = Γ(z+n) Γ(z) (the rising factorials of z). The generalized Stirling numbers S n,k ≡ S n,k (-α 2 , -α 1 , w 2 ) , k = 0, ..., n S n,k ≡ 0 if k > n, are defined by the identity [10] [w : α 2 ] n = n k=0 S n,k [ww 2 : α 1 ] k .

  ]]. The limiting random variable Z admits the decomposition into two independent factors: Z = BZ 0 where B d ∼Beta(w 1 /α, w 2 /α) and Z 0 d ∼ M L (α, w) is a two-parameters Mittag-Leffler distribution with moments

4 .

 4 A related random walk as a nucleation/aggregation model In this Section, we study a new random walk model based on the S n,k > 0 under the regimes (A) or (B) (C). It is related to Gibbs partitions models in Pitman's course [[17], pages 25 : 29].

w 2 α 2 τ

 2 (z) with τ (z) =log (1α 2 z) . Alternatively as a set of cycles (cocktail bars); see the Appendix. Particles failing to connect the clusters or nucleate are sent to a reservoir set of cycles where they sit cyclically at tables, with w 2 /α 2 marking the number of tables. The Taylor coefficientsr n := n! [z n ] r (z) = α n 2 [w 2 /α 2 ]n are the positive weights of the background reservoir forest, obeying the recurrence (31)

  with the summation below running over m k := (m 1 , ..., m k ) ∈ N k , with |m k | := k j=1 m j = n and N := {1, 2, ...}, [there are n-1 k-1

Proposition 6 .

 6 Let A n (m) be the number of boxes (species) visited m times by the n-sample and K n = n -A n (0) the number of visited boxes. For all integral sequences a m obeying m≥1 ma m = n and m≥1 a m = k (the star-sum * am:(n,k)

  3)!!. -Set of trees (forests) EGF's: E T (z) or E I (z) . -Set of sequenced trees EGF's: E S T (z) -1 or E S I (z) -1 , as a disconnected collection of trees attached to open chains. The EGF of fragmented permutations is exp θ [1/ (1z) -1]

1 - 1 2 in view of n- 1 b=0 2 c 2 and W n = a n 1 a n- 1 2 1 -

 1112211 We now come to trees: -Consider the plane tree EGF T (z) = 1 -√ 4z /2 solving T (z) = zS T (z) , withC n = n! [z n ] T (z) = (2n-2)! (n-1)! . Let w (τ n ) = n-1 b=0 w n b (τ n ) bbe the (multiplicative) weight of an unlabelled rooted tree τ n with n nodes (|τ n | = n) having n b (τ n ) nodes with outdegree (branching number) b. The weight w (τ n ) is the product over the n nodes x of τ n of the w bn(x) 's where b n (x) is the outdegree of x. Then W n = τ n w (τ n ) is the weight of all sizen such trees associated to the weight sequence w:= (w b ≥ 0, b ≥ 0) . The number of these trees is c n := C n /n! = [z n ] T (z) . Let T w (z) = n≥1 z n W n . Then T w (z) solves T w (z) = zg T w (z) where g (z) = b≥0 w b z b .By Lagrange inversion formula, we get the identityW n = [z n ] T w (zExamples of w are: w = (ε b ; b ≥ 0) with ε b ∈ {0, 1}.Only the branches of the tree with ε b = 1 contribute to its weight.-w b = a 1 a b 2 , a 1 , a 2 > 0 with g (z) = a 1 b≥0 (a 2 z) b = a 1 / (1a 2 z) . Note w (τ n ) = a n 1 a n-n b (τ n ) = n and n-1b=1 bn b (τ n ) = n -1 (the total tree length). As a result, W n = a n 1 a n-1 n where c n = C n /n! = (2n-2)! n!(n-1)! . In that separable case, each tree τ n has equal weight a n 1 a n-1c n . Note T w (z) = 1 -√ 4a 1 a 2 z / (2a 2 ) = T (a 1 a 2 z) /a 2 . -if w := (w b ≥ 0, b ≥ 0) is a probability sequence with b≥0 w b = 1, then T w (z)is the EGF of the total progeny of a Galton-Watson tree with branching mechanism g, solving T w (z) = zg T w (z) , T w (0) = 0. The numbers [z n ] T w (z) = W n are the (sub-)probabilities of a progeny with size n.-Consider the increasing plane tree EGF I (z) solving I ′ (z) = S I (z) , I (0) = 0, with I (z) = 1 -√ 1 -2z andC n = n! [z n ] I (z) = (2n -3)!! = 2 -(n-1) (2n -2)!/ (n -1)!. Let W n = τ n w (τ n ) (where w (τ n ) = n-1 b=0 w n b (τ n ) b) be the weight of all size n increasing labelled rooted trees (|τ n | = n) associated to the weight sequence w := (w b ≥ 0, b ≥ 0), with w b = z b g (z) the weight of an atom with out-degree b. Let then I w (z) = n≥1 z n W n /n!, with W n = n! [z n ] I w (z). Then I w (z) solves

1 mπ 1 ( 1 - 1 0g 1 -

 11111 Finally, we mention in passing a random version of increasing trees. With m > 0, w := (w b = mπ b ≥ 0, b ≥ 0) where (π b ) is a probability sequence ( b≥0 π b = 1), I w (z) is the EGF of the total progeny of a Galton-Watson increasing tree with branching mechanism g (z) = b≥0 π b z b and parameter m. If the total progeny is finite with probability (w.p.) 1, I w (1) = 1 and I ′ w (1) = m is the mean number of its nodes, fixing m = 1 0dz ′ g(z ′ ) . The numbers W n /n! = [z n ] I w (z) = m n τ n n-1 b=0 π n b (τ n) b/n! are then the probabilities of a progeny with size n. The EGF I w (z) solves w (z) , where P is a primitive of 1/g. By Lagrange inversion formula,n! [z n ] I w (z) = m n (n -1)! z n-1 z P (z) n = W n . With z * = sup (z > 0 : g (z) < ∞) ∈ [1, ∞], the convergence radius of I w (z) is z c = sup z > 0 : I w (z) < ∞ = 1 m z * 0 dz ′ g (z ′ ) , with I w (z c ) = z * owing to z c -′ ) then z c > 1. If so, I w (1) < ∞ and I w (z) is a generating function candidate. It is a defective probability generating function only if m < 1 0 dz ′ g(z ′ ) because then ρ := I w (1) < 1 (in view of 1 = ′ ) ): the model is supercritical, having ρ as its extinction probability. The total progeny is finite only w.p. ρ and I ′ w (1) = mg (ρ) is the mean number of its nodes on the extinction set. Fixing the parameter m to its critical upper value m c = 1 0 dz ′ g(z ′ ) entails I w (1) = 1, I ′ w (1) = m c > 1. The size of the corresponding tree is then finite w.p. 1, with mean value m c . Note that if m = 1 < m c = 1 0 dz ′ g(z ′ ) , then I w (1) < 1 for all pgf's g.When m = m c , the EGF I w (z) is proper. It admits a closed form integrable expression in the following cases (respectively binomial, Poisson and negativebinomial):g (z) = (π 0 + π 1 z) d and I w (z) = π 0 z/z c ) -1/(d-1) -1 , z c = 1/ 1π d-(z) = e -µ(1-z) and I w (z) = -1 µ log (1z/z c ) , z c = 1/ 1e -µ g (z) = q 1pz θ and I w (z) = 1 p (1z/z c ) 1/(θ+1) , z c = 1/ 1q θ+1 . With α, λ ∈ (0, 1), suppose g (z) = 1λ (1z) α with w b = αλ [1α] b-1 /b!, b≥1 and z * = 1 (a Sibuya branching mechanism with infinite mean µ). In that case, z c < ∞, with z c > 1 if and only if m < 1 0 dz ′ g(z ′ ) = m c . When m = m c > 1, z c = 1 and I w (1) = 1. There is no major impact of the mean value µ of the branching mechanism on the extinction possibility of the increasing tree.

(26) E w Kn 1 = Γ (1/α 2 + θ 2 ) Γ (w 1 /α 2 + θ 2 ) Γ (w 1 /α 2 + θ 2 + n) Γ (n) Γ (n) Γ (1/α 2 + θ 2 + n)

Data availability statement

There are no data associated with this paper.

The probability to observe a particular size-n tree τ n among all size-n trees is w (τ n ) /W n . The tilted probability to observe a tree of size n among all possible trees is

Examples of w are:

-

-

In that separable case, each tree τ n has equal weight a n 1 a n-1

For the generating functions

appearing in the above interpretation of φ (z) as the EGF (of the type I w (z)) of an increasing tree under condition (A), the formation of an increasing tree admits the following recursive tree evolution scheme (label 1 is assigned to the root): with probability p b (n

Whenever a connection to a node with outdegree b occurs, the number of nodes with out-degree b (respectively b + 1) decreases (increases) by one unit. In addition, a new node with outdgree-0 is always created whatever the degree of the node to which the new incoming atom connects to τ n .

For the three particular φ-models generated by the g's above, using n-1 b=0 n b (τ n ) = n and n-1 b=1 bn b (τ n ) = n -1 for any τ n , we get: 1) , with,

Note that for g (z) = (1α 1 z) -(α2/α1-1) , z * = 1/α 1 , z c = 1/α 2 and φ (z c ) = z * .

Declarations of interest

The authors have no conflicts of interest associated with this paper.