Local probabilities for random permutations without long cycles

We explore the probability that a permutation sampled from the symmetric group of order n uniformly at random has cycles of lengths not exceeding r. Asymptotic formulas valid in specified regions for the ratio n/r are obtained using the saddle point method combined with ideas originated in analytic number theory. Theorem 1 and its detailed proof are included to rectify formulas for small r which have been announced by a few other authors.


Introduction
The history on enumeration of decomposable structures missing large components starts from the papers by K. Dickman and N.G. de Bruijn dealing with natural numbers composed of small prime factors. After numerous continuations, this analytic theory is now extensively developed and well exposed in the book by G. Tenenbaum [19] and in more recent papers. By analogy, a similar theory was carried out for polynomials over a finite field (see, for example, [16], [5]) and generalized to the so-called additive arithmetical semigroups (see [21], [13], [14]). The survey [8] discusses the parallelism between the theories. In no way, the list does not pretend to be complete, however, it has influenced the present paper devoted to permutations. So far, the results on this particular class of structures do not reach the level of research achieved for natural numbers. We focus only on permutations comprising the symmetric group S n and seek asymptotic formulas for the probability ν(n, r) that a permutation sampled uniformly at random has cycles of lengths not exceeding r, where 1 ≤ r ≤ n, r ∈ N, and n → ∞. The goal is to cover the whole range for the parameter r.
As we have been able to check, the last formula without a detailed proof firstly appears in A.N. Timashov's paper [20]. Note that his reference to V.N. Sachkov's paper [17], in which formulas (23) and (24) really concern n!ν(n, r) for an arbitrary r, is misleading. These formulas have been also presented without a proof; containing a misprint, they go in contrast to (3) and even to (2). We have to note that, a year later, M. Lugo [12] also gave (3) leaving for a reader other cases of n!ν(n, r). Recently E. Schmutz kindly gave a reference to the manuscript by T. Amdeberhan and V.H. Moll [2] dealing with the same problem. Their Theorem 8.1 also contains errors. We feel obliged to present a correct formula in Theorem 1. Let Γ(z) be the Euler gamma-function, where z ∈ C. Avoiding numerous brackets, instead of O(·), we will use a complex quantity B, not the same at different places but always bounded by an absolute constant. Otherwise, stressing dependence on a parameter v in an estimate, we will write O v (·) with the extra index.
Our main results are the next two theorems. We prefer to present them as asymptotic formulas for the local probability P ℓ r (Z) = n having the Cauchy integral representation where α > 0 is to be chosen. In the saddle point method, we take α = x := x(n/r), where the function x(u) := x r (u) is the unique positive solution to the equation r j=1 x(u) j = ur, u ≥ 1.
Evidently, the problem concerns asymptotical behaviour of the nth power series coefficient of the function exp j≤r z j /j , belonging to the so-called Hayman's class of admissible functions (see [10]). B. Harris and L. Schoenfeld [9] extended Hyman's methodology in obtaining further asymptotical terms. In particular, it yields for arbitrary bounded r. Here x = x(n/r) and Actually, we owe to E. Schmutz whose Theorem 1 and the facts presented below it in [18] clarify the use of the general and fairly complicated expansion given in [9]. A.N. Timashov [20] mentions a Sachkov's result from 1986, extending formula (6) for r = o(log n). Unfortunately, we failed to find a relevant paper.
The above mentioned results deal with the case when the ratio n/r is large. In addition, there exists a vast literature dealing with the case when n/r is small. In fact, the problem is related to the limit distribution of the longest cycle length (say, L n (σ)) and other statistics of σ ∈ S n . So, V.L. Goncharov's result [6] from 1944 shows that uniformly in u ≥ 1. Here ρ(u) is the Dickman function defined as the continuous solution to the difference-differential equation with the initial condition ρ(u) = 1 for 0 ≤ u ≤ 1. Since ρ(u) ≤ Γ(u + 1) −1 , the error in the last estimate can dominate if u → ∞. Theorem 4.13 in [3], applied for permutations, deals with the relative error. Namely, it shows that if n/r → u ∈ (0, ∞). As a byproduct of enumeration of elements in an additive arithmetical semigroup missing large factors, the last relation (extended to a larger region for n/r) has appeared in the first author's paper [14]. The result is contained in Theorem 3 below. The present paper fills up missing details in its sketchy and indirect proof.
Actually, the result holds for all 1 ≤ r ≤ n. The bound for r is left to show the limitations of the applied approach. We will prove in Lemma 11 of Section 5 that if 1 ≤ r ≤ log n. Nevertheless, a direct proof of Theorem 1 by the use of (7) and Theorem 2 would be rather involved. We press more on the Lagrange-Bürmann inversion formulas, instead. Lemmas 4 and 5 below provide approximations of x and λ(x) for larger r. Then it is preferable to apply another technique giving even a sharper remainder term. To catch an idea, let us obtain explicit expressions of the main term of the probability examined in Theorem 2.
Corollary 1. If n ≥ 4 and √ n log n ≤ r ≤ cn(log n) −1 (log log n) −2 , then Consequently, the approximation involving the Dickman function holds in a rather wide region for n/r. This makes us even more greedy and motivates in searching for another approach to refine the corollary.
Having in mind an analogy with number theory, one can guess that the last approximation using Dickman's function is hardly further extendable. The next corollary justifies one expression for all possible r.
On the other hand, Theorem 3 gives a better remainder term than Br/n if n 2/3 log 1/3 n < r ≤ n.
The paper is organized as follows. Section 2 collects known and new auxiliary properties of the involved functions and the saddle point approximations. Theorem 2 and Corollary 1 are proved in Section 3. Section 4 is devoted to Theorem 3 and Corollary 2. A detailed proof of Theorem 1 is given in the last section.

Auxiliary Lemmas
Throughout the section, we assume that r ≥ 2 if it is not indicated otherwise. Let ξ(u), ρ(u), and x(u) := x r (u) be the functions defined above for u ≥ 1. Recall that they are positive and differentiable if u > 1. We will often use the abbreviation f = f (u) and f ′ = f ′ (u) for the values at the point u, where f (v), v > 1, is any of the involved functions. and Proof. To establish the effective bounds for all u > 1, it suffices to employ the strictly increasing function I ′ (v). Indeed, the lower bound follows from the inequality To prove the upper estimate, it suffices to repeat the same argument. The asymptotical formulas for ξ(u) and its derivative can be found in [11] or in the book [19].
The lemma is proved.
be the Laplace transform of ρ(v), s = −ξ(u) + iτ =: −ξ + iτ and τ ∈ R. Then Proof. This is Lemma 8.2 in Section III.5.4 of [19]. Denote a ∧ b := min{a, b} and a ∨ b := max{a, b} if a, b ∈ R. Recall that x := x(u) is the solution to the saddle point equation and λ(x) = r j=1 jx j . Lemma 4. If u ≥ 3, then If 3 ≤ u ≤ e r , then Moreover, for u > 1, Proof. By definition, x > 1 and u ≤ x r ≤ ru for u > 1. The well-known property of geometric and arithmetic means We have from the definition that Consequently, by (14) and by virtue of Similarly, The last two inequalities imply for r ≥ log u.
If r ≤ log u, we have The latter and (16) lead to relation (11).
To sharpen (11) for 3 ≤ u ≤ e r , we iterate once more and obtain This is the first relation in (12). Comparing the result and Lemma 1, we have the second one.
To prove (13), we first observe that Further, due to (14) and The lemma is proved. Using properties of differentiable functions, we improve the remainder term estimates.
Proof. One may skip the trivial case when r is bounded. From (5) and (13), for the differentiable function x(v), we have if v ≥ 3. The same holds if 1 ≤ v ≤ 3. Indeed, in this case it suffices to apply the trivial estimate λ(x(v)) ≥ r 2 /2 ≥ r 2 v/6. As a function of v, exp ξ(v)/r is also strictly increasing; therefore, given any u ≥ 1 and the value ξ = ξ(u), we can find w ≥ 1 such that where v is a point between the u and w, irrespective of their relative position on the real line.
If u ≥ 3, then by virtue of ξ ∼ log u as u → ∞ and log u ≤ r, we obtain from (18) again follows from (20) and (21).
To derive approximation (19) of the logarithmic derivative, we use similar arguments. First, given u ≥ 3, we define y > 1 such that x = e ξ(y)/r and claim that ξ = ξ(y) 1 + B/r .
Its behavior outside a vicinity of the point t = 0 is rather complicated; therefore, we consider it in a separate lemma. Denote where x = x(n/r). In particular, λ 1 = ur and λ 2 = λ(x).

So, we obtain
. Lemma 7 is proved.
Proof of Theorem 2. As it has been mentioned in the Introduction, it suffices to examine the case when r ≥ 4 and n is large. In the introduced notation, we have u ≥ c −1 (log n)(log log n) 2 and with t 0 := r −7/12 n −5/12 . Expanding the integrand in K 1 (n), we use relations e it = 1 + it − t 2 /2 − it 3 /6+Bt 4 if t ∈ R and e w = 1+B|w|e |w| if w ∈ C. Consequently, checking that λ 4 t 4 0 ≤ (r 3 n)(r −7/3 n −5/3 ) = (r/n) 2/3 ≤ 1 and using the abbreviation λ := λ 2 , we obtain Recall that u = n/r, λ 1 = n, λ k ≤ r k u if k ≥ 1, and, by Lemma 4, λ = λ(x) ∼ nr as n → ∞ because of u → ∞. We now see that Considering K 2 (n), we first observe that, by virtue of (31), ℜg r (t) ≤ −(2/π 2 )λt 2 if t 0 ≤ |t| ≤ 1/r. Therefore, the contribution of the integral over this interval to K 2 (n) equals B/u √ λ. Further, we apply Lemmas 4 and 7 to get It remains to prove that the quantity in the large curly braces does not exceed − log u + B if the bounds of r are as in Theorem 2. This is trivial, if 4 log u > r + 1 ≥ 5. If 4 log u ≤ r + 1 and n is sufficiently large, we have an estimate 1 4π 2 u 1−4/(r+1) log 2 u ≥ 3cu log 2 u ≥ 3 log n(log log n) 2 (log log n + 2 log log log n + B) 2 ∼ 3 log n which assures the desired bound K 2 (n) = B/u √ λ. Inserting the estimates of K j (n), j = 1, 2, into (32), we finish the proof of Theorem 2.
Inserting these estimates into (33), we deduce Observe also that, by Lemma 5, a relation holds if 1 < u ≤ n/ log n. Now, it suffices to apply the last two relations only for c −1 (log n)(log log n) 2 ≤ u ≤ n/ log n. Taking into account Lemma 2, we can present the formula in Theorem 2 in two ways: Corollary 1 is proved.

Proof of Theorem 3
The idea is to use the Cauchy integral (4)  where T (−s) is the function examined in Lemma 6. Observe that, under the conditions of Theorem 2, 1 ≤ u ≤ n/ log n, where n may be considered large. Let us introduce the following vertical line segments in the complex plane: Using Lemmas 1, 2, and 3 for the case |τ | ≥ πr > 1 + uξ, we obtain In the last step, we have used the fact that I(ξ) ∼ e ξ /ξ as ξ → ∞. The next task is to estimate J. If s ∈ ∆ then, by Lemma 6, T (−s) = B and exp{T (−s)} = 1 + BT (−s). Let us split J into the sum of three integrals J k over the strips ∆ k , where k = 0, 1 and 2, respectively. If s ∈ ∆ 0 then T (−s) = B(1 + u log u)/r. Therefore, using Lemmas 1, 2, and 3, now for the case |τ | ≤ π, we derive Further, To estimate L 1 , we combine the first estimate ofρ(s) given in Lemma 3 with Lemmas 1 and 2. So we obtain Similarly, the second estimate in Lemma 3 leads to Estimation of the integral L 3 is more subtle. It uses an estimate following from Lemma 6 and the asymptotic formula b(v) = 1 + v/2 + Bv 2 for |v| ≤ π √ 2. We have Using the two different estimates ofρ(s) on the line segments ∆ 11 := {s ∈ ∆ 1 : |ℑs| ≤ 1 + uξ} and ∆ 12 := ∆ 1 \ ∆ 11 given by Lemma 3, we proceed as follows: r .
Collecting the obtained estimates, we obtain The same holds for integral J 2 . Consequently, Proof. This is Lagrange-Bürmann Inversion Formula, presented, for instance on page 732 of a fairly concise book [4].
In this note, superpositions of series involving f (z) are used, therefore we recall more variants of the inversion formula. Let us stress that, by Lemma 8, f 1 = 1; therefore, z/f (z) and log z/f (z) have formal power series expansions.
Lemma 9. Let f (z) be as in Lemma 8 and j ∈ N. Then

Moreover,
Proof. Without a proof the first part of Lemma 9 is exposed as A.11 on pages 732-733 of [4]; an inaccuracy is left in the case N = j, however. For readers convenience, we provide a sketch of a proof.
Let N ∈ N 0 be fixed. The coefficients under consideration have expressions in terms of Φ k with 0 ≤ k ≤ N only; therefore, we may assume that Φ(u) is a polynomial of degree N . Then f (z) is well defined as an analytic function in a vicinity of the zero point. Thus, we may apply Cauchy's formula. Afterwards let δ and δ 1 be sufficiently small positive constants. Using a substitution z = u/Φ(u) and properties of the one-to-one conformal mapping of the vicinities of the zero points in the z-and u-complex planes, we obtain The same argument gives Finally, applying the previous substitution, we derive The lemma is proved. We will apply the lemmas in a very particular case. Then the first power series coefficients of implicitly defined functions attain a simple form. Let 1{·} stand for the indicator function. (I) If g(z) j =: For (37), apply the second part of Lemma 9 to obtain Similarly, (II) follows from the last formula in Lemma 9.
Having in mind that z r h(z) has a power series expansion in C[[z]], we may apply the same principles. Using (37), it is easy to check that Further, we observe that Hence

This implies that
if N ≥ −r + 1 and h −r = 1/r. To prove (IV ), we use relation (38) again. Differentiating it, we arrive at The assertion (IV ) now is evident. The lemma is proved.
Corollary 3. As above, let g(z) = z/y(z). Then g 0 = 1, g 1 = −1/r, Proof. Apply (I) of Lemma 10 for j = 1. If 2 ≤ N ≤ r − 1, the relevant sum has the only nonzero summand corresponding to the pair (l, m) = (0, N ). A formula for g r has two summands giving the expression. If N ≥ 2, then by Lemma 10 and Cauchy's inequality, The corollary is proved.
Proof. Again, if 1 ≤ N ≤ r − 1, it suffices to observe that the relevant sum (II) of Lemma 10 has the only nonzero summand corresponding to the pair (l, m) = (0, N ). A formula for b r has two subtracting summands.
If N ≤ r − 1, the given estimate follows from (39). If r < N ≤ 2r − 1, assertion (II) in Lemma 10 gives We have applied an inequality log Finally, by Cauchy's inequality, The corollary is proved. We now prove the promised expansion (7).
Lemma 11. If 2 ≤ r ≤ log n, then Proof. The equation defining x can be rewritten as This gives the relation y(n −1/r ) = x −1 , where y = y(z) has been explored in Lemma 10. Consequently, we may apply the expansions of g(z) given in (I) with respect to powers of z = n −1/r . The first coefficients have been calculated in Corollary 3. It remains to estimate the remainder. Using also the obtained estimates, we have if |z| ≤ e −1 . Consequently, we obtain Define the functions R(z) and K r (z) by We claim that R(z) = B(|z| + n|z| r+1 ) if |z| ≤ e −1 implying R(n −1/r ) = Bn −1/r .
Concluding Remark. The approach can be adopted for more general decomposable structures, in particular, for the so-called logarithmic classes of set constructions (see [3]). Using a different method, X. Gourdon [7] has established some results related to our Theorem 2. They concern the asymptotic distribution of size of the largest component.