On some densities in the set of permutations

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1 On some densities in the set of permutations Eugenijus Manstavičius Department of Mathematics and Informatics, Vilnius University and Institute of Mathematics and Informatics Vilnius, Lithuania Submitted: Aug 24, 2008; Accepted: Jun 26, 200; Published: Jul 0, 200 Mathematics Subject Classifications: 60C05, 05A6 Abstract The asymptotic density of random permutations with given properties of the kth shortest cycle length is examined. The approach is based upon the saddle point method applied for appropriate sums of independent random variables. Introduction Let n N, S n be the symmetric group of permutations acting on the set {, 2,..., n}, and S := S S 2 Set ν n for the uniform probability measure on S n. By ν n A := ν n A S n, we trivially extend it for all subsets of S. If the limit lim ν na =: da, n A S exists, da can be called the asymptotic density of A. Let S S be the class of A having an asymptotic density da. The triple {S, S, d} is far from being a probability space. However, the behavior of da m for some specialized subsets A m S as m are worth to be investigated. In this paper, we demonstrate that by taking sets connected to the ordered statistics of different cycle lengths. Recall that each σ S n can be uniquely up to the order written as a product of independent cycles. Let k j σ 0 be the number of cycles of length j, j n, in such decomposition. The structure vector is defined as kσ: = k σ,...,k n σ. The final version of this paper was written during the author s stay at Institute of Statistical Science of Academia Sinica, Taipei. We gratefully acknowledge the support and thank Professors Hsien-Kuei Hwang and Vytas Zacharovas for the warm hospitality. the electronic journal of combinatorics 7 200, #R00

2 Set l k := k + + nk n, where k := k,..., k n Z n +, then l kσ = n. Moreover, if l k = n, then the set {σ S n : kσ = k} agrees with the class of conjugate permutations in S n. If ξ j, j, are independent Poisson random variables r.vs given on some probability space {Ω, F, P }, Eξ j = /j, and ξ: = ξ,...,ξ n, then [2] ν n kσ = k = {l k = n} n j= j k j kj! = P ξ = k l ξ = n, where denotes the indicator function. Moreover, k σ,...k n σ, 0,..., ν n ξ,...,ξ n, ξ n+,... in the sense of convergence of the finite dimensional distributions. Here and in what follows we assume that n. The so-called Fundamental Lemma sheds more light than. We state it as the following estimate of the total variation distance. If kσ r := k σ,...,k r σ and ξ r := ξ,..., ξ r, then [] 2 k Z r + ν n kσr = k P ξr = k = Rn/m 2 for r n. Here and in the sequel Ru denotes an error term which has the upper bound Ru = Oe ulog u+ou 3 with some absolute constants in O. In the recent decade, a lot of investigations were devoted to the limit distributions of values of additive functions with respect to ν n. Given a real two dimensional sequence {h j k}, j, k, h j 0 0, such a function is defined as hσ = j n h j k j σ. The relevant references can be found in [2], [2], [5] and other papers. The family of additive functions sσ, y = { k j σ }, y n, j y is closely related to the ordered statistics j σ < j 2 σ < < j s σ of different cycle lengths appearing in the decomposition. Now s := sσ, n counts all such lengths. We have sσ, j k σ = k for each k s. The last relation and the laws of iterated logarithm for sσ, m led [4] to the following result. Denote Lu = log max{u, e} = L u,..., L r u = LL r u for u R. For 0 < δ < and r 2, set β rk ± δ = 2k L 2 k + 32 L 3k + L 4 k + + ± δl r k /2. the electronic journal of combinatorics 7 200, #R00 2

3 Theorem [4]. For arbitrary 0 < δ < and r 2, we have lim lim ν log j k σ k n max = 0 n n n k s β rk + δ and lim lim log j k σ k ν n max =. n n n k s β rk δ Thus, we may say that for almost all σ S n log j k σ k β rk + δ uniformly in k, n k s, where n arbitrarily slowly. This assertion is sharp in the sense that we can not change δ by δ. It can be compared with the following corollary of the invariance principle see [3], [4] where the convergence of distributions is examined. Theorem [4]. Uniformly in x R, ν n max log j kσ k x log n = x l e u 2lx2 /2 du + o. k s 2π Instead of j k σ, k s, one can deal with the sequence l Z J σ J 2 σ J w σ of all cycle lengths appearing in the decomposition of σ. The behavior of these ordered statistics is similar, however, some technical differences do arise in their analysis. Section 3.2 of the paper by D. Panario and B. Richmond [6] contains rather complicated asymptotical formulas for ν n J k σ = m as m, n if k is fixed. We are more interested into the case when k is unbounded therefore we now include V.F. Kolchin s result for the so-called middle region. Theorem [9]. Let 0 < α < be fixed, k = α log n + o log n, and n. Then ν n log Jk σ k + x k = 2π x Despite such variety of results, the frequency x e u2 /2 du + o =: Φx + o. ν n jk σ = m = ν n km σ, sσ, m = k, 4 where k m n, can further be examined. Observe that the event under frequency is described in terms of the first m components k j σ of the structure vector. If m = on, then by 2 its frequency can be approximated by an appropriate probability for independent random variables ξ j, j m. Introduce the independent Bernoulli r. vs η j, j, such that and set X y = j y η j where y n. Pη j = = e /j = Pη j = 0 the electronic journal of combinatorics 7 200, #R00 3

4 Proposition. For k m = on, = ν n jk σ = m = Pη m = PX m = k + Rn/m { e /m exp } k e /jr j j<m j < <j k <m r= +Rn/m, where the remainder term is estimated in 3. This simple corollary following from 2 motivates our interest to the probabilities P m k := PX m = k if k 0 and m 3. Of course, we can exclude the trivial cases P m k = 0 if k > m, { P m 0 = exp j m } e γ j m, and P m m = e /j Pm 0 C 0. m! j m Here m, γ is the Euler constant, and C 0 = + 2!j + 3!j + e /2j. 2 j Contemporary probability theory provides a lot of local theorems for sums of independent Bernoulli r. vs. Since λ m := EX m = j m e /j = log m + C + O, m C := γ + j e /j, j and, similarly, VarX m = log m + C + O, m where C is a constant, and m 3, the results on the so-called large deviations imply the approximations for P m k in the region k log m = olog m as m see [7], Chapter VIII or [7]. H.-K. Hwang s work [8] as well as many others can be used in this zone. However, we still have a great terra incognita if + ε log m k m where ε > 0. The present paper sheds some light to it. First, we prove some new asymptotic formulas for P m k which are nontrivial outside the region of classical large deviations. Further, we apply them by inserting into the equalities given in Proposition. The very idea goes back to the number-theoretical paper by P. Erdős and G. Tenenbaum [6]. the electronic journal of combinatorics 7 200, #R00 4

5 We now introduce some notation. Denote Fz, m = 0 k mq k mz k := j m + e /j z, z C. Then P m k = q k m/f, m. Let ρt, m satisfy the saddle point equation for 0 t m. Set x F x, m Fx, m = x a j m j + x = t, a j := e /j 5 Wt = Γt + t t e t 6 if t > 0 and W0 =, where Γt denotes the Euler gamma-function. Theorem. Let 0 < ε < be arbitrary, m 3, and 0 0 :=. Then Fρk, m, m P m k = + O F, m ρk, m k Wk log m uniformly in 0 k m ε. Further analysis of the involved quantities leads to interesting simpler formulae. Set m Lt, m = log + t/ log m. Corollary. If 0 k m ε, then and P m k = P m k = Fk/Lk, m, m F, m Lk, m k F, m k! Lk, m e { k } exp O, 7 log m k { k k! exp O log 2 m + }. 8 log m The first formula in the corollary implies an asymptotic expression only in the region k = olog m, however, it yields an effective estimate of P m k for 0 k m ε. The classical results for k log m = olog m are hidden in the second one. Instead of going into the details of that, we return to the cycle lengths and exploit Proposition. Set d k m = d j k σ = m = e /m P m k. 9 Theorem 2. Let m 3, 0 < ε <, and k m ε. Then d k+ m Lk, m = + O. 0 d k m k log m Moreover, max d km = k m and the maximum is achieved at m + O 2π log m log m k = k m = log m + O. the electronic journal of combinatorics 7 200, #R00 5

6 Having this and some other arguments in mind, we conjecture that d k m is unimodal for k m and all m 3. Going further, we can exploit an idea from the renewal theory. The event {σ : j k σ > y} occurs if and only if σ S n has less than k cycles with lengths in [, y]. Hence, by 2, d j k σ > y = d sσ, y < k = PX y < k. The last probability is traditionally examined as y and k = ky belonging to specified regions. This can be exploited. For instance, applying formula 6 from [4], we have sup d jk σ > y Π λy k = e /j 2 + O log y 3/2, k 2λ y 2πe where Π λy is the Poisson distribution with the parameter λ y defined above. The paper [4] and many other works published in the last decade provide even more exact approximations applicable for d j k σ > y. We now seek an asymptotical formula for it as k, where y = yk is a suitably chosen function of k. That may be ascribed to the renewal theory when the summands η j, j in X y are independent but non-identically distributed. The next our result resembles in its form the Kolchin s theorem. The very idea goes back to the numbertheoretical paper [5] by J.-M. De Koninck and G. Tenenbaum. j y Theorem 3. We have d log j k σ k + x k = Φx x2 3C 3 2πk uniformly in k and x R. e x2 /2 + O k 2 Finally, we observe that these results can be used to obtain asymptotical formulas for max m k d k m as k. We intend to discuss that in a forthcoming paper. 2 The saddle point method Since P m k = q k m/f, m, it suffices to analyze the Cauchy integral q k m = Fz, m dz. 3 2πi z k+ z =ρ Similar but more complicated integrals have been the main task in the work on the number of permutations missing long cycles see [0] and []. We now exploit this experience and the ideas coming from paper [6]. Henceforth let k 0, m 3, and 0 < ε <. For 0 t m ε, we have L := Lt, m log m. Here and in the sequel the symbol a b means a b and b a while the electronic journal of combinatorics 7 200, #R00 6

7 is an analog of O. The implicit constants in estimates depend at most on ε therefore the remainder O/L is just a shorter form of O/ logm. Observe that the functions x/a j + x, j m 2, are strictly increasing for x [0, = R + therefore the sum over j varies from 0 to the value m 0. This proves the existence of a unique ρt, m > 0 for 0 < t m and m 3. Moreover, ρ0, m = 0. The main task of this section is to prove the following proposition. Proposition 2. Let ρk, m be the solution to equation 5 for t = k. Then Fρk, m, m q k m = + O ρk, m k Wk L uniformly in 0 k m ε. The function Wt has been defined in 6. Firstly, we prove a few lemmas. Lemma. For all 0 t m ε, ρt, m = t + O. 4 Lt, m L Proof. By the definition, using the inequalities 0 < j a j 2 for all j and the abbreviation ρ := ρt, m, for t > 0, we obtain t ρ = j + ρ + a j m j m j + ρ j + ρ m du m + ρ = + O = log + O. 5 u + ρ + ρ By virtue of we have ρ m ε. Now 5 reduces to m ε t mρ m + ρ, t ρ = log m + O. 6 + ρ If ρ, then t/ρ = log m + O. The last equality is equivalent to 4. In the remaining case ρ m ε, where m mε is sufficiently large, by 6, log m + O t ρ ε log m + O. Hence ρ = Bt/ log m with some B = Bt, m, where 0 < cε B Cε for all m 3 and some constants cε and Cε depending on ε. Since log + ρ = log + t log m + Bt + log log m log m + t = log + t + O, log m the electronic journal of combinatorics 7 200, #R00 7

8 from 6 we obtain the desired formula 4. The lemma is proved. We set b j := a j = e /j and examine an analytic function ϕz which, for z < a or Rz > 0, is defined by ϕz := j m log + b j z = log Fz, m. Denote s r = dr ϕρe w dw r. w=0 Lemma 2. In the above notation, if k m ε, then If z + 2ρ/4ρ, then Moreover, s = k and ϕρ = k + Oρ = k + OL. 7 fz := ϕρz ϕρ = kz + OkL z 2. 8 s r = k + OL, r 2. 9 Proof. We will use the estimates 0 < b j min{2, e/j} and 0 < b j /j 2j 2 for j. It suffices to take sufficiently large m. In the proof of Lemma, we observed that ρ C εm ε. If ρ 2e, then ϕρ = + log + b j ρ 2eρ<j m = ρ 2eρ<j m j 2eρ b j + O ρ 2 2eρ<j m = ρ log m ρ + + Oρ = k + Oρ = k + OL. + O log3eρ/j j 2 j 2eρ In the last step we applied formula 4 and in the step before that we applied formula 6. The derived expression for ϕρ also holds in the easier case ρ < 2e. We omit the details. To prove formula 8, we observe that b j ρ z + b j ρ in the given region, therefore the function fz = j m 2ρ + 2ρ + 2ρ 4ρ log = 2 + b jρz + b j ρ the electronic journal of combinatorics 7 200, #R00 8

9 is analytic in it. Expanding the logarithm, we obtain fz = z b j ρ + b j m j ρ + O ρ 2 z 2 j 2 + ρ 2 j m ρ 2 z 2 = kz + O ρ + = kz + O kl z 2. To derive relations 9, it suffices to apply 8 and Cauchy s formula on the circumference w = c, where c > 0 is chosen so that e w ce c /2 + 2ρ/4ρ. The lemma is proved. Remark. The argument mentioned in the last step actually yields the Taylor expansion of fe w in the region w c. Lemma 3. There exists an absolute positive constant c such that uniformly in k m ε and τ π. Proof. We apply the identity Fρe iτ, m exp { c kτ 2 }Fρ, m + xe iτ 2 2x cosτ = + x 2 + x 2 with x = b j ρ eρ/j /4 for j > 4eρ and obtain Fρe iτ, m 2 Fρ, m 2 2b jρ cosτ + b 4eρ j m j ρ 2 { exp log 2b } jρ cos τ + b j ρ 2 { exp 4eρ<j m { exp 4 5 4eρ<j m cosτ } b j ρ cos τ + b j ρ 2 } b j ρ. + b j ρ 4eρ<j m Now, to involve k, using the definition of ρ we complete the sum in the exponent by the quantity b j ρ k + b j ρ = Oρ = O. L j 4eρ By virtue of the inequality cosτ 2τ 2 /π 2, we now obtain Fρe iτ, m 2 Fρ, m 2 exp { 8τ2 5π 2k + O L } the electronic journal of combinatorics 7 200, #R00 9

10 for k m ε. This implies the desired estimate if m mε is sufficiently large. For 3 m mε, the claim of Lemma 3 is trivial. The lemma is proved. Proof of Proposition 2. For k = 0, its claim is evident therefore we assume that k. Firstly, we separate a special case. If ρ /6, then z = implies z 2 2ρ+/4ρ. Thus estimate 8 is at our disposal. Moreover, as we have observed in the proof of Lemma, k/ρ = log m + O and L log m. So, using Lemma 2, we obtain q k m = = = Fρ, m exp{fw} dw ρ k 2πi w = w k+ Fρ, m e kw + O kl w 2 dw eρ k 2πi w = w k+ Fρ, m k k ke k π eρ k k! + O e k cos τ cosτdτ L π. By virtue of cosτ τ 2 for τ π, the last integral is of order k 3/2. This and Stirling s formula yield the desired asymptotic formula in the selected case. If /6 ρ m ε, then we can start with q k m = Fρ, m ρ k 2π π π exp{fe iτ ikτ}dτ. Using the expansion of fe iτ mentioned in Remark after Lemma 2, for τ c, we have Hence fe iτ = ikτ 2 s 2τ 2 i 6 s 3τ 3 + Okτ 4. exp{fe iτ ikτ} = e s 2τ 2 /2 is 3τ Okτ 4 + k 2 τ 6 for τ ck /3. Exploiting the symmetry of the term with τ 3 we have I : = exp{fe iτ ikτ}dτ 2π τ ck /3 = e s 2τ 2 /2 dτ + Ok 3/2 2π R τ >ck /3 = + Ok 3/2 = + OL. 2πs2 2πk In the last step we used 9 and the inequality k L following from ρ /6. Applying Lemma 3 we obtain I 2 : Fρe iτ, m = dτ ck /3 τ π Fρ, m ckτ2 dτ k 3/2. τ ck /3 e the electronic journal of combinatorics 7 200, #R00 0

11 Collecting the estimates of I and I 2, we see that q k m = Fρ, m I ρ k + OI 2 = Again Stirling s formula yields the desired result. The proposition is proved. Fρ, m ρ k 2πk + OL. 3 Proofs of Theorems and Corollaries The claim of Theorem follows from Proposition 2. We will discuss only the remaining statements. Proof of Corollary. The case k = 0 is trivial. If k, formula 7 follows from Proposition 2, 4, and 7. To prove 8, let us set ρ = ρk, m and L = Lk, m. Applying Proposition 2 we approximate Fρ, m by Fk/L, m. That is available because of the inequality k/ρl /2 following from 4 provided that m is sufficiently large. By 8 and 4, we have ϕρ = ϕ ρ k k k ρl ρl k k = ϕ k log L ρl k + O L k + O. L 2 k 2 ρl Inserting this into the equality in Proposition 2 we complete the proof of 8. Proof of Theorem 2. Observe that, for k t k m ε, we have Lt, m = Lk, m + O/ logm and Lk, m = Lk, m + O/m therefore afterwards, for different arguments, we may use L = Lk, m. Denote rt := ρt, m, r 0 = rk, and r = rk. Then, by Proposition 2, d k+ m d k m = q m k q m k = Fr, m Fr 0, m r k 0 r k k k + O e k L for k. Set We have d k+ m d k m Kt = log F rt, m t log rt. { k k } = exp K tdt + k log + O. k k L the electronic journal of combinatorics 7 200, #R00

12 By the definitions, F K t = r rt, m t F rt, m t log rt rt = log rt = log L log t + OL. Inserting this expression into the previous equality and integrating we obtain the desired result 0. To prove, we first restrict the region to k m where all obtained remainder term estimates contain absolute constants. Hence, by virtue of 0, we can find absolute positive constants C 2 and C 3 such that d k m is increasing for k log m C 2 and decreasing for log m + C 3 k m. If k = log m + O, then Lk, m = log m + O, ρ = ρk, m = + Olog m, and, by 8 applied with z = /ρ, ϕρ k log ρ ϕ / log m. This and Proposition 2 imply P m k = exp { ϕρ k log ρ ϕ } + O Wk log m = + O 2π log m log m for k = log m + O. The same holds for P m k. Recalling 9 we obtain max k d k m = m m 2π log m + O. 20 log m It remains to estimate d k m for m k m. Differentiating the function Fz, m we have the estimate q k m k e /j k! j m exp { k log2 log m + C 4 k log k + Ok } exp{ /3 m log m} which shows that the maximum of d k m = q k m /F, m over k m is given by 20. The theorem is proved. Proof of Theorem 3. Let gz = log + b j z b jz + Cz, Gz = e gz. + b j j the electronic journal of combinatorics 7 200, #R00 2

13 The function gz has an analytic continuation to the region C\[ e,, however, we prefer to use it as a function of real variable. Observe that Gu = e γ + Cu + O u 2. 2 for u [0, T], where T > 0 is an arbitrary fixed number and the constant in O depends on T. For such u, using elementary inequalities we can rewrite { Fu, y = exp log + b j u b ju + u bj + u } + b j + b j y j j j j y j y = Guy u + O/y, 22 where the remainder depends on T only. Let k be arbitrary, 0 l k, and let y 3 be such that k/0 log y 0k. Then Ll, y log y log and l Ll, y = l + O, log y log y where the constant in O is absolute. From formula 8 with y instead of m and 22 with u = l/ll, y, we obtain q l y = F, yp y l l Ll, y l = F Ll, y, y + O e l! k l { l log y } = G exp Ll, y Ll, y + l log Ll, y l + O l! k l log y l = G + O log y l! k for 0 l k with an absolute constant in the remainder term. In the exponent, we have used the second order approximation of the logarithmic function. If k/0 log y 0k, then recalling 2 and 22, we arrive at d j k σ > y = F, y = 0 l<k = 0 l<k 0 l<k log y l l!y q l y log y l e log y l! { l l } 2 + C log y + O k + log y log yk C k!y + O. k Consequently, if x k /6 and k 3, the last equality for y = e k+x k implies d log j k σ > k + x k = S k log y k + x k k e k x C k + x k k k! k k! e k x k + O, 23 k the electronic journal of combinatorics 7 200, #R00 3

14 where, by Lemma 2. from [5], S k log y := k + x k l e k x k l! 0 l k = Φx x2 e x2 /2 3 2πk + O. k For the other terms in 23, Stirling s formula yields k + x k k e k x k k! /2 x 3 = e x2 + O + O 2πk k k /2 = e x2 + O 2πk k provided that x k /6. The last quantity on the right-hand side is also equal to the factor of C in 23. Inserting these estimates into 23 we obtain the desired result 2 for x k /6 and k 3. If x k /6, by monotonicity of x d log j k σ > k + x k and the just proved relation, d log j k σ > k + x k d log j k σ > k + k 2/3 k. Consequently, formula 2 trivially holds in this region. Similarly, if x k /6, to verify equality 2, we can use the estimate d log j k σ k + x k d log j k σ k k 2/3 k. The theorem is proved. Acknowledgement. The author thanks an anonymous referee for his helpful advises how to improve the presentation of the paper. References [] R. Arratia and S. Tavaré, The cycle structure of random permutations, Ann. Probab., 992, 20, 3, [2] R. Arratia, A.D. Barbour and S. Tavaré, Logarithmic Combinatorial Structures: a Probabilistic Approach, EMS Monographs in Mathematics, EMS Publishing House, Zürich, [3] G.J. Babu and E. Manstavičius, Brownian motion and random permutations, Sankhyā, A, 999, 6, 3, [4] K. Borovkov and D. Pfeifer, On improvements of the order of approximation in the Poisson limit theorem, J. Appl. Probab., 996, 33, [5] J.-M. De Koninck and G. Tenenbaum, Sur la loi de répartition du k-ième facteur premier d un entier, Math. Proc. Cambridge Phil. Soc., 2002, 3, [6] P. Erdős and G. Tenenbaum, Sur les densités de certaines suites d entiers, Proc. London Math. Soc., 989, 59, 3, [7] H.-K. Hwang, Large deviations of combinatorial distributions. II. Local limit theorems, Ann. Appl. Probab., 998, 8, the electronic journal of combinatorics 7 200, #R00 4

15 [8] H.-K. Hwang, Asymptotics of Poisson approximation to random discrete distributions: an analytic approach, Advances in Appl. Probab., 999, 3, [9] V.F. Kolchin, A problem of the allocation of particles in cells and cycles of random permutations, Teor. Veroyatnost. i Primenen., 97, 6,, Russian. [0] E. Manstavičius, Semigroup elements free of large prime factors. In: New Trends in Probability and Statistics, vol 2. Analytic and Probabilistic Methods in Number Theory, F.Schweiger, E.Manstavičius, Eds, TEV/Vilnius, VSP/Utrecht, 992, [] E. Manstavičius, Remarks on the semigroup elements free of large prime factors, Lith. Math. J., 992, 32, 4, [2] E. Manstavičius, Additive and multiplicative functions on random permutations, Lith. Math. J., 996, 36, 4, [3] E. Manstavičius, The law of iterated logarithm for random permutations, Lith. Math. J., 998, 38, [4] E. Manstavičius, Iterated logarithm laws and the cycle lengths of a random permutation, Trends Math., Mathematics and Computer Science III, Algorithms, Trees, Combinatorics and Probabilities, M.Drmota et al Eds, Birkhauser Verlag, Basel, 2004, [5] E. Manstavičius, Asymptotic value distribution of additive function defined on the symmetric group, The Ramanujan J., 2008, 7, [6] D. Panario and B. Richmond, Smallest components in decomposable structures: Exp- Log class, Algorithmica, 200, 29, [7] V.V. Petrov, Sums of Independent Random Variables, Springer Verlag, Berlin, 975. the electronic journal of combinatorics 7 200, #R00 5