Asymptotics of the Eulerian numbers revisited: a large deviation analysis

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1 Asymptotics of the Eulerian numbers revisited: a large deviation analysis Guy Louchard November 204 Abstract Using the Saddle point method and multiseries expansions we obtain from the generating function of the Eulerian numbers A nk and Cauchy s integral formula asymptotic results in noncentral region. In the region k = n n α > α > /2 we analyze the dependence of A nk on α. This paper fits within the framework of Analytic Combinatorics. Keywords: Eulerian numbers Asymptotics Saddle point method Multiseries expansions Analytic Combinatorics. 200 Mathematics Subject Classification: 05A6 60C05 60F05. Introduction The Eulerian numbers A nk have been the object of renewed interest recently. They are defined by the recurrence A n+k = n k + 2A nk + ka nk where we chose A 00 = 0 A 0 =. They correspond for instance to runs in permutations. The double exponential generating function is given by We have gz w := n=0 k=0 k=0 A nk z n k w = n! A nk n! = hence we can define a random variable RV J n such that P[J n = k] = A nk. n! w w e w z w. From Flajolet and Sedgewick [5] ch.ix we know that the roots of the denominator are with h j w = fw + 2ijπ w j Z fw = w lnw. As w fw is close to whereas the other poles h j w with j 0 escape to infinity. This fact is consistent with the limit form gz = z which has only one simple pole at. If one Université Libre de Bruxelles Département d Informatique CP 22 Boulevard du Triomphe B-050 Bruxelles Belgium louchard@ulb.ac.be

2 restricts w to w 2 there is clearly at most one root of the denominator in z 2 given by fw. Thus we have for w close enough to with Rz w analytic in z 2 and gz w = fw + Rz w z [z n ]gz w = fw n+ + O2 n. Note that fw does not correspond to a discrete RV but if we set w = e t fe t n+ corresponds to a sum of n + independent RV uniformly distributed on [0 ]. In the rest of this paper we set m := n + µ := m. The corresponding mean and variance are given by M = m 2 σ 2 = m 2. The generating function of the mean and second moment are given respectively by 2 z 2 z z 3. Note that the exact mean and second moment generating functions are derived from gz w as 2 2z + z 2 2 z 2 6 2z + 5z 2 7z 3 + z 4 6 z 3. Of course asymptotically by classical singularity analysis exact and asymptotic moments are the same. ˆ In the central region k = M + xσ x = O J n is asymptotically normal. This has first been proved by David an Barton [3]. Without being exhaustive a very complete bibliography can be found in Janson [9] let us also mention Bender [] Carlitz et al. [2] Tanny [3]. The first two terms of a correction were given in Sira zdinov [4] and Nicolas [2]. A complete analysis is given in Gawronski and Neuschel [6]: There exists polynomials q ν ν such that for any l 0 as n uniformly for all k Z A nk 6 l /2 q ν x = n! πn + e x2 + n + ν + O n l 3/2 ν= q ν x = 2 ν ν H 2ν+2s x6 s B km 2m+2 m + 2m + 2! m= summing over all non-negative integers k... k ν with k + 2k νk ν and letting s = k + k k ν. A very simple proof is given in Janson [9]. ˆ As far as the large deviation is concerned let us mention Bender [] Hwang [8]. Esseen [4] improves Bender s result as follows: 2

3 Let a := k/n + uniformly in all 0 < k < n +. Let ta be the solution of Set a = eta e ta ta. ma = eta tae ata σ 2 a = ta 2 e ta e ta 2. Then A nk n! = ma n+ 2πn + σa + On. As noted by Esseen further terms can be obtained in this asymptotic. All these papers use the solution ρ of mwf w kfw = 0 2 which actually corresponds to the Saddle point of the Saddle point method see Sec.2. In this paper we are interested in the extreme large deviation case k = m m α /2 < α <. the choice of this range is justified in Sec.3. This range was already the object of our analysis of Stirling numbers of first and second kind see Louchard [0] []. Let us summarize the motivation of this paper: α is chosen such that m α is integer. ˆ Previous papers simply use ρ as the solution of 2. They don t compute the detailed dependence of ρ on α neither the precise behaviour of functions of ρ they use. ˆ We will use multiseries expansions: multiseries are in effect power series in which the powers may be non-integral but must tend to infinity and the variables are elements of a scale. The scale is a set of variables of increasing order. The series is computed in terms of the variable of maximum order the coefficients of which are given in terms of the next-to-maximum order etc. This is more precise than mixing different terms. Our work fits within the framework of Analytic Combinatorics. In Sec.2 we revisit the asymptotic expansion in the central region and in Sec.3 we analyze the non-central region k = m m α α > /2. We use Cauchy s integral formula and the Saddle point method. Sec.4 provides a justification of the Saddle point technique we use here. 2 Central region In this section as a warm-up we rederive the first terms of the asymptotics. We use the Saddle point technique for a good introduction to this method see Flajolet and Sedgewick [5] ch.v III. 3

4 Let ρ be the Saddle point and Ω the circle ρe iθ. By Cauchy s theorem A nk fz m dz n! 2πi Ω zk+ = π ρ k fρe iθ m e kiθ dθ 2π π π = ρ k e m lnfρeiθ kiθ dθ 2π π = fρ m π [ { ρ k exp m 2π π 2 κ 2θ 2 i }] 6 κ 3θ dθ 3 i κ i ρ = lnfρe u u. 4 u=0 See Good [7] for an ancient but neat description of this technique. Let us now turn to the Saddle point computation. ρ is the root of smallest modulus of 2 with k = M + xσ. This amounts to [ w lnw w lnw 2 w w ] 2 lnw The solution is given by the asymptotic expansion µ ρ := + 2x3/2 µ µ 2 x3/2 w µ = 0. 6 lnw + 6x2 µ x3 3 /2 5µ x4 5µ 4 + O µ 5. In the sequel we will only give the first terms of our expansions of course Maple knows much more. This leads to T = k lnρ = x3 /2 µ x 2 x3 3 /2 5µ x4 5µ 2 + O µ 3 T 2 = µ 2 lnfρ = x3 /2 µ + x2 2 + x3 3 /2 5µ + 3x4 T + T 2 = x2 2 x4 20µ 2 x6 050µ 4 + O µ 6 20µ 2 + O µ 3 T 3 = expt + T 2 + x2 2 = x4 20µ 2 + /050x6 + x 8 /800 µ 4 + O Now we turn to the integral in 3. The first κ i are given by κ[2] = 2 x2 20µ 2 + 2x4 525µ 4 + 2x6 κ[3] = x3/2 60µ + 79x3 3 /2 6300µ 3 + O µ 4 κ[4] = 20 + x2 42µ 2 + O 2625µ 6 + O µ 8 µ 5 µ 6. and similar expressions for the next κ i that we don t detail here. We proceed as in Flajolet and Sedgewick [5] ch.v III. Let us choose a splitting value θ 0 such that mκ 2 θ0 2 mκ 3θ0 3 0 n. For instance we can use θ 0 = µ /2. We must prove that the integral K mk = 2π θ0 θ 0 e m lnfρeiθ kiθ dθ 4

5 is such that K mk is exponentially small. This is done in Appendix 4. Now we use the classical trick of setting [ ] m κ 2 θ 2 /2! + κ l iθ l /l! = u 2 /2. Computing θ as a series in u this gives by Lagrange s inversion θ = 6 l=3 ]/ u i a[i]/µ = 3 [u 2 /2 + 3x2 2ix 5µ 2 + O µ 4 + u 2 5µ ix3 525µ 4 + O µ 6 + Ou 3 µ. This expansion is valid in the dominant integration domain u µ a θ 0 = µ /2. Setting dθ = dθ dudu we integrate on u = [.. ]: this extension of the range is justified as in Flajolet and Sedgewick [5] ch.v III. This gives T 4 := 3/2 2 /2 π /2 µ Now it remains to compute T 5 := T 3 T 4 = 3/2 2 /2 π /2 µ [ 3x / 57x µ x / ] µ 4 + O 20 µ 6. [ + x x2 0 3 / x µ x x x / ] µ 4 + O 20 µ 6 Note that the coefficient of the exponential term is asymptotically equivalent to the dominant term of 2πσ as expected. The first three terms correspond to. Note that our derivation is simpler then Nicolas computation in [2]. 3 Large deviation k = m m α > α > /2 m α integer m We have k = m m α m α integer. We set ε := m α ε = m α µ exp/ε. The multiseries scale is here {m α µ exp/ε}. Set τ = exp /ε. Our result can be summarized in the following local limit theorem: Theorem 3. With /2 < α < A nk n! = e m ε m T 5 + T 6 τ + T 7 τ 2 + Oτ 3 5 with T 5 = 2π εµ 2εµ 3 T 6 = µ 2 ε + 2ε 2 + T 7 = µ4 2 + ε 2 + ε 2 / µ 2 + O 8ε ε 3 ε 2 µ 2 + ε 4 0 3ε ε 2 ε + O µ 4 µ

6 Proof. We have The Saddle point equation 2 becomes To first order we have lnw /ε. So we set we now have to the next order k = m m α = m ε. w + + lnw + lnwεw lnwε = 0. 7 ρ = e ξ ξ = + η ε ερ η + ρ + ρ ε = 0 or hence This gives ηερ + ηε + ρ = 0 η ρ ε ε. ρ = e ξ = e /ε η/ε τ η ε hence η τ η ε τ ε ε + τ ε 2 = τ ε τ 2 ε 3. We derive by bootstrapping from 7 again we only provide a few terms here we use more terms in our expansions η = ε τ + ε 3 + ε 2 τ ε 2ε ε 4 4 ε ε 2 τ 3 + Oτ 4 ε and successively ρ = expη/ε i.e. τ ρ = τ ε 2 + 2ε 4 + ε 3 ε 2 τ + 2 2ε ε 5 3 ε ε 3 ε 2 τ 2 + Oτ 3 lnρ = + η i.e. ε lnρ = ε ε 2 τ + ε 4 + ε 3 ε 2 τ 2 + fρ = τ + ε ε + 2ε 3 + ε 2 ε lnfρ = lnε + ε + ε 2 + ε τ + Oτ ε ε 5 4 ε ε 3 ε 2 τ + Oτ 2 τ 3 + Oτ 4 For the first part of the Cauchy s integral we have T = lnfρ ε lnρ = lnε + τ + 2ε 2 τ 2 + Oτ 3 2 now we extract the dominant part lnε + T 2 = expµ 2 T lnε + = µ 2 τ + + ε2 2ε 2 µ2 + 2 µ4 τ 2 + Oτ 3. 6

7 Also κ[2] = ε ε τ + Oτ 2 κ[3] = 2ε 3 + 6ε 2 τ + Oτ 2. Again we must choose a splitting value θ 0 = m β β < 0 such that mκ 2 θ0 2 mκ 3θ0 3 This leads to β > 2 α β < 2 3 α 0 n. that is why we restrict the range to /2 < α <. We can then use θ 0 = m /2 α. We must prove that the integral K mk = 2π θ0 θ 0 e m lnfρeiθ kiθ dθ is such that K mk is exponentially small. This is done in Appendix 4. Proceeding further we derive θ = u i a[i]/µ = u εµ + iu2 3εµ 2 u3 36εµ 3 + iu4 270εµ 4 + Oτ. This expansion is valid in the dominant integration domain u θ 0µ a = m. Setting dθ = dθ dudu we integrate on u = [.. ] This gives successively [ T 3 = 2π εµ 2ε 2εµ ε 2 µ T 4 = T 3 T 2 = T 5 + T 6 τ + T 7 τ 2 + Oτ 3 with T 5 = 2π εµ 2εµ 3 T 6 = µ 2 ε + 2ε 2 + T 7 = µ4 2 + ε 2 + ε 2 + ] ε 5 6ε 4 24ε 3 2ε 2 µ 3 τ + Oτ 2 / µ 2 + O 8ε ε 3 ε 2 µ 2 + ε 4 0 3ε ε 2 ε + O This concludes the proof. Given some desired precision how many terms must we use in our expansions? It depends on α. For instance in T 7 we encounter terms like ε 2 µ 2 and ε 4. So we must compare α + with 4 α. The critical value is α = 2/3. µ 4 µ 2. To check the quality of our asymptotic we have chosen m [90 500] and α = 2/3. Figure shows lna nk circle and the ln of expression 5 line. Figure 2 shows the quotient of lna nk by the ln of expression 5 without the τ term in 5 line and with this term circle. Of course the good influence of the τ term is less effective for large m. Another way is to fix m to 00 for instance n = 000. The maximum value for k is m m /2 = 969. We must set k larger than the central domain for instance larger than m m/2 = 58. But note that the term T 6 start with two negative terms µ 2 ε so k must be large so that τ is small enough to compensate these negative terms. It appears that k = 860 is large enough in our case. So our α range is [/2 0.7]. Figure 3 shows lna nk circle and the ln of expression 5 line. Figure 4 shows the quotient of lna nk by the ln of expression 5 without the τ term in 5 line and with this term circle. 7

8 Figure : α = 2/3 lna nk circle and the ln of expression 5 line as function of m Figure 2: α = 2/3 quotient of lna nk by the ln of expression 5 as function of m without the τ term in 5 line and with this term circle 8

9 Figure 3: n = 000 lna nk circle and the ln of expression 5 line as function of k Figure 4: n = 000 quotient of lna nk by the ln of expression 5 as function of k without the τ term in 5 line and with this term circle 9

10 4 Appendix. Justification of the integration procedure 4. The central region We must analyze Rm lnfρe iθ kiθ but ρ k m/2 so to first order this leads to analyze e iθ 2 sinθ/2 R ln iθe iθ/2 = ln θ which has a dominant peak at The non-central region Now we must analyze Rm lnfρe iθ k lnρe iθ but k m so to first order this leads to analyze ρe iθ R ln lnρe iθ 2 = 2 2ρ cosθ + ρ 2 ln lnρ 2 + θ 2 2 which has a dominant peak at 0. References [] E.A. Bender. Central and local limit theorems applied to asymptotic enumeration. Journal of Combinatorial Theory Series A 5: [2] L. Carlitz D.C. Kurtz R. Scoville and O.P. Stackelberg. Asymptotic properties of Eulerian numbers. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 23: [3] F.N. David and D.E. Barton. Combinatorial Chance. Charles Griffin and co. London 962. [4] C. G. Esseen. On the application of the theory of probability to two combinatorial problems invoving permutations. In Proceedings of the Seventh Conference on Probability Theory Braşov pages [5] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press [6] W. Gawronski and T. Neuschel. Euler-frobenius numbers. Integral Transforms and Special Functions 240: [7] I. J. Good. Saddle-point methods for the multinomial distribution. Annals of Mathematical Statistics 284: [8] H.K. Hwang. Large deviations for combinatorial distributions I: Central limit theorems. Advances in Applied Probability 6: [9] S. Janson. Euler-Frobenius numbers and rounding. Online Journal of Analytic Combinatorics [0] G. Louchard. Asymptotics of the stirling numbers of the first kind revisited: A saddle point approach. Discrete Mathematics and Theoretical Computer Science 22: [] G. Louchard. Asymptotics of the stirling numbers of the second kind revisited: A saddle point approach. Applicable Analysis and Discrete Mathematics 72:

11 [2] J.L. Nicolas. An integral representation for Eulerian numbers. In Sets graphs and numbers Budapest 99 volume 60 pages [3] S. Tanny. A probabilistic interpretation of Eulerian numbers. Duke Mathematical Journal 40: [4] S.H. Sira zdinov. Asymptotic expression for Euler numbers in Russian. Izv. Akad. Nauk UZSSR Ser. Fiz.-Mat. Nauk 6: