Flajolet, P. and Soria, M. (1990). Gaussian limiting distributions for the number of components in combinatorical structures. Journal of Combinatorial Theory, Series A, 53:165--182.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....properties of a class of combinatorial structures. This line of investigation, initiated by Bender [13] in the early seventies and then continued by Bender, Canfield, Richmond, Williamson and Gao (cf. 20, 14, 15, 40] was recently further developed, most notably, by Flajolet and Soria [36, 37] and Hwang [53] In particular, the uniformity provided by the powerful singularity analysis of Flajolet and Odlyzko [35] played an important role. In this section, we start with the bivariate generating functions of several di#erent types of the parameter number of components in partitional ....

....uniformity provided by the powerful singularity analysis of Flajolet and Odlyzko [35] played an important role. In this section, we start with the bivariate generating functions of several di#erent types of the parameter number of components in partitional complex and multiset constructions (cf. [36, 38, 43]) Then we review an analytic scheme originally introduced by Flajolet and Soria [36, 37] and studied in detail in Hwang [53, Ch. 5] for which we will be able to, in particular, apply our Poisson approximation formul of Theorems 1 and 2. We then state the restriction of this scheme and then go on ....

[Article contains additional citation context not shown here]

Flajolet, P. and Soria, M. (1990) Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A 53, 165--182.

....Given a finite field F q , where q is a prime power. Assume that all q monic polynomials of degree n are equally likely. Let Y n denote the number of irreducible factors (counted with multiplicity) in the prime factorization of a random polynomial. Defining P n (y) q Yn ) then (see [16]) P n (y)z 1 Gamma yz j Delta GammaI j ; 4) where I n z log 1 1 Gamma qz (n) being the Mobius function: n) 0 if n is not square free; n) Gamma1) if n = p 1 Delta Delta Delta p k with distinct prime numbers p 1 ; p k . From this generating ....

P. Flajolet, M. Soria, Gaussian limiting distributions for the number of components in combinatorial structures, Journal of Combinatorial Theory, Series A, 53 (1990), 165-182.

....properties of a class of combinatorial structures. This line of investigation, initiated by Bender [13] in the early seventies and then continued by Bender, Canfield, Richmond, Williamson and Gao (cf. 20, 14, 15, 40] was recently further developed, most notably, by Flajolet and Soria [36, 37] and Hwang [53] In particular, the uniformity provided by the powerful singularity analysis of Flajolet and Odlyzko [35] played an important role. In this section, we start with the bivariate generating functions of several different types of the parameter number of components in partitional ....

....provided by the powerful singularity analysis of Flajolet and Odlyzko [35] played an important role. In this section, we start with the bivariate generating functions of several different types of the parameter number of components in partitional complex and multiset constructions (cf. [36, 38, 43]) Then we review an analytic scheme originally introduced by Flajolet and Soria [36, 37] and studied in detail in Hwang [53, Ch. 5] for which we will be able to, in particular, apply our Poisson approximation formulae of Theorems 1 and 2. We then state the restriction of this scheme and then go ....

[Article contains additional citation context not shown here]

Flajolet, P. and Soria, M. (1990) Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A 53, 165--182.

....Given a finite field F q , where q is a prime power. Assume that all q monic polynomials of degree n are equally likely. Let Y n denote the number of irreducible factors (counted with multiplicity) in the prime factorization of a random polynomial. Defining P n (y) q Yn ) then (see [16]) P n (y)z yz I , 4) where I n z log 1 qz (n) being the Mobius function: n) 0 if n is not square free; n) 1) if n = p 1 p k with distinct prime numbers p 1 , p k . From this generating function, we have the recurrence P 0 (y) 1 and P n ....

P. Flajolet, M. Soria, Gaussian limiting distributions for the number of components in combinatorial structures, Journal of Combinatorial Theory, Series A, 53 (1990), 165-182.

....of papers dealing with the value distribution problems of various maps F q [X] R when the polynomials f are taken at random . Usually, the probability measure n ( q Gamman #ff : ffif = n; g; where ffif : deg f , is applied. We mention here the investigations [1] [5], 7 13] 17 20] 25] On the other hand, there exists a parallel theory investigating the value distribution of the maps Sn R, where Sn denotes the symmetric group of order n, when a permutation oe 2 Sn is taken with the equal probability 1=n (see, for instance, 3] 6] 10] 12] 14] ....

P.Flajolet, M.Soria, Gaussian limiting distributions for the number of components in combinatorial structures, J. Combinatorial Theory, Ser. A 53(1990), 165--182.

....original work [32] Currently the most powerful and general results are those of Gao and Richmond [155] They apply to general multivariate problems, not only two variable ones. Other papers that deal with central and local limit theorems or other multivariate problems with small singularities are [38, 42, 65, 96, 142, 143, 183, 227]. 14. Mellin and other integral transforms When the best generating function that one can obtain is an infinite sum, integral transforms can sometimes help. There is a large variety of integral transforms, such as those of 135 Fourier and Laplace. The one that is most commonly used in asymptotic ....

P. Flajolet and M. Soria, Gaussian limiting distributions for the number of components in combinatorial structures, J. Combinatorial Theory, Series A, 53 (1990), pp. 165--182.

....monic polynomials over a finite field of cardinality q (irreducible, general) functional 117 118 graphs (connected, general) in either the labelled or the unlabelled case. In fact, the examples just cited all belong to an interesting class called the exp log class that was introduced in [4]. Definition 1. A pair (I; F) is said to have the exp log property if I(z) has a unique dominant singularity ae of the logarithmic type, I(z) z ae a log 1 1 Gamma z=ae c 0 O( 1 Gamma z=ae) ffl ) 2) for some ffl 0, where a is called the multiplier. Accordingly, one has F (z) e ....

....is the Buchstab function (u) classically defined by the difference differential equation u (u) 1 (1 u 2) u (u) 0 = u Gamma 1) u 2) The Prime Number Theorem for exp log classes derives immediately from basic singularity analysis theorems. The Gaussian law was established in [4] by means of characteristic functions, thanks to the uniformity afforded by singularity analysis; it is an analogue of the classical Erdos Kac theorem for the number of prime divisors of integers. The Dickman law is known originally from number theory [12] and it holds as well for the cycle ....

Flajolet (Philippe) and Soria (Mich`ele). -- Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A, vol. 53, 1990, pp. 165--182. 122

.... cycles is the coefficient of y k in t n (y) where t n (y) is called the tree polynomial of order n [46] and is generated by 1 (1 Gamma T (z) y = X n0 t n (y) z n n : 2:3) One application of these functions is to derive the limiting distribution of cycles in random mappings [29]. Chaotic maps of the unit interval using floating point arithmetic can be studied in this way; an elementary discussion that looks only at the expected length of the longest cycle can be found in [18] Iterated exponentiation The problem of iterated exponentiation is the evaluation of h(z) ....

P. Flajolet and M. Soria, "Gaussian Limiting Distributions for the Number of Components in Combinatorial Structures", J. Combinatorial Theory, Series A, 53 (1990) 165--182.

....R 1 be the radius of convergence of C(x) If R = 0, the mass of the distribution is concentrated at 1 because Cn=An 1 by Theorem 1. If C(R) diverges, the distribution is often normal. Arguments for normality usually rely on analytic properties of the generating function as in Flajolet and Soria [13]. When there is a logarithmic singularity on the circle of convergence, Hwang [17] obtained refinements which are similar to what happens when C(R) converges: The labeled case leads to a shifted Poisson and unlabeled case is more complicated. Compton [10] obtained some results in these cases, and ....

P. Flajolet and M. Soria, Gaussian limiting distributions for the number of components in combinatorial structures, J. Combin. Theory, Ser. A 53 (1990) 165--182.

....estimates for the larger coefficients rather than tail probabilities. Unfortunately, multivariate generating functions have proven to be recalcitrant subjects for asymptotic analysis. When A(x, y) has small singularities, methods akin to Darboux s Theorem may be useful. See Flajolet and Soria [5] and Gao and Richmond [6] for examples. See Odlyzko [12] for an extensive discussion of asymptotic methods. In order to study a variety of single variable functions with large singularities, Hayman [10] defined a class of admissible functions in such a way that (a) class members have useful ....

P. Flajolet and M. Soria, Gaussian limiting distributions for the number of components in combinatorial structures, J.Combin.TheorySer.A53 (1990) 165--182.

....of combinatorics will recognize in eqn. 10 the tree function T (x) GammaW Gamma1 ( Gammax) used in the enumeration of trees and graphs on sets of labeled vertices [Wright, 1977; Janson et al. 1993] and in computing the distribution of cycles in random mappings M BRAND: STRUCTURE LEARNING 22 [Flajolet and Soria, 1990]. Connections to dynamical stability via the W function and to sparse graph enumeration via the T function are very intriguing and may lead to arguments as to whether the entropic prior is optimal for learning concise sparse models. We offer a tantalizing clue, reworking and solving a partial ....

Flajolet, P. and Soria, M. (1990). Gaussian limiting distributions for the number of components in combinatorical structures. Journal of Combinatorial Theory, Series A, 53:165--182.

.... cycles is the coefficient of y k in t n (y) where t n (y) is called the tree polynomial of order n (see [47] and is generated by 1 (1 Gamma T (z) y = X n0 t n (y) z n n : 2:3) One application of these functions is to derive the limiting distribution of cycles in random mappings [29]. Chaotic maps of the unit interval using floating point arithmetic can be studied in this way; an elementary discussion that looks only at the expected length of the longest cycle can be found in [18] Iterated exponentiation The problem of iterated exponentiation is the evaluation of h(z) z ....

P. Flajolet and M. Soria, "Gaussian Limiting Distributions for the Number of Components in Combinatorial Structures", J. Combinatorial Theory, Series A, 53 (1990) 165--182.

....transformer . We find, for finite families and for the other classical families, that path length is on average n log n, that the expected number of leaves is asymptotic to ffn, for some constants ; ff dependent upon OE. A variation of this scheme in line with Bender s work [2] and with [11, 12] leads to limit distributions. For instance, the distribution of nodes in strata of a tree or the number of leaves both asymptotically conform to a Gaussian law. Most existing works (with the notable exception of Meir and Moon s studies [31, 32] appeal to special recurrence relation, often based ....

....(z) Gamma Y 0 (z) Delta u : The integral form of L(z; u) is then obtained by the variation of parameter method. In the case of a polynomial OE, using Lemma 2, we determine log Y 0 (z) ffi 1) log 1 1 Gamma z=ae C O[ 1 Gamma z=ae) 2ffi ] A theorem of Flajolet and Soria [11] states that a bivariate scheme of the form exp(uL(z) for some function L(z) with a dominant logarithmic singularity induces Gaussian distributions in the asymptotic limit. It applies here to (Y 0 (z) u = exp(u log Y 0 (z) In the case of L(z; u) in Eq. 22) the integral is convergent ....

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Flajolet, P., and Soria, M. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A 53 (1990), 165--182.

....objects and combine these cycles into a set, then ask for the number of cycles of the set that have a majority of elements of one type, or an equal number of elements of each type. It should be possible to extend the distribution results on the number of components presented by Flajolet and Soria [5] to study the number of components of a given type (good, bad or neutral) for various combinatorial constructs. Acknowledgments We thank P. Flajolet for information on Bessel functions and G. Louchard for information on Gaussian processes. 7. APPENDIX We give in this part some mathematical ....

P. Flajolet and M. Soria. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory (A), 53(2):165-- 182, March 1990.

....estimates for the larger coefficients rather than tail probabilities. Unfortunately, multivariate generating functions have proven to be recalcitrant subjects for asymptotic analysis. When A(x; y) has small singularities, methods akin to Darboux s Theorem may be useful. See Flajolet and Soria [5] and Gao and Richmond [6] for examples. See Odlyzko [12] for an extensive discussion of asymptotic methods. In order to study a variety of single variable functions with large singularities, Hayman [10] defined a class of admissible functions in such a way that (a) class members have useful ....

P. Flajolet and M. Soria, Gaussian limiting distributions for the number of components in combinatorial structures, J. Combin. Theory Ser. A 53 (1990) 165--182.

....the complexity of the generator comes with it for free This important technique is not only theoretically interesting but was later implemented in MapleV4 under the name CombStruct. 16. Gaussian limiting distributions The collaboration with Mich ele Soria concentrates on limiting distributions [90, 97, 114]. The flavour of the general results is perhaps best described by giving a few examples. The bivariate generating function for the number of cycles in permutations with u marking the number of cycles is X n;k n k u k z n n = exp i u log 1 1 Gamma z j : From that form alone one ....

P. Flajolet and M. Soria. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A, 53:165--182, 1990.

.... b n , where b k is formed by the product of all the irreducible factors of a 6 P. FLAJOLET, X. GOURDON, D. PANARIO Phase Properties Refs. Sec. 1) Fraction of irreducibles is 1 n . Eq. 8) 4, 40] The number of irreducible factors has mean log n and a limiting Gaussian law. Eq. 13) [4, 7, 21, 40] ERF (Sec. 2) Non squarefree part has size O(1) on average and with high probability. Thm. 2.1; 4, 9, 18] Algorithmic cost of ERF is G(n) on average and with high probability Thm 2.2; 18] DDF (Sec. 3) Largest degree is (n) with Dickman distribution and mean g n where g : ....

....n. The bivariate generating function appears already in (10) Differentiating with respect to u, setting u = 1 and then analysing the singularity at u = 1 provides moment estimates [40, Ex. 4.6.2. 5] methods that build further on singularity analysis even give access to a limiting distribution [7, 21]. ANALYSIS OF POLYNOMIAL FACTORIZATION 13 Fact. Let Xn be the random variable that represents the number of irreducible factors in a random polynomial of degree n. Then the mean E(Xn ) and variance Var(Xn ) satisfy E(Xn ) log n O(1) Var(Xn ) log n O(1) 13) In addition the distribution ....

Flajolet, P., and Soria, M. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A 53 (1990), 165--182.

....as n 1. Thus, a common schema covers a variety of seemingly unrelated phenomena. In this way, we find Gaussian limit laws for the number of cycles in a random permutation, the number of factors of a random polynomial over GF (q) or the number of components in a random mapping of large size [35]. First results along these lines were derived by Bender, Canfield and Richmond [4, 6, 8] A classification of some major schemas and their associated laws is given in Soria s thesis [71] Even for a structure as complicated as random trains, it is the case that all probability distributions of ....

Flajolet, P., and Soria, M. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A 53 (1990), 165--182.

....derangement, the number of components in a random mapping, the number of irreducible factors in a random polynomial over a finite field. The occurrence of a common structural analytic scheme explains the origin of a limiting normal distribution for these rather diverse combinatorial structures [35]. Such questions can be pushed much further and one might aim at a complete characterization of limit distributions that occur inside elementary structures of the LI, UI, LR, UR classes. The problems are naturally more complicated since we are then dealing with bivariate problems. For recursive ....

Flajolet, P., and Soria, M. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A 53 (1990), 165--182.

....and gcd s whose costs can be taken under the standard form product: 1 n 2 ; gcd: 2 n 2 : 2. Summary of results It is well known [1, 16] that a random polynomial of degree n is irreducible with probability tending to 0 and has close to log n factors on average and with a high probability [4, 10]. Thus, the factorization of a random polynomial over a finite field is almost surely nontrivial. The first phase ERF of our factorization chain classically starts with the elimination of repeated factors, a simplified form of squarefree factorization described in Section 4. Theorem 1 quantifies ....

Flajolet, P., and Soria, M. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A 53 (1990), 165--182.

....due to the existence of integral representations. In the sequel, we shall see that expressions qualitatively similar to (17) though much less explicit, hold in higher dimensions. Asymptotic normality for Cn and Dn would result from these developments using the main theorem of Flajolet and Soria [14]. A derivation is however not given here as it is subsumed by the more general treatment valid for all dimensions that we are going to expose now. 5 The singularity perturbation method The architecture of the proof of the main theorem asserting asymptotic normality of the distribution of search ....

....(ff(u) d Gamma 2 d u = 0: Forms belonging to the general type (19) were already encountered when d = 1, see Eq. 11) and when d = 2, see Eq. 17) Asymptotic normality of coefficients is known to hold for a closely related class of bivariate functions exhibiting a similar singular behaviour [14]. As z 1, the dominant term in the expansion of Phi(u; z) is the one corresponding to the root 2u 1=d which has maximal real part. In particular when the parameter u is close to 1, this is the principal determination of 2 d p u. From the shape (19) of singular elements, we thus expect the ....

[Article contains additional citation context not shown here]

Philippe Flajolet and Mich`ele Soria. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A, 53:165--182, 1990.

....of classical results appear in [8, 26] Bender [1] first recognized that such limit distributions could be established for general combinatorial schemas under analytic conditions of a general character. This line of investigation was later pursued by Bender, Canfield, Richmond, Compton, and others [4, 2, 7, 12]. In a way, the situation parallels that of the central limit theorem in probability theory. There, we know that the common scheme of taking sums of many random variables leads, under wide sets of conditions, to a general asymptotic law, a normal distribution in the limit. Here, we show how ....

....These are analytic functions of two complex variables of the form 1 P (u; z) X n;k0 P n;k u k z n : 1) We are thus facing a double inversion problem. In some cases, real variable methods may be used, see in particular Compton s work [7] The approach taken here (as well as in [1, 4, 2, 12]) relies instead on complex variable methods. It consists of a two stage process. ffl First, we consider u as a parameter and solve a parameterized single variable inversion problem, by estimating p n (u) j X k0 P n;k u k = 1 2i I P (z; u) dz z n 1 ; 2) asymptotically for large n ....

[Article contains additional citation context not shown here]

Philippe Flajolet and Mich`ele Soria. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A, 53:165--182, 1990.

No context found.

Flajolet, P. and Soria, M. (1990). Gaussian limiting distributions for the number of components in combinatorical structures. Journal of Combinatorial Theory, Series A, 53:165--182.

No context found.

Flajolet, P. and Soria, M. (1990). Gaussian limiting distributions for the number of components in combinatorical structures. Journal of Combinatorial Theory, Series A, 53:165--182.

No context found.

Philippe Flajolet and Michele Soria. Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theory Ser. A, 53(2):165-182, 1990.

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