E.R. Canfield. Central and local limit theorems for the coefficients of polynomials of binomial type. J. Comb. Theory A, 23:275--290, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a Gibbs(1 ) fragmentation process of [n] What is the least such n Harper [160] also proved a local limit theorem for such a sequence provided the central limit theorem holds. Hence both kinds of Stirling numbers admit local normal approximations. See also Bender [28, 29] and Canfield [69] for more general analytic methods to obtain central and local limit theorems for combinatorial sequences. Canfield [69] gives nice sufficient conditions for central and local limit theorems for coefficients of polynomial of binomial type. Similar results may also be derived using classical ....

....for such a sequence provided the central limit theorem holds. Hence both kinds of Stirling numbers admit local normal approximations. See also Bender [28, 29] and Canfield [69] for more general analytic methods to obtain central and local limit theorems for combinatorial sequences. Canfield [69] gives nice sufficient conditions for central and local limit theorems for coefficients of polynomial of binomial type. Similar results may also be derived using classical analytic techniques like the saddle point approximation [280] and Hayman s criterion [164] 2 Exchangeable random ....

E.R. Canfield. Central and local limit theorems for the coefficients of polynomials of binomial type. J. Comb. Theory A, 23:275--290, 1977.

....n = mg valid for the widest possible range of m, but to show that for m lying in the interval n Sigma O(oe n ) very precise asymptotic formulae can be obtained. These formulae are in close connection with our results in [17] Although local limit theorems receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results, especially Theorem 1, seem rarely discussed in a systematic manner. Recall that a lattice random variable X is said to be of maximal span h if X takes only values of the form b hk, k 2 Z, for some constants b and h 0; and there does not exist b and h h such ....

.... Gamma log( 1 Gamma e ) 1 Gamma e ) In general, since the mean and the variance of Xi n and n (defined by (26) are different (due to the large factor m ) asymptotic formula for one provides large deviations (from the mean) for the other. This observation has formerly been applied in [3] for polynomials of binomial type. 15 Example 4. Random mapping patterns. Random mapping patterns are equivalence classes of random mappings and, structurally, they are multisets of cycles of rooted unlabeled trees. Namely, S(z A ; where S(z) is the generating function for cycles of ....

E. R. Canfield, Central and local limit theorems for the coefficients of polynomials of binomial type, Journal of Combinatorial Theory, Series A, 23, 275--290 (1977).

....tools for asymptotic analysis make it possible a systematic treatment of the statistical 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 ....

Canfield, E. R. (1977) Central and local limit theorems for the coefficients of polynomials of binomial type. Journal of Combinatorial Theory, Series A 23, 275--290.

.... 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 various components can be characterized in their asymptotic form: Non classical laws as ....

Canfield, E. R. Central and local limit theorems for the coefficients of polynomials of binomial type. Journal of Combinatorial Theory, Series A 23 (1977), 275--290.

....classes of combinatorial structures, regarding the mean number of components in random structures or the existence of asymptotic probability (not necessarily 0 or 1) in first or second order logical theories. Another category of results stems from the original observations of Bender and Canfield [4, 14] that certain general combinatorial schemes like sequence or set formation, whose translation into generating functions is 1 1 Gamma uC(z) or exp(uC(z) 39) lead to Gaussian distributions under quite general analytic conditions on the series C. These schemes give bivariate generating ....

Canfield, E. R. Central and local limit theorems for the coefficients of polynomials of binomial type. Journal of Combinatorial Theory, Series A 23 (1977), 275--290.

....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 ....

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E. Rodney Canfield. Central and local limit theorems for the coefficients of polynomials of binomial type. Journal of Combinatorial Theory, Series A, 23:275--290, 1977.

....are zero. The literature on log concavity is vast, and we mention only a few selections; the bibliographies of these will lead the interested reader to many other works. A standard reference is [5] especially Chapter 8. Combinatorial inequalities in particular are the subject of [1] and [9] In [2] it is shown that if the coefficients of the power series g(u) are log concave then s(n; k) u n ]g(u) k is log concave in k for fixed n; as a corollary the coefficients of the polynomial Pn (x) u n =n ] exp(xg(u) are strictly log concave. In [6] consideration is given to the question ....

E. R. Canfield, Central and local limit theorems for the coefficients of polynomials of binomial type, J. Combin. Theory Ser. A 23 (1977) 275--290.

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E. R. Canfield. Central and local limit theorems for the coefficients of polynomials of binomial type. Journal of Combinatorial Theory, Series A, 23, 275--290 (1977).

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E. R. Canfield. Central and local limit theorems for the coefficients of polynomials of binomial type. Journal of Combinatorial Theory, Series A, 23:275--290, 1977.

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