On the asymptotic enumeration of accessible automata

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Main result
Let A n be the set of non-isomorphic accessible (also called initially connected) complete and deterministic automata with n states over a k-letter alphabet.By using suitable combinatorial transformations, Bassino and Nicaud (2007) prove that the order of magnitude of the cardinality of A n is related to the Stirling number of the second kind kn n .Then they reformulate the asymptotic estimate of |A n | due to Korshunov (1978) in terms of this Stirling number.Our purpose is to state a simplification of this formula.
First we need to recall a few definitions.For m, n non-negative integers, the Stirling number of the second kind, denoted by n m , is the number of ways of partitioning a set of n elements into m non-empty subsets.Let W 0 denote the principal branch of the so-called Lambert W function (which is the inverse of the function x → x e x ).See Graham et al. (1994) and Corless et al. (1996) for more details about these definitions.Also define the constants Except for v k , these constants are considered in Bassino and Nicaud (2007).Notice that v k is the unique root in (0, 1) of the function The next result, due to Bassino and Nicaud (2007), reformulates in terms of the Stirling numbers the asymptotic estimate for |A n | previously obtained by Korshunov (1978).
Theorem 1 (Bassino and Nicaud (2007)) The number |A n | of accessible complete and deterministic automata with n states over a k-letter alphabet satisfies where Our main result presents a simplified formula for E k .
Theorem 2 For any k ≥ 2, Remark 1 We observe that 0 < E k < 1 for all k ≥ 2. Since the principal branch of W is analytic at 0, we have that

Proof of Theorem 2
The proof relies on the theory of Lagrange series, which constitutes a fundamental tool for the solution of implicit equations and for the reversion of series.We refer to Henrici (1988) for an extensive treatment of this subject.To recall the two basic forms of Lagrange series, suppose that the dependence between the variables y and z is implicitly given by the equation where ϕ is a regular function in a neighborhood of the point z = 0 and ϕ(0 where the primes denote derivatives. To prove Theorem 2, we study a classical application of this theory, namely, the solutions of Problems 212 and 216 in Pólya and Szegő (1998, Chapter 5 of Part III).Let a and b be real numbers and consider the equation 1 − x + y x b = 0, that is, y = (x − 1) x −b .We define z = x − 1 and apply the formulae in (1) with ϕ(z) = (1 + z) b and f (z) = (1 + z) a .Notice that Hence, (1) yields and with y = (x − 1) x −b .Next, we consider the case b > 1 and investigate the neighborhood in R of the point x = 1 in which ( 2) and ( 3) hold.
For a = 1, the power series appearing in (2) is called the generalized binomial series.Thus, B b is the solution of the functional equation x(y) = 1 + y x(y) b that satisfies x(0) = 1.Moreover, ( 2) and ( 3) can be rewritten as In Graham et al. (1994, Sections 5.4 and 7.5), B b is introduced as a formal power series, these properties are proved and used to derive some combinatorial identities.See also Heggie and Nicklason (1996) for a representation of the function B b as a contour integral in the complex plane.Now let b > 1 be fixed.We are interested in studying the set of values of x for which the identity B b ((x − 1) x −b ) = x holds.To this end, we observe that by the ratio test the series in ( 4 We are ready to prove our main result.
Proof of Theorem 2: Note that .

k,
we conclude from Lemma 2 that the numerator of E k equals k − (k − 1) Table 1 exhibits the approximate values of v k and E k for k = 2, . . ., 7. Values of v k and E k .