No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Maximum Stirling numbers of the second kind
Abstract
Say an integer n is exceptional if the maximum Stirling number of the second kind S(n,k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x 1/2+ε ), for any ε>0. We derive a similar result for partitions of n into exactly k integers.
... [19]). Denote the set of them by E. E. R. Canfield and C. Pomerance [19] showed showed that |{n ≤ x : n ∈ E}| x (3/5)+ε , and G. Kemkes, D. Merlini and B. Richmond [56] improved this to |{n ≤ x : n ∈ E}| x (1/2)+ε . ...
... E. R. Canfield and C. Pomerance [19] used estimates for integer points close to a curve similar to the results in Section 4, whereas G. Kemkes, D. Merlini and B. Richmond [56] used work of E. Bombieri and J. Pila [17] on estimates for integer points on a curve. ...
We survey various developments in Number Theory that were inspired by classical papers by Roth [On the gaps between squarefree numbers, J. London Math. Soc. 26 (1951) 263–268] and by Halberstam and Roth [On the gaps between consecutive k-free integres, J. London Math. Soc. 26 (1951) 268–273].
... we show examples in Sect. 2. Searching these databases is valuable also for cases where the student can compute the integral, as it enables finding new connections with other mathematical fields. See also [7]. ...
We describe working session for the study of 1-parameter families of definite integrals in a technology-rich environment. The joint usage of paper-and-pencil work together with a Computer Algebra System and eventually a web based database may lead to closed forms for the integrals, to the derivation of combinatorial identities, and other kinds of output. This joint usage applied to parametric integrals lead to mathematical expressions rather than graphic representation. This permits a new learning process in the teaching of mathematics, physics and engineering. We begin with a short survey of classical cases, using telescopic methods. Then we show a new example where the integrals depend not only on an integer parameter, but also on a real variable. In this last case, the study of the parametric integral involves the study of a recurrence differential equation.
... S(m, n) = S(m-1, n-1) + m * S(m, n-1) [Dic03] A Stirling2 stream object implemented in Figure 3.1 is, in theory, capable of producing all of Stirling numbers of the second kind starting from some initial values of m and n. (1,1); cout << "n/m"; for ( int col = 1; col <= 8; col++ ) cout << '\t' << col << " "; cout << endl; for ( int n = 1; n <= 8; n++ ){ cout << n << " \t"; for ( int m = 1; m <= n; m++ ){ cout << s->head << '\t'; s = s->tail(); } cout << endl; } } ...
We show that the ability of a lazy language, like Haskell, to allow procedures to lazily generate a stream of tokens can be added to ANSI C++ merely by writing code in a style which uses classes to implement function closures. Coding in this style provides an easy way to handle infinite streams in C++, results in application layer implementations that closely resemble a problem's specification, and can be applied to a wide variety of problems in computer science.
... for some index K * n . It is conjectured by Wegner that S(n, K * n ) > S(n, K * n + 1), that is, the maximum is unique for n > 2 (see also [5,7]). There are good estimations for the mode K * n , it asymptotically equals to n/ log(n) (see [8] for a short proof). ...
It is known that the S(n,k) Stirling numbers as well as the ordered
Stirling numbers k!S(n,k) form log-concave sequences. Although in the first
case there are many estimations about the mode, for the ordered Stirling
numbers such estimations are not known. In this short note we study this
problem and some of its generalizations.
... In this section we relate the integral F n (1) with the Stirling numbers of the second type. The rational of this is the fact that the Stirling numbers of the second type have the exponential generating function; see [11] for a review of these numbers, and [10] and [4] for graphical representations. ...
We compute closed forms for some parametric integrals. The tools used are MacLaurin developments and other infinite series. Finally, Stirling numbers appear.
... Determining the value of K n is an old problem ([9] [10] [6] [1] [5] [11] [15] [13] [2]). A related longstanding conjecture ([15] [3] [12]) is that there exists no n > 2 such that S(n, K n ) = S(n, K n + 1). ...
Let K_n denote the smaller mode of the nth row of Stirling numbers of the second kind S(n, k). Using a probablistic argument, it is shown that for all n>=2, [exp(w(n))]-2<=K_n<=[exp(w(n))]+1, where [x] denotes the integer part of x, and w(n) is Lambert's W-function. Comment: accepted by Discrete Mathematics
ResearchGate has not been able to resolve any references for this publication.