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267
result(s) for
"Lah"
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Probabilistic Lah numbers and Lah-Bell polynomials
Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. The aim of this paper is to study the probabilistic Lah numbers associated with Y and the probabilistic Lah-Bell polynomials associated with Y, as probabilistic versions of the Lah numbers and the Lah-Bell polynomials, respectively. We derive some properties, explicit expressions, recurrence relations and certain identities for those numbers and polynomials. In addition, we treat the special cases that Y is the Poisson random variable with parameter α > 0 and the Bernoulli random variable with probability of success p.
Journal Article
Note on$ r $ -central Lah numbers and$ r $ -central Lah-Bell numbers
The$ r $ -Lah numbers generalize the Lah numbers to the$ r $ -Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The$ r $ -Lah number counts the number of partitions of a set with$ n+r $elements into$ k+r $ordered blocks such that$ r $distinguished elements have to be in distinct ordered blocks. In this paper, the$ r $ -central Lah numbers and the$ r $ -central Lah-Bell numbers ( $ r\\in \\mathbb{N} $ ) are introduced parallel to the$ r $ -extended central factorial numbers of the second kind and$ r $ -extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the$ r $ -central Lah numbers and the$ r $ -central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.
Journal Article
Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials
The nth r-extended Lah–Bell number is defined as the number of ways a set with n+r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.
Journal Article
Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks
Let ξ1,ξ2,… be a sequence of independent copies of a random vector in Rd having an absolutely continuous distribution. Consider a random walk Si:=ξ1+⋯+ξi, and let Cn,d:=conv(0,S1,S2,…,Sn) be the convex hull of the first n+1 points it has visited. The polytope Cn,d is called k-neighborly if for any indices 0≤i1<⋯
Journal Article
The r-Dowling–Lah Polynomials
The notions of
r
-Bell polynomials and their generalization, the
r
-Dowling polynomials are due to Mező and Cheon, Jung. Recently, Nyul and Rácz defined the
r
-Lah polynomials, which are close relatives of
r
-Bell polynomials. In the present paper, we introduce the Dowling type generalization of
r
-Lah polynomials, the
r
-Dowling–Lah polynomials. We give a comprehensive study of them using the results of the author and Nyul on
r
-Whitney–Lah numbers, which are the coefficients of these polynomials.
Journal Article
On the magnitude of the roots of some well-known enumerative polynomials
We present estimations of the roots of r-Dowling, r-Lah and r-Dowling–Lah polynomials. It is known that these polynomials have simple, real and non-positive roots. We give bounds for them and we also compute the real magnitude of the roots via computational methods.
Journal Article
A Lightweight Hybrid Scheme for Hiding Text Messages in Colour Images Using LSB, Lah Transform and Chaotic Techniques
Data security can involve embedding hidden images, text, audio, or video files within other media to prevent hackers from stealing encrypted data. Existing mechanisms suffer from a high risk of security breaches or large computational costs, however. The method proposed in this work incorporates low-complexity encryption and steganography mechanisms to enhance security during transmission while lowering computational complexity. In message encryption, it is recommended that text file data slicing in binary representation, to achieve different lengths of string, be conducted before text file data masking based on the lightweight Lucas series and mod function to ensure the retrieval of text messages is impossible. The steganography algorithm starts by generating a random key stream using a hybrid of two low-complexity chaotic maps, the Tent map and the Ikeda map. By finding a position vector parallel to the input image vector, these keys are used based on the previously generated position vector to randomly select input image data and create four vectors that can be later used as input for the Lah transform. In this paper, we present an approach for hiding encrypted text files using LSB colour image steganography by applying a low-complexity XOR operation to the most significant bits in 24-bit colour cover images. It is necessary to perform inverse Lah transformation to recover the image pixels and ensure that invisible data cannot be retrieved in a particular sequence. Evaluation of the quality of the resulting stego-images and comparison with other ways of performing encryption and message concealment shows that the stego-image has a higher PSNR, a lower MSE, and an SSIM value close to one, illustrating the suitability of the proposed method. It is also considered lightweight in terms of having lower computational overhead.
Journal Article
APPLICATION OF THE FINK IDENTITY TO JENSEN-TYPE INEQUALITIES FOR HIGHER ORDER CONVEX FUNCTIONS
The focus of this paper is the application of the Fink identity in obtaining Jensen-type inequalities for higher order convex functions. In addition to the basic form, we establish superadditivity and monotonicity relations that correspond to the Jensen inequality in this setting. We also obtain the corresponding Lah-Ribarič inequality. The obtained results are valid for functions of even degree of convexity. With this method, we derive some new bounds for the differences of power means, as well as some new Hölder-type inequalities.
Journal Article