Number Of Combinations Research Articles

Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

A Memetic Algorithm Based on an NSGA-II Scheme for Phylogenetic Tree Inference

Phylogenetic inference allows building a hypothesis about the evolutionary relationships between a group of species, which is usually represented as a tree. The phylogenetic inference problem can be seen as an optimization problem, searching for the most qualified tree among all the possible topologies according to a selected criterion. These criteria can be based on different principles. Due to the combinatorial number of possible topologies, diverse heuristics and meta-heuristics have been proposed to find approximated solutions according to one criterion. However, these methods may result in several phylogeny trees which could be in conflict with one another. In order to deal with this problem, models based on multiobjective optimization with different configurations have been used. In this paper, we propose an ad-hoc multiobjective memetic algorithm (MO-MA) to infer phylogeny using two objectives: 1) maximum parsimony and 2) likelihood. Several population operators and local search strategies are proposed and evaluated in order to measure their contribution to the algorithm. Additionally, we perform a comparison among different configurations and tree rearrangement strategies. The results show that the proposed MO-MA is able to identify a Pareto set of solutions that include new trees which were nondominated by solutions from the current state of the art single-objective optimization tools. Furthermore, the MO-MA improves the results presented in the literature for multiobjective approaches in all of the studied data sets. These results make our proposal a good alternative for phylogenetic inference.

An expansion for the number of partitions of an integer

We consider p(n) the number of partitions of a natural number n, starting from an expression derived by Baez-Duarte in (Adv Math 125(1):114–120, 1997) by relating its generating function f(t) with the characteristic functions of a family of sums of independent random variables indexed by t. The asymptotic formula for p(n) follows then from a local central limit theorem as $$t\uparrow 1$$ suitably with $$n\rightarrow \infty $$. We take further that analysis and compute formulae for the terms that compose that expression, which accurately approximate them as $$t\uparrow 1$$. Those include the generating function f and the cumulants of the random variables. After developing an asymptotic series expansion for the integral term, we obtain an expansion for p(n) that can be simplified as follows: for each $$N>0$$, $$\begin{aligned} p(n)=\frac{2\pi ^2}{3\sqrt{3}} \,\frac{\text{ e }^{r_n}}{(1+2\,r_n)^2}\, \left( 1-\sum _{\ell = 1}^N\,\frac{D_\ell }{(1+2\,r_n)^\ell }+\,\mathcal R_{N+1}\,\right) . \end{aligned}$$The coefficients $$D_\ell $$ are positive and have simple expressions as finite sums of combinatorial numbers, $$r_n=\sqrt{\frac{2\pi ^2}{3}\,(n-\frac{1}{24})+\frac{1}{4}}$$ and the remainder satisfies $$n^{N/2}\,{\mathcal {R}}_{N+1} \rightarrow 0$$ as $$n\rightarrow \infty $$. The cumulants are given by series of rational functions and the approximate formulae obtained could be also of independent interest in other contexts.

Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment

We investigate node degrees in a network grown from a seed by hooking self-similar components under two models of randomness: a uniform attachment model and a model based on preferential attachment. We study two degree profiles: a local profile tracking the evolution of the degree of a particular node over time, and a global profile concerned about counts of the number of nodes of a particular degree.For the local profile, under uniform attachment growth, we have the exact mean, variance and probability distribution in terms of standard combinatorial numbers like generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: The early nodes have an asymptotically normal distribution, intermediate nodes have a Poisson distribution and the late nodes have a degenerate distribution. In contrast, under preferential attachment, the moments of the degree of a node contain Stirling numbers of the second kind and (under appropriate scaling) the profile has a gamma-type limit law.As for the global profile, we use Pólya urns to derive strong laws. Four regimes arise according to the structure of the seed. Within these regimes, we identify a few degenerate cases. Barring these degenerate cases, we uncover an asymptotically normal joint multivariate distribution for nodes of very small degrees.

The $$\varvec{r}$$-alternating permutations

The classical notion of zigzag or alternating permutations is extended by an additional parameter r so that we require the elements in the set $$\{1, \ldots , r\}$$ to appear in decreasing order in the zigzag permutation. This will lead to a generalized Andre theorem and to an interesting class of new combinatorial numbers. We also study a refinement according to the first element and its associated combinatorial sequence, called the r-Entringer numbers. Finally, we give several combinatorial identities for this new sequence.

Identities, inequalities for Boole-type polynomials: approach to generating functions and infinite series

The main purpose and motivation of this work is to investigate and provide some new identities, inequalities and relations for combinatorial numbers and polynomials, and for Peters type polynomials with the help of their generating functions. The results of this paper involve some special numbers and polynomials such as Stirling numbers, the Apostol–Euler numbers and polynomials, Peters polynomials, Boole polynomials, Changhee numbers and the other well-known combinatorial numbers and polynomials. Finally, in the light of Boole’s inequality (Bonferroni’s inequalities) and bounds of the Stirling numbers of the second kind, some inequalities for a combinatorial finite sum are derived. We mention an open problem including bounds for our numbers. Some remarks and observations are presented.

On a family of special numbers and polynomials associated with Apostol-type numbers and polynomials and combinatorial numbers

In this article, we examine a family of some special numbers and polynomials not only with their generating functions, but also with computation algorithms for these numbers and polynomials. By using these algorithms, we provide several values of these numbers and polynomials. Furthermore, some new identities, formulas and combinatorial sums are obtained by using relations derived from the functional equations of these generating functions. These identities and formulas include the Apostol-type numbers and polynomials, and also the Stirling numbers. Finally, we give further remarks and observations on the generating function including ?-Apostol-Daehee numbers, special numbers, and finite sums.

Notes on two kinds of special values for the Bell polynomials of the second kind

In the paper, by methods and techniques in combinatorial analysis and the theory of special functions, the authors discuss two kinds of special values for the Bell polynomials of the second kind for two special sequences, find a relation between these two kinds of special values for the Bell polynomials of the second kind, and derive an identity involving the combinatorial numbers.

Formulas derived from moment generating functions and Bernstein polynomials

The purpose of this paper is to provide some identities derived by moment generating functions and characteristics functions. By using functional equations of the generating functions for the combinatorial numbers y1 (m,n,?), defined in [12, p. 8, Theorem 1], we obtain some new formulas for moments of discrete random variable that follows binomial (Newton) distribution with an application of the Bernstein polynomials. Finally, we present partial derivative formulas for moment generating functions which involve derivative formula of the Bernstein polynomials.

Solutions to All‐Colors Problem on Graph Cellular Automata

The All‐Ones Problem comes from the theory of σ+‐automata, which is related to graph dynamical systems as well as the Odd Set Problem in linear decoding. In this paper, we further study and compute the solutions to the “All‐Colors Problem,” a generalization of “All‐Ones Problem,” on some interesting classes of graphs which can be divided into two subproblems: Strong‐All‐Colors Problem and Weak‐All‐Colors Problem, respectively. We also introduce a new kind of All‐Colors Problem, k‐Random Weak‐All‐Colors Problem, which is relevant to both combinatorial number theory and cellular automata theory.

Computational Enumeration of Colorings of Hyperplanes of Hypercubes for all Irreducible Representations and Applications

We obtain the generating functions for the combinatorial enumeration of colorings of all hyperplanes of hypercubes for all irreducible representations of the hyperoctahedral groups. The computational group theoretical techniques involve the construction of generalized character cycle indices of all irreducible representations for all hyperplanes of the hypercube using the M\"{o}bius function, polynomial generators for all cycle types and for all hyperplanes. This is followed by the construction of the generating functions for colorings of all (n-q)-hyperplanes of the hypercube, for example, vertices (q=5), edges (q=4), faces (q=3), cells (q=2) and tesseracts (q=4) for a 5D-hypercube. Tables are constructed for the combinatorial numbers for coloring all hyperplanes of 5D-hypercubes for 36 irreducible representations. Applications to chirality, chemistry and biology are also pointed out.

Predicting growth rate from gene expression

Growth rate is one of the most important and most complex phenotypic characteristics of unicellular microorganisms, which determines the genetic mutations that dominate at the population level, and ultimately whether the population will survive. Translating changes at the genetic level to their growth-rate consequences remains a subject of intense interest, since such a mapping could rationally direct experiments to optimize antibiotic efficacy or bioreactor productivity. In this work, we directly map transcriptional profiles to growth rates by gathering published gene-expression data from Escherichia coli and Saccharomyces cerevisiae with corresponding growth-rate measurements. Using a machine-learning technique called k-nearest-neighbors regression, we build a model which predicts growth rate from gene expression. By exploiting the correlated nature of gene expression and sparsifying the model, we capture 81% of the variance in growth rate of the E. coli dataset, while reducing the number of features from >4,000 to 9. In S. cerevisiae, we account for 89% of the variance in growth rate, while reducing from >5,500 dimensions to 18. Such a model provides a basis for selecting successful strategies from among the combinatorial number of experimental possibilities when attempting to optimize complex phenotypic traits like growth rate.

Triangular sequences, combinatorial recurrences and linear difference equations

In this work we introduce the triangular double sequences of arbitrary order given by linear recurrences, that generalize some well-known recurrences that appear in enumerative combinatorics. In particular, we focussed on triangular sequences generated by two double sequences and establish their relation with the solution of linear three-term recurrences. We show through some simple examples how these triangular sequences appear as essential components in the expression of some classical orthogonal polynomials and combinatorial numbers.

Formal residue and computer-assisted proofs of combinatorial identities

The coefficient of $x^{-1}$ of a formal Laurent series $f(x)$ is called the formal residue of $f(x)$. Many combinatorial numbers can be represented by the formal residues of hypergeometric terms. With these representations and the extended Zeilberger algorithm, we generate recurrence relations for summations involving combinatorial sequences such as Stirling numbers and their $q$-analog. As examples, we give computer proofs of several known identities and derive some new identities. The applicability of this method is also studied.

Construction method for generating functions of special numbers and polynomials arising from analysis of new operators

The aim of this paper is to construct a new method related to a family of operators to define generating functions for special numbers and polynomials. With the help of this method, we investigate various properties of these special numbers and polynomials with their generating functions, functional equations, and differential equations. We also give some algorithms to calculate the values of these numbers and polynomials. Moreover, we introduce some fundamental properties of this operator. By applying integral method to this operator, we obtain integral formulas together with combinatorial sums. Moreover, by applying this operator, the Lagrange inversion formula and convolution formula to these generating functions, we derive some novel identities and relations including combinatorial sums and combinatorial numbers including these new numbers and polynomials, ie, the Apostol‐Bernoulli numbers, the Apostol‐Euler numbers, the Stirling numbers, the central factorial numbers, and the other special numbers and polynomials. In addition, we give some examples derived from relations between composita and this operator on the set of formal power series. Finally, we give complex form of the Fourier series for these generating functions. By using these Fourier series, we not only derive some series relations and trigonometric sums including trigonometric function and hyperbolic functions but also give an example for a differential equation.

Simultaneous task allocation and planning for temporal logic goals in heterogeneous multi-robot systems

This paper describes a framework for automatically generating optimal action-level behavior for a team of robots based on temporal logic mission specifications under resource constraints. The proposed approach optimally allocates separable tasks to available robots, without requiring a priori an explicit representation of the tasks or the computation of all task execution costs. Instead, we propose an approach for identifying sub-tasks in an automaton representation of the mission specification and for simultaneously allocating the tasks and planning their execution. The proposed framework avoids the need to compute a combinatorial number of possible assignment costs, where each computation itself requires solving a complex planning problem. This can improve computational efficiency compared with classical assignment solutions, in particular for on-demand missions where task costs are unknown in advance. We demonstrate the applicability of the approach with multiple robots in an existing office environment and evaluate its performance in several case study scenarios.

Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y 1 n , k ; λ . Finally, we make some remarks and observations regarding these identities and relations.

On the complexity of finding and counting solution-free sets of integers

Given a linear equation L, a set A of integers is L-free if A does not contain any ‘non-trivial’ solutions to L. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving L-free sets of integers. The main questions we consider involve deciding whether a finite set of integers A has an L-free subset of a given size, and counting all such L-free subsets. We also raise a number of open problems.

APPLYING THE ATTRIBUTED PREFIX TREE FOR MINING CLOSED SEQUENTIAL PATTERNS

Mining closed sequential patterns is one of important tasks in data mining. It is proposed to resolve difficult problems in mining sequential pattern such as mining long frequent sequences that contain a combinatorial number of frequent subsequences or using very low support thresholds to mine sequential patterns is usually both time- and memory-consuming. This paper applies the characteristics of closed sequential patterns and sequence extensions into the prefix tree structure to mine closed sequential patterns from the sequence database. The paper uses the parent–child relationship on prefix tree structure and each node on prefix tree is also added fields to determine whether that is a closed sequential pattern or not. Experimental results show that the number of sequential patterns is reduced significantly.

Partitions of the set of natural numbers and symplectic homology

We illustrate a somewhat unexpected relation between symplectic geometry and combinatorial number theory by proving Tamura’s theorem on partitions of the set of positive integers (a generalization of the more famous Rayleigh–Beatty theorem) using the positive $${\mathbb{S}^1}$$ -equivariant symplectic homology.