Vladimir Kruchinin

Tomsk State University of Control Systems and Radioelectronics | TUSUR

The presented research is devoted to the problem of developing new combinatorial generation algorithms for combinations. In this paper, we develop a modification of Ruskey’s algorithm for unranking m-combinations of an n-set in co-lexicographic order. The proposed modification is based on the use of approximations to make a preliminary search for t...

In this paper, we consider a three-parameter generalization of the Narayana numbers that is related to the powers of the generating function for the Narayana numbers. We find a multivariate generating function for these generalized Narayana numbers. For the composition of generating functions where the inner function is the generating function for...

In this paper, we study methods for obtaining explicit formulas for the coefficients of generating functions. To solve this problem, we consider the methods that are based on using the powers of generating functions. We propose to generalize the concept of compositae to the case of generating functions in two variables and define basic operations o...

The article describes the design of evaluation system for electronic educational-methodical complexes of disciplines (EEMCD) applied at the Faculty of Distance Learning of Tomsk State University of Control Systems and Radioelectronics (TUSUR). The design technique developed at TUSUR involves a tool system to evaluate the procedure. The following ma...

In this paper, we study the problem of developing new combinatorial generation algorithms. The main purpose of our research is to derive and improve general methods for developing combinatorial generation algorithms. We present basic general methods for solving this task and consider one of these methods, which is based on AND/OR trees. This method...

The aim of this paper is to study the Tepper identity, which is very important in number theory and combinatorial analysis. Using generating functions and compositions of generating functions, we derive many identities and relations associated with the Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Stirling numbers. M...

In this paper, we consider the second-order Eulerian triangle for 1 ≤ m ≤ n. Also we obtain two explicit formulas for the Eulerian numbers of the second kind and present their proofs. The obtained formulas are based on the use of binomial coefficients and the Stirling numbers of the second kind.

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude j and ending at a given altitude k, with additional constraints, for example, to never attain altitude 0 in-between. We first discuss the case of walks on the integers with steps \(-h, \dots , -1, +1, \dots ,...

In the paper, 2 explicit formulas for the Euler numbers of the second kind are obtained. Based on those formulas a exponential generating function is deduced. Using the generating function some well-known and new identities for the Euler number of the second kind are obtained.

In this paper, we study the properties of polynomials defined by generating functions of form F(t, x)α · G(t, α)x. We obtain new properties for those polynomials, which allow to obtain interesting identities. As application, using the results of paper we get the identities for the generalized Bernoulli polynomials.

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and ending at a given altitude $k$, with additional constraints such as, for example, to never attain altitude $0$ in-between. We first discuss the case of walks on the integers with steps $-h, \dots, -1,...

The aim of this paper is to show how to obtain expressions for coefficients of compositional inverse generating functions in explicit way. The method is based on the Lagrange inversion theorem and composita of generating functions. Also we give a method of obtaining expressions for coefficients of reciprocal generating functions and consider some e...

We present techniques for obtaining a generating function for the diagonal T2n,n of the triangle formed from the coefficients of a generating function G(x) raised to the power k. We obtain some relations between central coefficients and coefficients of the diagonal T2n,n, and we also give some examples.

Using notions of composita and composition of generating functions, we establish some explicit formulas for the Generalized Hermite polynomials, the Generalized Humbert polynomials, the Lerch polynomials, and the Mahler polynomials.

Using the notion of the composita, we obtain a method of solving iterative functional equations of the form $A^{2^n}(x)=F(x)$, where $F(x)=\sum_{n>0} f(n)x^n$, $f(1)\neq 0$. We prove that if $F(x)=\sum_{n>0} f(n)x^n$ has integer coefficients, then the generating function $A(x)=\sum_{n>0}a(n)x^n$, which is obtained from the iterative functional equa...

The powers of generating functions and its properties are analyzed. A new class of functions is introduced, based on the application of compositions of an integer $n$, called composita. The methods for obtaining reciprocal and reverse generating functions, and solutions of the functional equations $F(A(x))=G(x)$, where $A(x)$ is an unknown generati...

Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials, Bernoulli Polynomials of the Second Kind, Generalized Bernoulli polynomials, Euler Polynomials, Peters polyno...

We propose a method for obtaining expressions for polynomials based on a composition of generating functions. We obtain expressions for Chebyshev polynomials, Stirling polynomials, Narumi polynomials.

We present techniques for obtaining a generating function for the central coefficients of a triangle $T(n,k)$, which is given by the expression $[xH(x)]^k=\sum_{n\geqslant k} T(n,k)x^n$, $H(0)\neq 0$. We also prove certain theorems for solving direct and inverse problems.

New methods for derivation of Bell polynomials of the second kind are presented. The methods are based on an ordinary generating function and its composita. The relation between a composita and a Bell polynomial is demonstrated. Main theorems are written and examples of Bell polynomials for trigonometric functions, polynomials, radicals, and Bernou...

A new class of functions based on compositions of an integer $n$ and termed compositae is introduced. Main theorems are presented; compositae are written for polynomials, trigonometric and hyperbolic functions, radicals, exponential and log functions. A solution is proposed for the problems of derivation of compositions of ordinary generating funct...

In this paper we study the coefficients of the powers of an ordinary generating function and their properties. A new class of functions based on compositions of an integer $n$ is introduced and is termed composita. We present theorems about compositae and operations with compositae. We obtain the compositae of polynomials, trigonometric and hyperbo...

A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written for polynomials, trigonometric and hyperbolic functions, exponential and log functions. It is shown that the...

We present recurrence formulas for the number of partitions of a natural number n whose parts must be not less than m. A simple proof of Euler’s formula for the number of partitions is given. We construct the triangle of partitions, put forward conjectures concerning the properties of the triangle, and propose an algorithm for calculating the parti...

Not Available

Not Available