Dmitry Kruchinin

Tomsk State University of Control Systems and Radioelectronics | TUSUR · Department of Computer Control and Design Systems

In this paper, we study the combinatorial set of RNA secondary structures of length $n$ with $m$ base-pairs. For a compact representation, we encode an RNA secondary structure by the corresponding Motzkin word. For this combinatorial set, we construct an AND/OR tree structure, find a bijection between the combinatorial set and the set of variants o...

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...

In this paper, we study the combinatorial set of closed questionnaire answers with a specified minimum number of correct answers. For this combinatorial set, we obtain an explicit formula for its cardinality function. Using the obtained cardinality function, we construct the corresponding AND/OR tree structure and determine the bijecion rules for t...

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...

In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the generated number triangle to Catalan’s and Euler’s triangles. On the basis of properties of Catalan’s and Eule...

In this paper, we consider properties of coefficients of a composition of ordinary generating functions. Using the notion of a composita, we get some new properties for this composition. The obtained properties can be used for distinguishing prime numbers from composite numbers. As an application, the obtained results can be used for getting new pr...

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.

In this paper, the relevance of searching for computational methods for obtaining polynomials is shown. The authors propose to use a library for Wolfram Mathematica as such computational method. This library automates the method based on compositae of generating functions, which helped to obtain explicit formulas for many polynomials. Also, this me...

In Tomsk University of Control Systems and Radioelectronics (TUSUR) one of the main areas of research is information security. The work is carried out by a scientific group under the guidance of Professor Shelupanov. One of the directions is the development of a comprehensive approach to assessing the security of the information systems. This direc...

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 the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine,...

In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the results of the paper we get the known properties of the Bell numbers (Touchard’s Congruence for k = 0).

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...

Using the notion of composita and the Lagrange inversion theorem, we present techniques for solving the following functional equation B(x) = H(xB(x)(m)), where H(x), B(x) are generating functions and m is an element of N. Also we give some examples.

Using notions of composita and composition of generating functions, we show an easy way to obtain explicit formulas for some current polynomials. Particularly, we consider the Meixner polynomials of the first and second kinds.

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.

In this work, using notions of composita and composition of generating functions, we show an easy way to obtain explicit formulas for some current polynomials. Particularly, we consider the Mott polynomials. Also we introduce a generalization of the Mott polynomials.

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...

In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the result of paper we get the known properties of the Bell numbers(Touchard's Congruence for $k=0$) and new properties of the Euler numbers. Key words: expon...

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.

In this paper we obtained an original integer sequence based on the properties of the multinomial coefficient. We investigated a property of the sequence that shows connection with a primality testing. For any prime n the n-th term in the sequence is less by 1 than the number of partitions of n. We hypothesize the existence of an asymptotic algorit...

Obtained a new property of superposition of the generating functions ln(1/(1-F(x))), where F(x) - generating function with integer coefficients, which allows the construction a primality tests. The theorem which is based on compositions of positive numbers and its corollary are proved. Examples are given. Key words: Generating functions, superposit...

The paper discusses logarithmic generating functions and their properties. The theorem which is based on compositions of positive numbers and its conclusion are proved. Examples are given. Key words: Logarithmic generating functions, superposition of generating functions, composition of positive number.

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...