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Dmitry Kruchinin

Publications and source records attributed to Dmitry Kruchinin.

10 recordsLinked to original sources
arXiv2026

A Family of Generating Functions for Reciprocal Binomial Coefficients and Its Applications

A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained, including identities connecting reciprocal binomial coefficients with harmonic numbers and Fibonacci numbers. The application of the found functions for evaluating infinite numerical sequences involving reciprocal binomial coefficients is demonstrated.

math.CO↗
arXiv2023

Algorithms for ranking and unranking the combinatorial set of RNA secondary structures

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 of the AND/OR tree, and develop algorithms for ranking and unranking the variants of the AND/OR tree. The developed ranking and unranking algorithms have polynomial time complexity $O(m^2 (n - m))$ for $m < n - 2 m$ and $O(m (n - m)^2)$ for $m > n - 2 m$. In contrast to the existing algorithms, the new algorithms do not require preprocessing steps and have better time complexity.

cs.DS↗
arXiv2017

Explicit formulas for enumeration of lattice paths: basketball and the kernel method

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 $-h, \dots, -1, +1, \dots, +h$. The case $h=1$ is equivalent to the classical Dyck paths, for which many ways of getting explicit formulas involving Catalan-like numbers are known. The case $h=2$ corresponds to "basketball" walks, which we treat in full detail. Then we move on to the more general case of walks with any finite set of steps, also allowing some weights/probabilities associated with each step. We show how a method of wide applicability, the so-called "kernel method", leads to explicit formulas for the number of walks of length $n$, for any $h$, in terms of nested sums of binomials. We finally relate some special cases to other combinatorial problems, or to problems arising in queuing theory.

math.CO↗
arXiv2015

New properties for a composition of some generating functions for primes

In this paper, we consider properties of coefficients of a generating functions composition, where the outer function is a logarithmic generating function and the inner function is an ordinary generating function with integer coefficients. Using notions of composita and composition of generating functions, we get new properties for this composition. The properties can be used for distinguishing prime numbers from composite numbers. As an application, obtained results can be used to obtain new primality criteria. We obtain primality criteria for the Mersenne numbers, the Lucas numbers, the Pell-Lucas numbers, the Jacobsthal-Lucas numbers, and the Lucas sequences. Keywords: generating function, composition of generating function, composita, primality criterion.

math.CO↗
arXiv2013

Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$

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 equation $4A(A(x))=F(4x)$, has integer coefficients. Key words: iterative functional equation, composition of generating functions, composita.

math.CO↗
arXiv2012

Integer properties of a composition of exponential generating functions

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: exponential generating function, composition of generating functions, composita, primality, Touchard's Congruence, Bell numbers, Euler numbers.

math.NT↗
arXiv2012

Application of a composition of generating functions for obtaining explicit formulas of polynomials

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 polynomials, Narumi polynomials, Humbert polynomials, Lerch polynomials and Mahler polynomials.

math.NT↗
arXiv2012

The number of multinomial coefficients based on a set of partitions of n into k parts and divided by k evenly

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 algorithm of primality testing.

math.CO↗
arXiv2011

On a property of superposition of the generating functions ln(1/(1-F(x)))

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, superposition of generating functions, composition of a natural number.

math.CO↗