arXiv subjects

Max A. Alekseyev

Publications and source records attributed to Max A. Alekseyev.

At least 37 records · Page 2Linked to original sources

arXiv2018

On Enumeration of Dyck--Schröder Paths

We address the problem of enumerating paths in square lattices, where allowed steps include (1,0) and (0,1) everywhere, and (1,1) above the diagonal y=x. We consider two such lattices differing in whether the (1,1) steps are allowed along the diagonal itself. Our analysis leads to explicit generating functions and an efficient way to compute terms of many sequences in the Online Encyclopedia of Integer Sequences, proposed by Clark Kimberling almost two decades ago.

math.CO↗

arXiv2018

On p-adic approximation of sums of binomial coefficients

We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are divisible by arbitrarily large powers of prime $p$.

math.NT↗

arXiv2018

On partitions into squares of distinct integers whose reciprocals sum to 1

In 1963, Graham proved that all integers greater than 77 (but not 77 itself) can be partitioned into distinct positive integers whose reciprocals sum to 1. He further conjectured that for any sufficiently large integer, it can be partitioned into squares of distinct positive integers whose reciprocals sum to 1. In this study, we establish the exact bound for existence of such partitions by proving that 8542 is the largest integer with no such partition.

math.NT↗

arXiv2017

Making Walks Count: From Silent Circles to Hamiltonian Cycles

We illustrate the application of the matrix-transfer method for a number of enumeration problems concerning the party game Silent Circles, Hamiltonian cycles in the antiprism graphs, and simple paths and cycles of a fixed length in arbitrary graphs.

math.CO↗

arXiv2017

Estimation of the True Evolutionary Distance under the Fragile Breakage Model

The ability to estimate the evolutionary distance between extant genomes plays a crucial role in many phylogenomic studies. Often such estimation is based on the parsimony assumption, implying that the distance between two genomes can be estimated as the rearrangement distance equal the minimal number of genome rearrangements required to transform one genome into the other. However, in reality the parsimony assumption may not always hold, emphasizing the need for estimation that does not rely on the rearrangement distance. The distance that accounts for the actual (rather than minimal) number of rearrangements between two genomes is often referred to as the true evolutionary distance. While there exists a method for the true evolutionary distance estimation, it however assumes that genomes can be broken by rearrangements equally likely at any position in the course of evolution. This assumption, known as the random breakage model, has recently been refuted in favor of the more rigorous fragile breakage model postulating that only certain "fragile" genomic regions are prone to rearrangements. We propose a new method for estimating the true evolutionary distance between two genomes under the fragile breakage model. We evaluate the proposed method on simulated genomes, which show its high accuracy. We further apply the proposed method for estimation of evolutionary distances within a set of five yeast genomes and a set of two fish genomes.

q-bio.GN↗

arXiv2017

Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs

Genome rearrangements can be modeled as $k$-breaks, which break a genome at k positions and glue the resulting fragments in a new order. In particular, reversals, translocations, fusions, and fissions are modeled as $2$-breaks, and transpositions are modeled as $3$-breaks. While $k$-break rearrangements for $k>3$ have not been observed in evolution, they are used in cancer genomics to model chromothripsis, a catastrophic event of multiple breakages happening simultaneously in a genome. It is known that the $k$-break distance between two genomes (i.e., the minimum number of $k$-breaks required to transform one genome into the other) can be computed in terms of cycle lengths in the breakpoint graph of these genomes. In the current work, we address the combinatorial problem of enumerating genomes at a given $k$-break distance from a fixed unichromosomal genome. More generally, we enumerate genome pairs, whose breakpoint graph has a given distribution of cycle lengths. We further show how our enumeration can be used for uniform sampling of random genomes at a given $k$-break distance, and describe its connection to various combinatorial objects such as Bell polynomials.

q-bio.GN↗

arXiv2016

Weighted de Bruijn Graphs for the Menage Problem and Its Generalizations

We address the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses sit next to each other and no k consecutive persons are of the same gender. While the case of k=2 corresponds to the classical problème des ménages with a well-studied solution, no closed-form expression for the number of seating arrangements is known when k>=3. We propose a novel approach for this type of problems based on enumeration of walks in certain algebraically weighted de Bruijn graphs. Our approach leads to new expressions for the menage numbers and their exponential generating function and allows one to efficiently compute the number of seating arrangements in general cases, which we illustrate in detail for the ternary case of k=3.

math.CO↗

arXiv2016

Combinatorial Scoring of Phylogenetic Networks

Construction of phylogenetic trees and networks for extant species from their characters represents one of the key problems in phylogenomics. While solution to this problem is not always uniquely defined and there exist multiple methods for tree/network construction, it becomes important to measure how well the constructed networks capture the given character relationship across the species. In the current study, we propose a novel method for measuring the specificity of a given phylogenetic network in terms of the total number of distributions of character states at the leaves that the network may impose. While for binary phylogenetic trees, this number has an exact formula and depends only on the number of leaves and character states but not on the tree topology, the situation is much more complicated for non-binary trees or networks. Nevertheless, we develop an algorithm for combinatorial enumeration of such distributions, which is applicable for arbitrary trees and networks under some reasonable assumptions.

q-bio.PE↗

arXiv2016

Computing the inverses, their power sums, and extrema for Euler's totient and other multiplicative functions

We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power sums (e.g., cardinality) or extrema, without direct enumeration of the inverses. We illustrate our algorithm with Euler's totient function $φ(\cdot)$ and the $k$-th power sum of divisors $σ_k(\cdot)$. For example, we can establish that the number of solutions to $σ_1(x) = 10^{1000}$ is 15,512,215,160,488,452,125,793,724,066,873,737,608,071,476, while it is intractable to iterate over the actual solutions.

cs.DM↗

arXiv2015

A Computational Method for the Rate Estimation of Evolutionary Transpositions

Genome rearrangements are evolutionary events that shuffle genomic architectures. Most frequent genome rearrangements are reversals, translocations, fusions, and fissions. While there are some more complex genome rearrangements such as transpositions, they are rarely observed and believed to constitute only a small fraction of genome rearrangements happening in the course of evolution. The analysis of transpositions is further obfuscated by intractability of the underlying computational problems. We propose a computational method for estimating the rate of transpositions in evolutionary scenarios between genomes. We applied our method to a set of mammalian genomes and estimated the transpositions rate in mammalian evolution to be around 0.26.

q-bio.GN↗

arXiv2014

Solving the Tower of Hanoi with Random Moves

We prove the exact formulae for the expected number of moves to solve several variants of the Tower of Hanoi puzzle with 3 pegs and n disks, when each move is chosen uniformly randomly from the set of all valid moves. We further present an alternative proof for one of the formulae that couples a theorem about expected commute times of random walks on graphs with the delta-to-wye transformation used in the analysis of three-phase AC systems for electrical power distribution.

math.CO↗

arXiv2014

On the minimal teaching sets of two-dimensional threshold functions

It is known that a minimal teaching set of any threshold function on the twodimensional rectangular grid consists of 3 or 4 points. We derive exact formulae for the numbers of functions corresponding to these values and further refine them in the case of a minimal teaching set of size 3. We also prove that the average cardinality of the minimal teaching sets of threshold functions is asymptotically 7/2. We further present corollaries of these results concerning some special arrangements of lines in the plane.

math.CO↗

arXiv2014

On integral points on biquadratic curves and near-multiples of squares in Lucas sequences

We describe an algorithmic reduction of the search for integral points on a curve y^2 = ax^4 + bx^2 + c with nonzero ac(b^2-4ac) to solving a finite number of Thue equations. While existence of such reduction is anticipated from arguments of algebraic number theory, our algorithm is elementary and to best of our knowledge is the first published algorithm of this kind. In combination with other methods and powered by existing software Thue equations solvers, it allows one to efficiently compute integral points on biquadratic curves. We illustrate this approach with a particular application of finding near-multiples of squares in Lucas sequences. As an example, we establish that among Fibonacci numbers only 2 and 34 are of the form 2m^2+2; only 1, 13, and 1597 are of the form m^2-3; and so on. As an auxiliary result, we also give an algorithm for solving a Diophantine equation k^2 = f(m,n)/g(m,n) in integers m,n,k, where f and g are homogeneous quadratic polynomials.

math.NT↗

arXiv2013

On the number of permutations with bounded run lengths

In this work we obtain recurrent formulae for the number of permutations with either increasing or monotonic (i.e., both increasing and decreasing) runs of bounded length. Our formulae allow one to efficiently compute the number of such permutations. In particular, we use the formulae to find and correct a few miscalculations in the classic 1966 book by David, Kendall, and Barton. We further use our formulae to derive differential equations for the corresponding exponential generating functions. In the case of increasing runs, we solve these equations and obtain closed-form expressions for the generating functions.

math.CO↗

arXiv2012

BayesHammer: Bayesian clustering for error correction in single-cell sequencing

Error correction of sequenced reads remains a difficult task, especially in single-cell sequencing projects with extremely non-uniform coverage. While existing error correction tools designed for standard (multi-cell) sequencing data usually come up short in single-cell sequencing projects, algorithms actually used for single-cell error correction have been so far very simplistic. We introduce several novel algorithms based on Hamming graphs and Bayesian subclustering in our new error correction tool BayesHammer. While BayesHammer was designed for single-cell sequencing, we demonstrate that it also improves on existing error correction tools for multi-cell sequencing data while working much faster on real-life datasets. We benchmark BayesHammer on both $k$-mer counts and actual assembly results with the SPAdes genome assembler.

q-bio.QM↗

arXiv2012

On pairwise distances and median score of three genomes under DCJ

In comparative genomics, the rearrangement distance between two genomes (equal the minimal number of genome rearrangements required to transform them into a single genome) is often used for measuring their evolutionary remoteness. Generalization of this measure to three genomes is known as the median score (while a resulting genome is called median genome). In contrast to the rearrangement distance between two genomes which can be computed in linear time, computing the median score for three genomes is NP-hard. This inspires a quest for simpler and faster approximations for the median score, the most natural of which appears to be the halved sum of pairwise distances which in fact represents a lower bound for the median score. In this work, we study relationship and interplay of pairwise distances between three genomes and their median score under the model of Double-Cut-and-Join (DCJ) rearrangements. Most remarkably we show that while a rearrangement may change the sum of pairwise distances by at most 2 (and thus change the lower bound by at most 1), even the most "powerful" rearrangements in this respect that increase the lower bound by 1 (by moving one genome farther away from each of the other two genomes), which we call strong, do not necessarily affect the median score. This observation implies that the two measures are not as well-correlated as one's intuition may suggest. We further prove that the median score attains the lower bound exactly on the triples of genomes that can be obtained from a single genome with strong rearrangements. While the sum of pairwise distances with the factor 2/3 represents an upper bound for the median score, its tightness remains unclear. Nonetheless, we show that the difference of the median score and its lower bound is not bounded by a constant.

q-bio.GN↗

arXiv2011

On convergence of the Flint Hills series

It is not known whether the Flint Hills series $\sum_{n=1}^{\infty} \frac{1}{n^3\cdot\sin(n)^2}$ converges. We show that this question is closely related to the irrationality measure of $π$, denoted $μ(π)$. In particular, convergence of the Flint Hills series would imply $μ(π) \leq 2.5$ which is much stronger than the best currently known upper bound $μ(π)\leq 7.6063...$. This result easily generalizes to series of the form $\sum_{n=1}^{\infty} \frac{1}{n^u\cdot |\sin(n)|^v}$ where $u,v>0$. We use the currently known bound for $μ(π)$ to derive conditions on $u$ and $v$ that guarantee convergence of such series.

math.CA↗

arXiv2010

Weighted genomic distance can hardly impose a bound on the proportion of transpositions

Genomic distance between two genomes, i.e., the smallest number of genome rearrangements required to transform one genome into the other, is often used as a measure of evolutionary closeness of the genomes in comparative genomics studies. However, in models that include rearrangements of significantly different "power" such as reversals (that are "weak" and most frequent rearrangements) and transpositions (that are more "powerful" but rare), the genomic distance typically corresponds to a transformation with a large proportion of transpositions, which is not biologically adequate. Weighted genomic distance is a traditional approach to bounding the proportion of transpositions by assigning them a relative weight α > 1. A number of previous studies addressed the problem of computing weighted genomic distance with α \leq 2. Employing the model of multi-break rearrangements on circular genomes, that captures both reversals (modelled as 2-breaks) and transpositions (modelled as 3-breaks), we prove that for α \in (1,2], a minimum-weight transformation may entirely consist of transpositions, implying that the corresponding weighted genomic distance does not actually achieve its purpose of bounding the proportion of transpositions. We further prove that for α \in (1,2), the minimum-weight transformations do not depend on a particular choice of α from this interval. We give a complete characterization of such transformations and show that they coincide with the transformations that at the same time have the shortest length and make the smallest number of breakages in the genomes. Our results also provide a theoretical foundation for the empirical observation that for α < 2, transpositions are favored over reversals in the minimum-weight transformations.

q-bio.GN↗