arXiv subjects

Khaydar Nurligareev

Publications and source records attributed to Khaydar Nurligareev.

12 recordsLinked to original sources

arXiv2026

The Distribution of the Deepest Leaves in Binary Trees

We study the extreme local structure of plane binary trees through the limiting distribution of the number of leaves at maximum depth. For every $m\geq1$, we determine the limiting probability $κ^{[m]}$ that the deepest level contains $2m$ leaves. The resulting sequence is encoded by a generating function $K(y)$, obtained from a bivariate refinement of the Catalan iteration. We show that, at the critical point of this iteration, a parabolic recurrence governs the marked generating functions. Using continuous iteration theory and a Fatou coordinate, we transform this recurrence into the functional equation $K((1+y/4)^2)=K(y)+4$, which describes the limiting law. Its singular behavior yields the exponential tail $κ^{[m]}\sim4^{-m+1}$. Numerical evaluation gives, e.g., $κ^{[1]}\approx0.7009$, $κ^{[2]}\approx0.2226$ and an average number of deepest leaves equal to $\hatκ=4κ^{[1]}\approx2.8037$.

math.CO↗

arXiv2026

Asymptotics for graphically divergent series: dense digraphs and 2-SAT formulae

We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of expansion coefficients into a special bivariate generating function, which we call a coefficient generating function. We show that coefficient generating functions possess certain general properties that make it possible to express asymptotics in a short closed form. Also, in most scenarios, we indicate a combinatorial meaning of the involved coefficients. Applications of our method include asymptotics of connected graphs, irreducible tournaments, strongly connected digraphs, 2-SAT formulae and contradictory strongly connected implication digraphs. Moreover, due to its flexibility, the method allows to treat a wide range of structural variations, including fixing the numbers of connected, irreducible, strongly connected and contradictory components, as well as source-like, sink-like and isolated ones, or adding weights and marking variables.

math.CO↗

arXiv2026

Growing Binary Trees

This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled structures and the analysis of polynomial iterates, we extend previous models of increasing trees with label repetitions by allowing growth branches to terminate. This mechanism enables a direct connection between dynamic evolutionary processes and classical unlabeled binary trees. We provide a combinatorial outlook for this model, linking our new approach to essential but traditionally complex parameters such as tree height, the maximum number of leaves at the deepest level (for a given tree size), and the overall tree profile. Our approach reveals structural links with Mandelbrot polynomials and coding theory. Furthermore, we leverage these structural insights to develop an efficient, iterative uniform random sampler for binary trees with a prescribed profile, achieving optimal complexity in both time and space and in random bit consumption.

math.CO↗

arXiv2026

Asymptotic probability of irreducibles III: Anti-SEQ

In this paper, we study the structure of the complete asymptotic expansion of the probability that a large combinatorial object is connected or consists of a given number of connected components. For rapidly growing labeled families of structures, the coefficients involved in these expansions are possibly negative integers. Using species theory, we interpret these coefficients as the difference between the counting sequences of two derivative species of structures. In particular, we show that this difference can be viewed as the counting sequence of the virtual species obtained with the help of an "anti-$\mathrm{SEQ}$" operator applied to the initial family of structures. Applications include $P$-angulated discrete surfaces, quadratic square-tiled surfaces, and non-orientable graph encoded manifolds, which were not reachable with our previous methods. Moving on to the weighted species, we establish the whole structure of the asymptotic expansion of the probability that a graph is connected in the Erdős-Rényi model $G(n,p)$. Here, the asymptotic coefficients are polynomials in $\frac{p}{1-p}$ and can be described both in terms of simple graphs and irreducible tournaments with ties. We also provide general asymptotic results for sequence and cycle decomposition, as well as the complete asymptotic expansion of the probability that a random labeled tournament with ties is irreducible.

math.CO↗

arXiv2026

Brick Wall Excursions: Combinatorial Interpretation of Random Flight Moments

We study the expected distance of short uniform random walks in arbitrary dimensions with unit steps in random directions. It is known that for dimensions $d=2$ and $d=4$, all the moments of an $m$-step walk are integer. While for $d=2$, the $n$th moment can be interpreted as the number of abelian squares of length $2n$ over an alphabet with $m$ letters, for $d=4$ no interpretation was known. The goal of this paper is to provide such an interpretation, both for $d=2$ and $d=4$, in terms of $2n$-step lattice paths in dimension $m-1$. Our construction relies on a bijection between Dyck paths with a prescribed number of peaks and words of a certain type. In addition, this bijection allows us to derive closed formulas for the number of lattice paths provided with certain statistics.

math.CO↗

arXiv2025

Asymptotic probability of irreducibles II: sequence

This paper is devoted to the structure of the complete asymptotic expansion of the probability that a large combinatorial object is irreducible or consists of a given number of irreducible parts, where irreducibility is understood in terms of combinatorial construction SEQ, labeled or unlabeled. We show that for rapidly growing (i.e. gargantuan) combinatorial classes, the coefficients that appear in this expansion are integers and can be interpreted as linear combinations of the counting sequences of three closely related combinatorial classes. We apply this general asymptotic result to labeled and unlabeled (multi-)tournaments, as well as to (multi-)permutations and (multi-)matchings. We also explore the limits of our approach with respect to other combinatorial constructions.

math.CO↗

arXiv2025

Endhered patterns in matchings and RNA

An endhered (end-adhered) pattern is a subset of arcs in matchings, such that the corresponding starting points are consecutive and the same holds for the ending points. Such patterns are in one-to-one correspondence with the permutations. We focus on the occurrence frequency of such patterns in matchings and native (real-world) RNA structures with pseudoknots. We present combinatorial results related to the distribution and asymptotic behavior of the pattern 21, which corresponds to two consecutive base pairs frequently encountered in RNA, and the pattern 12, representing the archetypal minimal pseudoknot. We show that in matchings these two patterns are equidistributed, which is quite different from what we can find in native RNAs. We also examine the distribution of endhered patterns of size 3, showing how the patterns change under the transformation called endhered twist. Finally, we compute the distributions of endhered patterns of size 2 and 3 in native secondary RNA structures with pseudoknots and discuss possible outcomes of our study.

math.CO↗

arXiv2024

Asymptotics of self-overlapping permutations

In this work, we study the concept of self-overlapping permutations, which is related to the larger study of consecutive patterns in permutations. We show that this concept admits a simple and clear geometrical meaning, and prove that a permutation can be represented as a sequence of non-self-overlapping ones. The above structural decomposition allows us to obtain equations for the corresponding generating functions, as well as the complete asymptotic expansions for the probability that a large random permutation is (non-)self-overlapping. In particular, we show that almost all permutations are non-self-overlapping, and that the corresponding asymptotic expansion has the self-reference property: the involved coefficients count non-self-overlapping permutations once again. We also establish complete asymptotic expansions of the distributions of very tight non-self-overlapping patterns, and discuss the similarities of the non-self-overlapping permutations to other permutation building blocks, such as indecomposable and simple permutations, as well as their associated asymptotics.

math.CO↗

arXiv2024

Asymptotic probability for connectedness

We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of other derivative combinatorial classes. The general result applies to rapidly growing combinatorial structures, which we call gargantuan, that also admit a sequence decomposition. The result is then applied to several models of graphs, of surfaces (square-tiled surfaces, combinatorial maps), and to geometric models of higher dimension (constellations, graph encoded manifolds). The corresponding derivative combinatorial classes are irreducible (multi)tournaments, indecomposable (multi)permutations and indecomposable perfect (multi)matchings.

math.CO↗

arXiv2022

Watermelons on the half-plane

We study the watermelon probabilities in the uniform spanning forests on the two-dimensional semi-infinite square lattice near either open or closed boundary to which the forests can or cannot be rooted, respectively. We derive universal power laws describing the asymptotic decay of these probabilities with the distance between the reference points growing to infinity, as well as their non-universal constant prefactors. The obtained exponents match with the previous predictions made for the related dense polymer models using the Coulomb Gas technique and Conformal Field Theory, as well as with the lattice calculations made by other authors in different settings. We also discuss the logarithmic corrections some authors argued to appear in the watermelon correlation functions on the infinite lattice. We show that the full account for diverging terms of the lattice Green function, which ensures the correct probability normalization, provides the pure power law decay in the case of semi-infinite lattice with closed boundary studied here, as well as in the case of infinite lattice discussed elsewhere. The solution is based on the all-minors generalization of the Kirchhoff matrix tree theorem, the image method and the developed asymptotic expansion of the Kirchhoff determinants.

math-ph↗

arXiv2022

Asymptotics for connected graphs and irreducible tournaments

We compute the whole asymptotic expansion of the probability that a large uniform labeled graph is connected, and of the probability that a large uniform labeled tournament is irreducible. In both cases, we provide a combinatorial interpretation of the involved coefficients.

math.CO↗

arXiv2022

Decompositions of functions defined on finite sets in $\mathbb{R}^d$

A finite subset $M \subset \mathbb{R}^d$ is basic, if for any function $f \colon M \to \mathbb{R}$ there exists a collection of functions $f_1, \ldots, f_d \colon \mathbb{R} \to \mathbb{R}$ such that for each element $(x_1, \ldots, x_d)\in M$ we have $f(x_1, \ldots, x_d) = f_1(x_1) + \ldots + f_d(x_d)$. For certain finite sets, we prove a criterion for a set to be basic, and we show that it cannot be extended to the general case. In addition, we interpret the above criterion in terms of doubly-weighted graphs and give an estimation for the number of elements in certain basic and non-basic subsets.

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