arXiv subjects

Sergey Kitaev

Publications and source records attributed to Sergey Kitaev.

At least 19 recordsLinked to original sources

arXiv2026

Wilf Equivalence for Length-Three Patterns and Flat POPs, and a Conjecture of Qiu and Remmel

It is well known that, for each classical pattern $\tau$ of length 3, the number of $\tau$-avoiding permutations of length $n$ is the $n$th Catalan number, and numerous bijections between different length-three avoidance classes have been constructed and studied. In this paper, we refine this classical problem by studying Wilf equivalence among permutations that simultaneously avoid a classical pattern of length three and a flat partially ordered pattern. Partially ordered patterns (POPs) provide a flexible framework for encoding families of classical permutation patterns. For $\ell\geq 3$ and $1\leq x\leq\ell$, let $P_{\ell,x}$ be the length-$\ell$ POP in which the entry at position $x$ is required to be smaller than all the other entries, while no relations are imposed among the remaining entries. Such POPs are called flat POPs. We classify the Wilf equivalences among all pairs $(\tau,P_{\ell,x})$, where $\tau$ is a classical pattern of length three. For every $\ell\geq4$, the resulting $6\ell$ pairs form exactly $2\ell-1$ Wilf equivalence classes, while the exceptional case $\ell=3$ gives four classes. Our proofs combine the derivation of explicit formulas and recurrence relations with the construction of bijections. Moreover, we introduce novel prime-divisor arguments to distinguish the remaining candidate classes, reducing the problem to showing that a certain Diophantine equation has no solutions for $\ell\ge 3{,}274$, where the bound $3{,}274$ is not claimed to be sharp. Finally, by extending our work on POPs, we resolve a conjecture of Qiu and Remmel concerning the distribution of quadrant marked mesh patterns on 132-avoiding permutations and correct an error in their paper that is crucial to the proof.

math.CO↗

arXiv2026

Involution $h$ on Catalan structures

We define an involution $h$ on Catalan structures through an abstract framework, prove an equidistribution theorem for four canonical statistics and present a generating function carrying these. This framework encompasses all combinatorial structures with a decomposition mirroring the first-return decomposition of Dyck paths. The fixed points of~$h$ are counted by Catalan numbers. Canonical bijections transport the equidistribution to eight well known concrete families, identifying the canonical statistics with native ones on each. In addition to its primary structure, each Catalan structure has a derived \emph{secondary structure}, and $h$~interchanges primary and secondary structure. The involution factors as $h = \rev \circ \corev \circ \rev$, where $\rev$ and $\corev$ are two simpler involutions, and the composition $M = h \circ \rev$ coincides with Donaghey's automorphism on plane trees. This yields $M^{-1} = \rev \circ M \circ \rev$ and a period theorem: Iterating the secondary structure construction produces a sequence that repeats with period equal to the order of~$M$. It is an open problem to describe $h$ and the canonical statistics explicitly on most of the more than two hundred known families of Catalan structures.

math.CO↗

arXiv2026

On mesh patterns of short length: Equidistribution and enumeration

The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds for the numbers of distribution-equivalence and Wilf-equivalence classes drop to $106$ and $47$, respectively. Combined with the known lower bounds of 105 and 46, conjectured to be exact, these results leave both classifications hinging on a single distribution-equivalence question conjectured in 2019, whose resolution would at once settle the remaining Wilf-equivalence case. We further conjecture that this unresolved equidistribution also holds for involutions, a subclass of all permutations. We also determine the distributions of three additional classes of length-$2$ mesh patterns through a detailed structural analysis. Our work combines bijective techniques with generating-function methods, yielding new insights into the structure and enumeration of short mesh patterns.

math.CO↗

arXiv2026

Equidistribution of mesh patterns of short length

We study the equidistribution of mesh patterns of length 2. We show that the number of equidistribution equivalence classes lies between 105 and 108, and conjecture that it is exactly 105. As a consequence, we obtain an upper bound of 49 Wilf-classes, improving the previously known bound of 56 due to Hilmarsson et al., and reducing the problem to three remaining conjectural equivalences (with the actual number conjectured to be 46). Our approach combines bijective constructions, generating functions, recurrence relations, and structural symmetries. We establish several new equidistribution results, including four previously unknown distribution classes, connect numerous patterns to known distributions in the literature, and resolve seven open pattern-avoidance enumeration problems posed by Hilmarsson et al. This work provides a near-complete classification of mesh patterns of length 2 and unifies several previously isolated results within a coherent framework.

math.CO↗

arXiv2026

The edit distance of word-representable and comparability graphs

In this paper, we establish that the maximum edit distance of an $n$-vertex graph from the hereditary property of word-representable graphs is $n^2/8-o(n^2)$. In addition, we establish that the maximum edit distance of an $n$-vertex graph from the hereditary property of poset comparability graphs is $5n^2/32-o(n^2)$. In fact, we determine the edit distance function over all edge densities $p\in [0,1]$ for the property of word-representable graphs, for the property of $k$-word-representable graphs for each $k\geq 2$, and for the property comparability graphs. The latter has a peculiar structure that requires an infinite sequence of colored regularity graphs.

math.CO↗

arXiv2026

Simultaneous avoidance of length-4 patterns in ascent sequences

Ascent sequences form a central class of combinatorial objects, as they are in bijection with several important families such as (2+2)-free posets, Stoimenow matchings, and other Fishburn objects, and are enumerated by the Fishburn numbers. We study pattern avoidance in ascent sequences for the five patterns of length 4: $0101$, $0102$, $0112$, $0120$, and $0121$. These patterns arise naturally from recent work on pattern avoidance in related families of Fishburn objects, including Stoimenow matchings and (2+2)-free posets. We enumerate ascent sequences avoiding any subset of these patterns, with the exception of the sets $\{0120\}$, $\{0121\}$, and $\{0120,0121\}$, for which the enumeration remains open. Our results reveal that the corresponding avoidance classes fall into $16$ Wilf equivalence classes and exhibit a wide range of enumerative behaviour, including connections to classical sequences such as the Catalan and Fibonacci numbers, as well as polynomial formulas and rational generating functions; several of the sequences we obtain appear to be new. Our methods combine structural decompositions with generating-tree techniques and, in several cases, rely on reductions to shorter patterns via restricted growth functions. This work contributes to the broader study of pattern avoidance across Fishburn families and highlights further connections between ascent sequences and other combinatorial structures.

math.CO↗

arXiv2026

Counting permutations avoiding two flat partially ordered patterns

Partially ordered patterns (POPs) play an important role in the study of permutation patterns, providing a convenient framework for describing large families of classical patterns. The problem of enumerating permutations that avoid POPs has therefore attracted considerable attention in the literature. In particular, Gao and Kitaev resolved many counting problems for POP-avoiding permutations of lengths 4 and 5, linking the enumeration to a wide range of other combinatorial objects. Motivated by their work, we initiate the study of permutations that simultaneously avoid two POPs belonging to the class of flat POPs. We establish a connection between permutations avoiding such POPs and the $k$-Fibonacci numbers. Moreover, we provide a bijection between permutations avoiding these POPs and certain restricted permutations, which allows us to use the method developed by Balti\'{c} to derive the generating function for permutations avoiding these POPs. Finally, we obtain enumerative results for separable permutations avoiding these two POPs, of lengths up to 5, with respect to six statistics, thereby extending the results of Gao et al. on the avoidance of a single flat POP in separable permutations. Notably, when both patterns are of length 5, the respective generating function is a rational function, with the sum in the numerator (resp., denominator) containing 293 (resp., 17) monomials.

math.CO↗

arXiv2026

On shortening universal words for multi-dimensional permutations

A universal word (u-word) for $d$-dimensional permutations of length $n$ is a 2-dimensional word with $d-1$ rows, any size $n$ window of which is order-isomorphic to exactly one permutation of length $n$, and all permutations of length $n$ are covered. It is known that u-words (in fact, even u-cycles, a stronger claim) for $d$-dimensional permutations exist. In this paper, we use the idea of incomparable elements to prove that u-words of length $(n!)^{d-1}+n-1-i(n-1)$, for $d\geq 2$ and $$0\leq i\leq \frac{2^{d-1}}{n-1}\left[(1+(n-1)!)^{d-1}-\left(1+\frac{(n-1)!}{2}\right)^{d-1}\right],$$ for $d$-dimensional permutations of length $n$ exist, which generalizes the respective result of Kitaev, Potapov and Vajnovszki for ``usual'' permutations ($d=2$).

math.CO↗

arXiv2026

Stable patterns on permutations of multisets

In this paper, we study patterns on permutations of multisets whose multivariate distribution generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a pattern as stable. Although various stability results are already implicit in existing enumerative work, we explicitly summarize them here and provide bijective proofs. These bijections offer new combinatorial insight into the symmetry of the generating functions. We also establish instability results. In particular, we provide a complete characterization of stable classical patterns, showing that the only such patterns are those of length one or two. For consecutive patterns, we reprove the stability of all monotone patterns and also identify a large class of unstable patterns. We conjecture that monotone patterns are the only stable consecutive patterns. As an application, we use stability to derive recurrence relations for the ascent distribution over permutations of restricted multisets, yielding a generalization of Eulerian numbers.

math.CO↗

arXiv2026

Eulerian-type polynomials over matchings and matching permutations

Claesson and Linusson [Proc. Am. Math. Soc., 139 (2011), 435-449] observed that there are n! matchings on [2n] with no left-nestings. Inspired by this result, this paper is devoted to exploring a deeper connection between matchings and permutations. We first discover that a quadruple statistic over matchings corresponds to the well known quadruple statistic (exc,drop,fix,cyc) over permutations, where exc, drop, fix and cyc are the excedance, drop, fixed point and cycle statistics, respectively. By introducing matching permutations, we provide a symmetric expansion of a five-variable neighbor polynomial of matchings, which encodes a great deal of neighbor information. As an application, we discover the e-positivity of NCA-polynomials, which implies that the left-nesting number, the left-crossing number and the neighbor alignment number are distributed symmetrically over all matchings on [2n]. We also establish the relationship between the five-variable neighbor polynomials and the trivariate second-order Eulerian polynomials, which generalizes the related results of Claesson and Linusson, Cameron and Killpatrick as well as Chen and Fu.

math.CO↗

arXiv2025

Distribution of statistics on separable permutations restricted by a flat POP

Finding distributions of statistics in pattern-avoiding permutations has attracted significant attention in the literature. In particular, Chen, Kitaev, and Zhang derived functional equations for the joint distributions of any subset of classical minima and maxima statistics, as well as for the joint distributions of ascents and descents in separable permutations. Meanwhile, partially ordered patterns (POPs) have also been extensively studied. Notably, so-called flat POPs played a key role, via the notion of shape-Wilf-equivalence, in proving a conjecture on pattern-avoiding permutations. In this paper, we study flat POP-avoiding separable permutations, where the maximum element in a flat POP receives the largest label. Avoiding such a POP imposes restrictions on the position of the maximum element in a separable permutation, forcing it to be positioned to the left. We establish a system of functional equations describing the joint distribution of six classical statistics in the most general case, extending the work of Chen, Kitaev, and Zhang. As a specialization, when the POP has length 3, we recover a joint distribution result of Han and Kitaev on permutations avoiding classical patterns of length 3. As another specialization, for the flat POP of length 4, we derive an explicit rational generating function that captures the distribution of six statistics, with a numerator containing 100 monomials and a denominator containing 19 monomials.

math.CO↗

arXiv2025

Stoimenow matchings avoiding multiple Catalan patterns simultaneously

Motivated by Vassiliev's knot invariants, Stoimenow introduced a special class of matchings, now known as Stoimenow matchings. These matchings have since been linked to various combinatorial structures enumerated by the Fishburn numbers. In a recent paper, a problem posed by Bevan et al. was addressed concerning the identification of subsets of Stoimenow matchings counted by the Catalan numbers. Five such subsets were presented, each defined by the avoidance of a single pattern, referred to as a Catalan pattern, within Stoimenow matchings. In the present paper, we extend this line of research by enumerating all cases of simultaneous avoidance of sets of Catalan patterns in Stoimenow matchings. This comprehensive analysis reveals connections to nine integer sequences listed in the OEIS.

math.CO↗

arXiv2025

Catalan structures arising from pattern-avoiding Stoimenow matchings and other Fishburn objects

In connection with Vassiliev's knot invariants, Stoimenow introduced in 1998 a class of matchings, also known as regular linearized chord diagrams. These matchings are linked to various combinatorial structures, all of which are associated with the Fishburn numbers. In this paper, we address a problem posed by Bevan et al.\ in 2025 concerning the identification of subsets of Stoimenow matchings that are counted by the Catalan numbers. We present five solutions in terms of pattern-avoiding matchings. We also consider four infinite families of patterns that generalize four of the five forbidden patterns appearing in the solution to the problem we solved and prove that the matchings avoiding them are equinumerous. Finally, we establish numerous results on distributions and joint equidistribution of statistics over these Catalan-counted subsets of Fishburn structures, namely Stoimenow matchings, $(2+2)$-free posets, ascent sequences, and Fishburn permutations, notably expressing some of them in terms of Narayana numbers and others in terms of ballot numbers.

math.CO↗

arXiv2025

On the word-representability of $K_m$-$K_n$ graphs

Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. Several papers in the literature have explored the word-representability of split graphs, in which the vertices can be partitioned into a clique and an independent set. In this paper, we initiate the study of the word-representability of graphs in which the vertices can be partitioned into two cliques. We provide a complete characterization of such word-representable graphs in terms of forbidden subgraphs when one of the cliques has a size of at most four. In particular, if one of the cliques is of size four, we prove that there are seven minimal non-word-representable graphs.

math.CO↗

arXiv2025

On the representation number of grid graphs and cylindric grid graphs

The representation number of a graph is the minimum number of copies of each vertex required to represent the graph as a word, such that the letters corresponding to vertices $x$ and $y$ alternate if and only if $xy$ is an edge in the graph. It is known that path graphs, circle graphs, and ladder graphs have representation number 2, while prism graphs have representation number 3. In this paper, we extend these results by showing that generalizations of the aforementioned graphs -- namely, the $m \times n$ grid graphs and $m \times n$ cylindrical grid graphs -- have representation number $3$ for $m \geq 3$ and $m \geq 2$, respectively, and $n\geq 3$. Furthermore, we discuss toroidal grid graphs in the context of word-representability, which leads to an interesting conjecture.

math.CO↗

arXiv2025

A bijection between $321$- and $213$-avoiding permutations preserving $t$-stack-sortability

We construct a bijection between $321$- and $213$-avoiding permutations that preserves the property of $t$-stack-sortability. Our bijection transforms natural statistics between these two classes of permutations and proves a refinement of an enumerative conjecture posed by Zhang and Kitaev. This work contributes further to the long-standing line of research on bijections between length-3 pattern avoiding permutations. Increasing binary trees lie at the heart of our approach.

math.CO↗

arXiv2025

On fourteen equidistribution conjectures of Lv and Zhang and monotone mesh patterns with corner shadings

Three complementation-like involutions are constructed on permutations to prove, and in some cases generalize, all remaining fourteen joint symmetric equidistribution conjectures of Lv and Zhang. Further enumerative results are obtained for several classes of (mesh) pattern-avoiding permutations, where the shadings of all involved mesh patterns are restricted to an opposing pair of corners.

math.CO↗

arXiv2025

Topological weight and structural diversity of polydisperse chromatin loop networks

Current biophysical models for transcriptionally active chromatin view this as a polymer with sticky sites, mimicking transcription units such as promoters and enhancers which interact via the binding of multivalent complexes of chromatin-binding proteins. It has been demonstrated that this model spontaneously leads to microphase separation, resulting in the formation of a network of loops with transcription units serving as anchors. Here, we demonstrate how to compute the topological weights of loop networks with an arbitrary 1D pattern of transcription units along the fibre (or `polydisperse' loop networks), finding an analogy with networks of electric resistors in parallel or in series. We also show how the BEST (de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte) theorem in combinatorics can be used to find the combinatorial multiplicity of any class of loop networks. Our results can be used to compute the structural diversity, or Shannon entropy, of loop networks: we show that this quantity depends on the 1D patterning of transcription units along the chain, possibly providing a pathway to control transcriptional noise in eukaryotic genes.

physics.bio-ph↗