Combinatorial Sequences Research Articles

The Log-Balancedness of Combinatorial Sequences

In this paper, we discuss the log-balancedness of combinatorial sequences. We consider operators on sequences that preserve the log-balancedness property. We also give a sufficient condition for the log-balancedness of the product of two sequences. As applications, we prove that some combinatorial sequences are log-balanced. In addition, we discuss the reverse ultra log-concavity of some sequences involving the log-balanced sequence.

2 × 2 minors of recursive matrices related to generalized Dyck paths

A Dyck path of semilength n is a path from (0,0) to (2n,0), such that the only allowed steps are U=(1,1) and D=(1,−1). The constraint on a Dyck path is that it never falls below the x-axis. In this paper, we give the concept of (a1,b1;a2,b2)-Dyck paths, which generates the recursive matrix corresponding to π=(a1,a2,a1,a2,⋯), σ=(0,0,0,0,⋯) and τ=(b1,b2,b1,b2,⋯). Some new combinatorial identities related to 2×2 minors and 2×2 permanents of the Dπ,σ,τ are investigated. When the weighted parameters (a1,b1;a2,b2) are specialized, several interesting identities are obtained about Catalan numbers and other combinatorial sequences. Based on Motzkin paths, alternating sums of 2×2 minors of the odd Dyck matrix are studied.

Efficient DNA-based data storage using shortmer combinatorial encoding

Data storage in DNA has recently emerged as a promising archival solution, offering space-efficient and long-lasting digital storage solutions. Recent studies suggest leveraging the inherent redundancy of synthesis and sequencing technologies by using composite DNA alphabets. A major challenge of this approach involves the noisy inference process, obstructing large composite alphabets. This paper introduces a novel approach for DNA-based data storage, offering, in some implementations, a 6.5-fold increase in logical density over standard DNA-based storage systems, with near-zero reconstruction error. Combinatorial DNA encoding uses a set of clearly distinguishable DNA shortmers to construct large combinatorial alphabets, where each letter consists of a subset of shortmers. We formally define various combinatorial encoding schemes and investigate their theoretical properties. These include information density and reconstruction probabilities, as well as required synthesis and sequencing multiplicities. We then propose an end-to-end design for a combinatorial DNA-based data storage system, including encoding schemes, two-dimensional (2D) error correction codes, and reconstruction algorithms, under different error regimes. We performed simulations and show, for example, that the use of 2D Reed-Solomon error correction has significantly improved reconstruction rates. We validated our approach by constructing two combinatorial sequences using Gibson assembly, imitating a 4-cycle combinatorial synthesis process. We confirmed the successful reconstruction, and established the robustness of our approach for different error types. Subsampling experiments supported the important role of sampling rate and its effect on the overall performance. Our work demonstrates the potential of combinatorial shortmer encoding for DNA-based data storage and describes some theoretical research questions and technical challenges. Combining combinatorial principles with error-correcting strategies, and investing in the development of DNA synthesis technologies that efficiently support combinatorial synthesis, can pave the way to efficient, error-resilient DNA-based storage solutions.

Identities and periodic oscillations of divide-and-conquer recurrences splitting at half

We study divide-and-conquer recurrences of the formf(n)=αf(⌊n2⌋)+βf(⌈n2⌉)+g(n)(n⩾2), with g(n) and f(1) given, where α,β⩾0 with α+β>0; such recurrences appear often in the analysis of computer algorithms, numeration systems, combinatorial sequences, and related areas. We show under an optimum (iff) condition on g(n) that the solution f always satisfies a simple identityf(n)=nlog2⁡(α+β)P(log2⁡n)−Q(n), where P is a periodic function and Q(n) is of a smaller order than the dominant term. This form is thus not only an identity but also an asymptotic expansion. Explicit forms for the continuity of the periodic function P are provided, together with a few other smoothness properties. We show how our results can be easily applied to many dozens of concrete examples collected from the literature, and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary, but leads to the strongest types of results for all examples to which our theory applies.

[formula omitted]-linear schemes for sequences modulo [formula omitted]

Many interesting combinatorial sequences, such as Apéry numbers and Franel numbers, enjoy the so-called Lucas property modulo almost all primes p. Modulo prime powers pr such sequences have a more complicated behaviour which can be described by matrix versions of the Lucas property called p-linear schemes. They are generalizations of finite p-automata. In this paper we construct such p-linear schemes and give upper bounds for the number of states which, for fixed r, do not depend on p.

Identification of two novel papillomaviruses in belugas.

Papillomaviruses (PVs) can cause hyperplasia in the skin and mucous membranes of humans, mammals, and non-mammalian animals, and are a significant risk factor for cervical and genital cancers. Using next-generation sequencing (NGS), we identified two novel strains of papillomavirus, PV-HMU-1 and PV-HMU-2, in swabs taken from belugas (Delphinapterus leucas) at Polar Ocean Parks in Qingdao and Dalian. We amplified the complete genomes of both strains and screened ten belugas and one false killer whale (Pseudorca crassidens) for the late gene (L1) to determine the infection rate. In Qingdao, 50% of the two sampled belugas were infected with PV-HMU-1, while the false killer whale was negative. In Dalian, 71% of the eight sampled belugas were infected with PV-HMU-2. In their L1 genes, PV-HMU-1 and PV-HMU-2 showed 64.99 and 68.12% amino acid identity, respectively, with other members of Papillomaviridae. Phylogenetic analysis of combinatorial amino acid sequences revealed that PV-HMU-1 and PV-HMU-2 clustered with other known dolphin PVs but formed distinct branches. PVs carried by belugas were proposed as novel species under Firstpapillomavirinae. The discovery of these two novel PVs enhances our understanding of the genetic diversity of papillomaviruses and their impact on the beluga population.

Totally non-negativity of a family of change-of-basis matrices

Let a=(a1,a2,…,an) and e=(e1,e2,…,en) be real sequences. Denote by Me→a the (n+1)×(n+1) matrix whose (m,k) entry (m,k∈{0,…,n}) is the coefficient of the polynomial (x−a1)⋯(x−ak) in the expansion of (x−e1)⋯(x−em) as a linear combination of the polynomials 1,x−a1,…,(x−a1)⋯(x−am). By appropriate choice of a and e the matrix Me→a can encode many familiar doubly-indexed combinatorial sequences, such as binomial coefficients, Stirling numbers of both kinds, Lah numbers and central factorial numbers.In all four of these examples, Me→a enjoys the property of total non-negativity — the determinants of all its square submatrices are non-negative. This leads to a natural question: when, in general, is Me→a totally non-negative?Galvin and Pacurar found a simple condition on e that characterizes total non-negativity of Me→a when a is non-decreasing. Here we fully extend this result. For arbitrary real sequences a and e, we give a condition that can be checked in O(n2) time that determines whether Me→a is totally non-negative. When Me→a is totally non-negative, we witness this with a planar network whose weights are non-negative and whose path matrix is Me→a. When it is not, we witness this with an explicit negative minor.

Degenerate Harmonic Numbers and Polynomials with Combinatorial Sequences

Degenerate Harmonic Numbers and Polynomials with Combinatorial Sequences

Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution

We investigate the combinatorial sequences $A(M, n)$ introduced by W. G. Brown (1964) and W. T. Tutte (1980) appearing in enumeration of convex polyhedra. Their formula is $$A(M, n) = \frac{2 (2M+3)!}{(M+2)! M!}\,\frac{(4n+2M+1)!}{n! (3n + 2M + 3)!} $$ with $n, M =0, 1, 2, \ldots$, and we conceive it as Hausdorff moments, where $M$ is a parameter and $n$ enumerates the moments. We solve exactly the corresponding Hausdorff moment problem: $A(M, n) = \int_{0}^{R} x^{n} W_{M}(x) d x$ on the natural support $(0, R)$, $R = 4^{4}/3^{3}$, using the method of inverse Mellin transform. We provide explicitly the weight functions $W_{M}(x)$ in terms of the Meijer G-functions $G_{4, 4}^{4, 0}$, or equivalently, the generalized hypergeometric functions ${_{3}F_{2}}$ (for $M=0, 1$) and ${_{4}F_{3}}$ (for $M \geq 2$). For $M = 0, 1$, we prove that $W_{M}(x)$ are non-negative and normalizable, thus they are probability distributions. For $M \geq 2$, $W_{M}(x)$ are signed functions vanishing on the extremities of the support. By rephrasing this problem entirely in terms of Meijer G representations we reveal an integral relation which directly furnishes $W_M(x)$ based on ordinary generating function of $A(M, n)$ as an input. All the results are studied analytically as well as graphically.

A benchmark for evaluating Arabic word embedding models

AbstractModelling the distributional semantics of such a morphologically rich language as Arabic needs to take into account its introflexive, fusional, and inflectional nature attributes that make up its combinatorial sequences and substitutional paradigms. To evaluate such word distributional models, the benchmarks that have been used thus far in Arabic have mimicked those in English. This paper reports on a benchmark that we designed to reflect linguistic patterns in both Contemporary Arabic and Classical Arabic, the first being a cover term for written and spoken Modern Standard Arabic, while the second for pre-modern Arabic. The analogy items we included in this benchmark are chosen in a transparent manner such that they would capture the major features of nouns and verbs; derivational and inflectional morphology; high-, middle-, and low-frequency patterns and lexical items; and morphosemantic, morphosyntactic, and semantic dimensions of the language. All categories included in this benchmark are carefully selected to ensure proper representation of the language. The benchmark consists of 45 roots of the trilateral, all-consonantal, and semivowel-inclusive types; six morphosemantic patterns (’af‘ala; ifta‘ala; infa‘ala; istaf‘ala; tafa‘‘ala; and tafā‘ala); five derivations (the verbal noun, active participle, and the contrasts in Masculine-Feminine; Feminine-Singular-Plural; Masculine-Singular-Plural); and morphosyntactic transformations (perfect and imperfect verbs conjugated for all pronouns); and lexical semantics (synonyms, antonyms, and hyponyms of nouns, verbs, and adjectives), as well as capital cities and currencies. All categories include an equal proportion of high-, medium-, and low-frequency items. For the purpose of validating the proposed benchmark, we developed a set of embedding models from different textual sources. Then, we tested them intrinsically using the proposed benchmark and extrinsically using two natural language processing tasks: Arabic Named Entity Recognition and Text Classification. The evaluation leads to the conclusion that the proposed benchmark is truly reflective of this morphologically rich language and discriminatory of word embeddings.

SoftMatch: Comparing Scanpaths Using Combinatorial Spatio-Temporal Sequences with Fractal Curves.

Recent studies matching eye gaze patterns with those of others contain research that is heavily reliant on string editing methods borrowed from early work in bioinformatics. Previous studies have shown string editing methods to be susceptible to false negative results when matching mutated genes or unordered regions of interest in scanpaths. Even as new methods have emerged for matching amino acids using novel combinatorial techniques, scanpath matching is still limited by a traditional collinear approach. This approach reduces the ability to discriminate between free viewing scanpaths of two people looking at the same stimulus due to the heavy weight placed on linearity. To overcome this limitation, we here introduce a new method called SoftMatch to compare pairs of scanpaths. SoftMatch diverges from traditional scanpath matching in two different ways: firstly, by preserving locality using fractal curves to reduce dimensionality from 2D Cartesian (x,y) coordinates into 1D (h) Hilbert distances, and secondly by taking a combinatorial approach to fixation matching using discrete Fréchet distance measurements between segments of scanpath fixation sequences. These matching “sequences of fixations over time” are a loose acronym for SoftMatch. Results indicate high degrees of statistical and substantive significance when scoring matches between scanpaths made during free-form viewing of unfamiliar stimuli. Applications of this method can be used to better understand bottom up perceptual processes extending to scanpath outlier detection, expertise analysis, pathological screening, and salience prediction.

On the Tutte and Matching Polynomials for Complete Graphs

Let T(G; X, Y) be the Tutte polynomial for graphs. We study the sequence ta,b(n) = T(Kn; a, b) where a, b are integers, and show that for every μ ∈ ℕ the sequence ta,b(n) is ultimately periodic modulo μ provided a ≠ 1 mod μ and b ≠ 1 mod μ. This result is related to a conjecture by A. Mani and R. Stones from 2016. The theorem is a consequence of a more general theorem which holds for a wide class of graph polynomials definable in Monadic Second Order Logic. This gives also similar results for the various substitution instances of the bivariate matching polynomial and the trivariate edge elimination polynomial ξ(G; X, Y, Z) introduced by I. Averbouch, B. Godlin and the second author in 2008. All our results depend on the Specker-Blatter Theorem from 1981, which studies modular recurrence relations of combinatorial sequences which count the number of labeled graphs.

Congruence Properties of Combinatorial Sequences via Walnut and the Rowland-Yassawi-Zeilberger Automaton

Certain famous combinatorial sequences, such as the Catalan numbers and the Motzkin numbers, when taken modulo a prime power, can be computed by finite automata. Many theorems about such sequences can therefore be proved using Walnut, which is an implementation of a decision procedure for proving various properties of automatic sequences. In this paper we explore some results (old and new) that can be proved using this method.

On some combinatorial sequences associated to invariant theory

We study the enumerative and analytic properties of some sequences constructed using tensor invariant theory. The octant sequences are constructed from the exceptional Lie group G2 and the quadrant sequences from the special linear group SL(3). In each case we show that the corresponding sequences are related by binomial transforms. The first three octant sequences and the first four quadrant sequences are listed in the On-Line Encyclopedia of Integer Sequences (OEIS). These sequences all have interpretations as enumerating two-dimensional lattice walks but for the octant sequences the boundary conditions are unconventional. These sequences are all P-recursive and we give the corresponding recurrence relations. In all cases the associated differential operators are of third order and have the remarkable property that they can be solved to give closed formulae for the ordinary generating functions in terms of classical Gaussian hypergeometric functions. Moreover, we show that the octant sequences and the quadrant sequences are related by the branching rules for the inclusion of SL(3) in G2.

Combinatorics of NC-probability spaces with independent constants

The boolean and monotone notions of independence lack the property of independent constants. We address this problem from a combinatorial point of view (based on cumulants defined from weights on set-partitions, in the general framework of operator-valued probability spaces). We show that if the weights are singleton inductive (SI), then all higher-order cumulants involving constants vanish, just as in the free and classical case. Our combinatorial considerations lead rather directly to mild variations of boolean and monotone probability theories which are closely related to the usual notions. The SI-boolean case is related to c-free and Fermi convolutions. We also describe some standard combinatorial aspects of the SI-boolean and cyclic-boolean lattices, such as their Möbius functions, featuring well-known combinatorial integer sequences.

On the sequences of polynomials and their generating functions

In this article, we will give first of all, an identity having interesting applications on polynomials and some combinatorial sequences. Secondly, we will refer two interesting formulas on generating functions of polynomials. Our results are illustrated in fact, by some comprehensive examples.

Counting of Distinct Equivalence Classes of Circuits in PSL2, Z‐Space

Graham Higman was the first who studied the transitive actions of the extended modular group over PL(Fq) = Fq∪{∞} graphically and named it as coset diagram. In these sorts of graphs, a closed path of edges and triangles is known as a circuit. Coset diagrams evolve through the joining of these circuits. In a coset diagram, a circuit is termed as a length‐l circuit if its one vertex is fixed by , and it is denoted by (π1, π2, π3, …, πl). In this study, we shall formulate combinatorial sequences and find the number of distinct equivalence classes of a length‐6 circuit (π1, π2, π3, π4, π5, π6) for a fixed number of triangle Δ of class Π.

A Two-Step TLS-Based Browser fingerprinting approach using combinatorial sequences

We propose a two-step TLS-based fingerprinting approach using combinatorial sequences and properties of TLS handshake messages. Our approach combines fingerprinting based on attributes of the initial ClientHello message with the observed behavior of TLS clients when presented with permuted handshake messages in order to enhance the granularity of the derived fingerprints without increasing the required number of exchanged messages. We conduct a detailed evaluation against 21 browsers and TLS clients on two operating systems. The results show a significant increase in the entropy of the achieved splittings, allowing for a more precise identification of the TLS client than permitted by either of the underlying approaches in isolation.

Hankel determinants of linear combinations of moments of orthogonal polynomials, II

We present a formula that expresses the Hankel determinants of a linear combination of length d+1 of moments of orthogonal polynomials in terms of a dtimes d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

Stieltjes moment sequences for pattern-avoiding permutations

A small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences, $Av(123\ldots k)$, are Stieltjes moment sequences, and we explicitly find the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool. We show that the generating functions of the sequences $\, Av(1234)$ and $\, Av(12345)$ correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian $\, _2F_1$ hypergeometric function, respectively to an order-two linear differential operator acting on the square of a classical modular form given as a pullback of a $\, _2F_1$ hypergeometric function. We demonstrate that the density function for the Stieltjes moment sequence $Av(123\ldots k)$ is closely, but non-trivially, related to the density attached to the distance traveled by a walk in the plane with $k-1$ unit steps in random directions. Finally, we study the challenging case of the $Av(1324)$ sequence and give compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, we show how rigorous lower bounds on the growth constant of this sequence can be constructed, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give an estimate of the (unknown) growth constant.