Combinatorial Sequences Research Articles

Using the “Freshman's Dream” to Prove Combinatorial Congruences

Recently, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition of their approach and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums. In fact, we even combine these two generalizations. We conclude with some super-challenges.

Two triple binomial sum supercongruences

In a recent article, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences. At the end, they propose some supercongruences as conjectures. Here we prove one of them, including a new companion enumerating Abelian squares, and we leave some remarks for the others.

Factorial transformation for some classical combinatorial sequences

Factorial transformation known from Euler’s time is a very powerful tool for summation of divergent power series. We use factorial series for summation of ordinary power generating functions for some classical combinatorial sequences. These sequences increase very rapidly, so OGFs for them diverge and mostly unknown in a closed form. We demonstrate that factorial series for them are summable and expressed in known functions. We consider among others Stirling, Bernoulli, Bell, Euler and Tangent numbers. We compare factorial transformation with other summation techniques such as Pade approximations, transformation to continued fractions, and Borel integral summation. This allowed us to derive some new identities for GFs and express their integral representations in a closed form.

The Monotonicity and Log-Behaviour of Some Functions Related to the Euler Gamma Function

AbstractThe aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functionsis considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequenceis proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functionsis also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

Log-convex and Stieltjes moment sequences

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.

Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences

Many combinatorial sequences (e.g. the Catalan and the Motzkin numbers) may be expressed as the constant term of , for some Laurent polynomials P(x) and Q(x) in the variable x with integer coefficients. Denoting such a sequence by , we obtain a general formula that determines the congruence class, modulo p, of the indefinite sum , for any prime p, and any positive integer r, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our computers) to automatically discover and prove congruence theorems for such partial sums. Moreover, we show that in many cases, the set of the residues is finite, regardless of the prime p.

Combinatorial Proofs of Addition Formulas

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schröder paths.

Very large scale droplet microfluidic integration (VLDMI) using genetic algorithm

Droplet microfluidics is likely to play a central role in the development of lab-on-a-chip technologies and as a result, significant research is directed toward this field. Understanding the spatiotemporal dynamics of discrete droplets inside microfluidic devices and the design of microfluidic devices for specific tasks are some of the dominant research topics. These works have since resulted in the development of microfluidic devices with functionalities, such as sorting, merging, synchronization, storing etc. However, the anticipated application of microfluidic devices to more complex problems will require more integrated devices that can incorporate the above functionalities on a single chip. In the current work, we present a genetic algorithm optimization-based design tool for discovering very large-scale integration of discrete microfluidic networks for a given objective function. The application of the algorithm is demonstrated through a combinatorial sequencing problem, where the objective is to achieve three different droplet combinatorial sequences for three different droplet types. Multiple fascinating, but nonobvious designs were discovered for this application. It is difficult to imagine such devices being designed using trial and error experimental procedure, which has been the main route for obtaining microfluidic device designs. With advances in technologies for fabrication of microfluidic devices, the current tool can be a significant step toward drastically cutting down on the laborious trial-and-error design process and help in developing droplet microfluidics-based lab-on-a-chip platforms cheaper and faster.

Maps Preserving Moment Sequences

For any sequence s of real numbers, we consider the class \(\mathcal {L}\) of maps (from \(\mathbb {R}^{\mathbb {N}_0}\) to \(\mathbb {R}^{\mathbb {N}_0}\)) that linearly combine a finite or infinite number of elements of s to obtain the new values of the transformed sequence. We characterize those maps in \(\mathcal {L}\) that transform moment sequences into moment sequences in terms of the existence of a stochastic process fulfilling appropriate requirements. Then, well-known stochastic processes are used to construct significant examples of such preserving mappings. As application, we also show that some celebrated numerical sequences (including several important combinatorial sequences) are actually transformed moment sequences.

On a combinatorial problem in botanical epidemiology

In botanical epidemiology, one of the major problems is to study the spread of diseases within crops. Several approaches to study the patterns and temporal evolution of diseases have been discussed in the literature, e.g., statistical techniques or variogram analysis. Recently, AlSharawi, Burstein, Deadman and Umar determined the total number of ways to have ℓ isolated infected individuals among m infected plants in a row of n plants. They discussed this in a straightforward combinatorial fashion and considered several associated points, like expectation value and variance of the number of isolated infected plants. In the present paper, we derive their results with the help of generating function techniques and use this method to extend the discussion to plants arranged in a circle as well as in two rows. The dependence of the expected number of isolated infected plants on the proportion of infected plants is considered for large n in all these cases as well as in the case of an arbitrary number of rows and a simple asymptotic behavior is found. Connections to several combinatorial sequences are established.

Zeta Functions and the Log Behaviour of Combinatorial Sequences

AbstractIn this paper, we use the Riemann zeta functionζ(x) and the Bessel zeta functionζμ(x) to study the log behaviour of combinatorial sequences. We prove thatζ(x) is log-convex forx> 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n≥ 1 is log-convex, whereBnis thenth Bernoulli number. We introduce the functionθ(x) = (2ζ(x)Γ(x + 1))1/x, whereΓ(x)is the gamma function, and we show that logθ(x) is strictly increasing forx≥ 6. This confirms a conjecture of Sun stating that the sequenceis strictly increasing. Amdeberhanet al. defined the numbersan(μ)= 22n+1(n+ 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence{an(μ)}n≥1is log-convex forμ= 0 andμ= 1. By proving thatζμ(x)is log-convex forx >1 andμ >-1, we show that the sequence{an(≥)}n>1 is log-convex for anyμ >- 1. We introduce another functionθμ,(x)involvingζμ(x)and the gamma functionΓ(x)and we show that logθμ(x)is strictly increasing forx >8e(μ+ 2)2. This implies thatBased on Dobinski’s formula, we prove thatwhereBnis thenth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property ofand Holder’s inequality in probability theory.

A new graph invariant arises in toric topology

In this paper, we introduce new combinatorial invariants of any finite simple graph, which arise in toric topology. We compute the $i$-th (rational) Betti number of the real toric variety associated to a graph associahedron $P_{\B(G)}$. It can be calculated by a purely combinatorial method (in terms of graphs) and is denoted by $a_i(G)$. To our surprise, for specific families of the graph $G$, our invariants are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers.

Context-free grammars for triangular arrays

We consider context-free grammars of the form \(G = \{ f \to f^{b_1 + b_2 + 1} g^{a_1 + a_2 } ,g \to f^{b_1 } g^{a_1 + 1} \} \), where ai and bi are integers subject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T(n, k)}0≤k≤n satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let Tn(x) = Σk=0nT(n, k)xk. Based on the differential operator with respect to G, we define a sequence of linear operators Pn such that Tn+1(x) = Pn(Tn(x)). Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Branden, we obtain a necessary and sufficient condition for the operator Pn to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh. Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers.

Automatic congruences for diagonals of rational functions

In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of Denef and Lipshitz, for computing a finite automaton for the sequence modulo $p^\alpha$, for all but finitely many primes $p$. This method gives completely automatic proofs of known results, establishes a number of new theorems for well-known sequences, and allows us to resolve some conjectures regarding the Ap\'ery numbers. We also give a second method, which applies to an algebraic sequence modulo $p^\alpha$ for all primes $p$, but is significantly slower. Finally, we show that a broad range of multidimensional sequences possess Lucas products modulo $p$.

Congruences for Taylor expansions of quantum modular forms

Abstract Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have proven refined results, for example, extending all of these congruences to arbitrary powers of the primes involved. Here, we take a different perspective and explain the general theory of such congruences in the context of an important class of quantum modular forms. As one example, we obtain an infinite series of combinatorial sequences connected to the ‘half-derivatives’ of the Andrews-Gordon functions and with Kashaev’s invariant on (2m+1,2) torus knots, and we prove conditions under which the sequences satisfy linear congruences modulo at least 50% of primes.

Grammatical analysis as a distributed neurobiological function.

Language processing engages large‐scale functional networks in both hemispheres. Although it is widely accepted that left perisylvian regions have a key role in supporting complex grammatical computations, patient data suggest that some aspects of grammatical processing could be supported bilaterally. We investigated the distribution and the nature of grammatical computations across language processing networks by comparing two types of combinatorial grammatical sequences—inflectionally complex words and minimal phrases—and contrasting them with grammatically simple words. Novel multivariate analyses revealed that they engage a coalition of separable subsystems: inflected forms triggered left‐lateralized activation, dissociable into dorsal processes supporting morphophonological parsing and ventral, lexically driven morphosyntactic processes. In contrast, simple phrases activated a consistently bilateral pattern of temporal regions, overlapping with inflectional activations in L middle temporal gyrus. These data confirm the role of the left‐lateralized frontotemporal network in supporting complex grammatical computations. Critically, they also point to the capacity of bilateral temporal regions to support simple, linear grammatical computations. This is consistent with a dual neurobiological framework where phylogenetically older bihemispheric systems form part of the network that supports language function in the modern human, and where significant capacities for language comprehension remain intact even following severe left hemisphere damage. Hum Brain Mapp 36:1190–1201, 2015. © 2014 The Authors Human Brain Mapping Published by Wiley Periodicals, Inc.

Log-convexity of Aigner–Catalan–Riordan numbers

Let T=[tn,k]n,k≥0 be an infinite lower triangular matrix defined byt0,0=1,tn+1,0=∑j=0nzjtn,j,tn+1,k+1=∑j=knaj,ktn,j for n,k≥0, where all zj,aj,k are nonnegative and aj,k=0 unless j≥k≥0. We show that the sequence (tn,0)n≥0 is log-convex if the coefficient matrix [ζ,A] is TP2, where ζ=[z0,z1,z2,…]′ and A=[ai,j]i,j≥0. This gives a unified proof of the log-convexity of many well-known combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the Schröder numbers, the Bell numbers, and so on.

On monotonicity of some combinatorial sequences

We conrm Sun's conjecture that ( n+1 p Fn+1= n p Fn)n>4 is strictly decreasing to the limit 1, where (Fn)n>0 is the Fibonacci se- quence. We also prove that the sequence ( n+1 p Dn+1= n p Dn)n>3 is strict- ly decreasing with limit 1, where Dn is the n-th derangement number. For m-th order harmonic numbers H (m) n = P n=1 1=k m (n = 1; 2; 3;:::), we show that ( n+1 q H (m) n+1 = n q H (m) n )n>3 is strictly increasing.

SpeedyGenes: an improved gene synthesis method for the efficient production of error-corrected, synthetic protein libraries for directed evolution.

The de novo synthesis of genes is becoming increasingly common in synthetic biology studies. However, the inherent error rate (introduced by errors incurred during oligonucleotide synthesis) limits its use in synthesising protein libraries to only short genes. Here we introduce SpeedyGenes, a PCR-based method for the synthesis of diverse protein libraries that includes an error-correction procedure, enabling the efficient synthesis of large genes for use directly in functional screening. First, we demonstrate an accurate gene synthesis method by synthesising and directly screening (without pre-selection) a 747 bp gene for green fluorescent protein (yielding 85% fluorescent colonies) and a larger 1518 bp gene (a monoamine oxidase, producing 76% colonies with full catalytic activity, a 4-fold improvement over previous methods). Secondly, we show that SpeedyGenes can accommodate multiple and combinatorial variant sequences while maintaining efficient enzymatic error correction, which is particularly crucial for larger genes. In its first application for directed evolution, we demonstrate the use of SpeedyGenes in the synthesis and screening of large libraries of MAO-N variants. Using this method, libraries are synthesised, transformed and screened within 3 days. Importantly, as each mutation we introduce is controlled by the oligonucleotide sequence, SpeedyGenes enables the synthesis of large, diverse, yet controlled variant sequences for the purposes of directed evolution.

Mixed volumes, log-concave sequences and B-splines

In this paper, a series of problems emerged in discrete geometry and combinatorics related to spline functions are systematically studied. For example, in discrete geometry, the spline representations of cube slicing and mixed volumes of polytopes are considered. With the geometric interpretations of B-splines, the volume of cube slicing can be considered as an equivalent problems in spline theory. Based on the connection, a simple proof for Laplace and P olya's formulas in cub slicing is given by spline theory. A class of mixed volumes are given by the relations between splines and the mixed volumes. A class of log-concave sequence are derived by the connection. Therefore, the open problem proposed by Schmidt and Simion is partially solved. In combinatorics, the combinatorial polynomials and log-concavity for some combinatorial sequences are investigated by spline theory. With the well developed spline theory, not only the existing results in discrete geometry and combinatorics have been veri ed, but also a novel method for solving related discrete mathematics problems has been studied. Splines as functions of a continuous nature provide an analysis method in combinatorial enumerations which are usually considered as counting discrete objects. This method provides a novel analysis method for studying discrete objects.