Combinatorial Counting Research Articles

The Second Omega Index

Background: The omega index has been recently introduced to identify a variety of topological and combinatorial aspects of a graph with a new viewpoint. As a continuing study of the omega index, by considering the incidence of edges and vertices to the adjacency of the vertices, in this paper, we have introduced the second omega index Ω2 and then computed it over some well-known graph classes. Methods: Many combinatorial counting methods have been utilized in the proofs. The edge partition is frequently used throughout the work. Naturally, some graph theoretical lemmas are also used. Results: In particular, trees having the smallest and largest Ω2 have been constructed. Finally, the second omega index of some derived graphs, such as line graphs, subdivision graphs, and vertex-semitotal graphs, has been presented. Conclusion: Omega invariant has already been explored in many papers. It has been defined in terms of vertex degrees. Vertices correspond to the atoms in a molecule and a calculation, which only depends on the atomic parameters, is not even comparable with a calculation containing both atoms and chemical bonds between them. With this idea in mind, we have evaluated some mathematical properties of the second omega index, which has great potential in chemical applications related to the number of cycles in the molecular graph.

An impossible combinatorial counting method in distance geometry

The Distance Geometry Problem asks for a geometric representation of a given weighted graph in RK so that vertices are points and edges are segments with lengths equal to the corresponding weights. Two problem variants are defined by a vertex order given as part of the input, which allows the use of a branching algorithm based on K-lateration: find two possible positions for the next vertex j using the positions of K predecessors and their distances to j, then explore each position recursively, pruning positions at need. Whereas the first variant only requires the K predecessors to exist, the second variant also requires them to be contiguous and immediately preceding j. For this variant, fixed-parameter tractability of the algorithm can be established by means of a solution counting method that only depends on the graph edges (rather than their weights). Only in the first variant, however, it is possible to efficiently construct a suitable vertex order directly from the graph. Since both fixed-parameter tractability and efficient vertex order construction are desirable properties, one would need an analogous solution counting method for the first variant. In this paper we prove that such a counting method cannot be devised for the first variant.

The endpoint of partial deconfinement

We study the matrix quantum mechanics of two free hermitian N × N matrices subject to a singlet constraint in the microcanonical ensemble. This is the simplest example of a theory that at large N has a confinement/deconfinement transition. In the microcanonical ensemble, it also exhibits partial deconfinement with a Hagedorn density of states. We argue that the entropy of these configurations, based on a combinatorial counting of Young diagrams, are dominated by Young diagrams that have the VKLS shape. When the shape gets to the maximal depth allowed for a Young diagram of SU(N), namely N, we argue that the system stops exhibiting the Hagedorn behavior. The number of boxes (energy) at the transition is N2/4, independent of the charge of the state.

Improvement and comparison the performance of fuzzing testing algorithms for applications in Google Thread Sanitizer

It is difficult to imagine modern information systems without the use of multithreading. The use of multithreading can both improve the performance of the system as in whole so as slow down the execution of multithreaded applications due to the occurrence of multithreaded programming errors. To find such errors in C/C++ languages, there exists a Google Thread Sanitizer compiler module. The order of execution of threads can change every time the program is started for execution and can affect the appearance of such errors. To repeatedly change the order of execution of threads during the execution of the program, Google Thread Sanitizer has a fuzzing testing module that allows you to increase the probability of finding errors. But all the thread scheduling algorithms in this module are presented in the form of sequential execution of threads which can lead to a significant slowdown in Google Thread Sanitizer as well as can affect the testing of applications that depends on timers (waiting for network events, deadline for operations, ...). To speed up the work of fuzzing schedulers, a method for parallelizing independent transitions is proposed. From the point of view of multithreaded programming errors, it is only important to change the shared state between threads, and local calculations do not affect the reproduction of multithreaded errors. The changes of shared states themselves occur at synchronization points (places in the code where threads are switched according to the principle of cooperative multitasking). The method suggests ordering only the change of shared states at synchronization points, and performing local calculations in parallel, due to which parallelization is achieved. For the analysis of theoretical complexity of the algorithm, the method of combinatorial counting is used. A new approach to the organization of fuzzing testing based on the method of parallelization of independent transitions is proposed the implementation of which, according to theoretical and practical estimates, shows a noticeable acceleration of the work of fuzzing schedulers. According to the results of the experiment, it was revealed that for the algorithm of iterating through all execution variants, the acceleration of execution reaches 1.25 times for two threads. For an arbitrary number of threads, an estimate is presented in the form of a formula. The proposed approach allows fuzzing tests to cover multithreaded applications for which execution time is important — applications with reference to timers which improve the quality of the software.

Coprime Ehrhart Theory and Counting Free Segments

Abstract A lattice polytope is free (or empty) if its vertices are the only lattice points it contains. In the context of valuation theory, Klain [16] proposed to study the functions $\alpha _i(P;n)$ that count the number of free polytopes in $nP$ with $i$ vertices. For $i=1$, this is the famous Ehrhart polynomial; for $i> 3$, the computation is likely impossible; and for $i=2,3$ it is computationally challenging. In this paper, we develop a theory of coprime Ehrhart functions that count lattice points with relatively prime coordinates and use it to compute $\alpha _2(P;n)$ for unimodular simplices. We show that the coprime Ehrhart function can be explicitly determined from the Ehrhart polynomial and we give some applications to combinatorial counting.

On properties of translation groups in the affine general linear group with applications to cryptography

The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well.

Contracting projected entangled pair states is average-case hard

An accurate calculation of the properties of quantum many-body systems is one of the most important yet intricate challenges of modern physics and computer science. In recent years, the tensor network ansatz has established itself as one of the most promising approaches enabling striking efficiency of simulating static properties of one-dimensional systems and abounding numerical applications in condensed matter theory. In higher dimensions, however, a connection to the field of computational complexity theory has shown that the accurate normalization of the two-dimensional tensor networks called projected entangled pair states (PEPS) is #P-complete. Therefore, an efficient algorithm for PEPS contraction would allow to solve exceedingly difficult combinatorial counting problems, which is considered highly unlikely. Due to the importance of understanding two- and three-dimensional systems the question currently remains: Are the known constructions typical of states relevant for quantum many-body systems? In this work, we show that an accurate evaluation of normalization or expectation values of PEPS is as hard to compute for typical instances as for special configurations of highest computational hardness. We discuss the structural property of average-case hardness in relation to the current research on efficient algorithms attempting tensor network contraction, hinting at a wealth of possible further insights into the average-case hardness of important problems in quantum many-body theory.

Quasimodularity and large genus limits of Siegel-Veech constants

Quasimodular forms were first studied systematically in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms. We apply this to prove conjectures of Eskin and Zorich on the large genus limits of Masur-Veech volumes and of Siegel-Veech constants. In Part I we connect the geometric definition of Siegel-Veech constants both with a combinatorial counting problem and with intersection numbers on Hurwitz spaces. We also introduce certain modified Siegel-Veech weights whose generating functions will later be shown to be quasimodular. Parts II and III are devoted to the study of the (quasi) modular properties of the generating functions arising from weighted counting of torus coverings. These two parts contain little geometry and can be read independently of the rest of the paper. The starting point is the theorem of Bloch and Okounkov saying that certain weighted averages, called q q -brackets, of shifted symmetric functions on partitions are quasimodular forms. In Part II we give an expression for the growth polynomials (a certain polynomial invariant of quasimodular forms) of these q q -brackets in terms of Gaussian integrals, and we use this to obtain a closed formula for the generating series of cumulants that is the basis for studying large genus asymptotics. In Part III we show that the even hook-length moments of partitions are shifted symmetric polynomials, and we prove a surprising formula for the q q -bracket of the product of such a hook-length moment with an arbitrary shifted symmetric polynomial as a linear combination of derivatives of Eisenstein series. This formula gives a quasimodularity statement also for the ( − 2 ) (-2) -nd hook-length moments by an appropriate extrapolation, and this in turn implies the quasimodularity of the Siegel-Veech weighted counting functions. Finally, in Part IV these results are used to give explicit generating functions for the volumes and Siegel-Veech constants in the case of the principal stratum of abelian differentials. The generating functions have an amusing form in terms of the inversion of a power series (with multiples of Bernoulli numbers as coefficients) that gives the asymptotic expansion of a Hurwitz zeta function. To apply these exact formulas to the Eskin-Zorich conjectures on large genus asymptotics (both for the volume and the Siegel-Veech constant) we provide in a separate appendix a general framework for computing the asymptotics of rapidly divergent power series.

Algebraic Independence of Sequences Generated by (Cyclotomic) Harmonic Sums

Indefinite nested sums are important building blocks to assemble closed forms for combinatorial counting problems or for problems that arise, e.g., in particle physics. Concerning the simplicity of such formulas an important subtask is to decide if the arising sums satisfy algebraic relations among each other. Interesting enough, algebraic relations of such formal sums can be derived from combinatorial quasi-shuffle algebras. We will focus on the following question: can one find more relations if one evaluates these sums to sequences and looks for relations within the ring of sequences. In this article we consider the sequences of the rather general class of (cyclotomic) harmonic sums and show that their relations coincide with the relations found by their underlying quasi-shuffle algebra. In order to derive this result, we utilize the quasi-shuffle algebra and construct a difference ring with the following property: (1) the generators of the difference ring represent (cyclotomic) harmonic sums, (2) they generate within the ring all (cyclotomic) harmonic sums, and (3) the sequences produced by the generators are algebraically independent among each other. This means that their sequences do not satisfy any polynomial relations. The proof of this latter property is obtained by difference ring theory and new symbolic summation results. As a consequence, any sequence produced by (cyclotomic) harmonic sums can be formulated within our difference ring in an optimal way: there does not exist a subset of the arising sums in which the sequence can be formulated as polynomial expression.

Hankel determinants for generating functions of Catalan-like numbers

Let be the generating functions of Catalan-like numbers . We show that the corresponding Hankel determinants satisfy a three-term recurrence relation. As applications, we express the Hankel determinants of some classical combinatorial counting coefficients, including the Catalan numbers, the central binomial coefficients, the large and little Schröder numbers, in a unified way.

New upper bounds for parent-identifying codes and traceability codes

In the last two decades, parent-identifying codes and traceability codes are introduced to prevent copyrighted digital data from unauthorized use. They have important applications in the scenarios like digital fingerprinting and broadcast encryption schemes. A major open problem in this research area is to determine the upper bounds for the cardinalities of these codes. In this paper we will focus on this theme. Consider a code of length N which is defined over an alphabet of size q. Let $$M_{IPPC}(N,q,t)$$ and $$M_{TA}(N,q,t)$$ denote the maximal cardinalities of t-parent-identifying codes and t-traceability codes, respectively, where t is known as the strength of the codes. We show $$M_{IPPC}(N,q,t)\le rq^{\lceil N/(v-1)\rceil }+(v-1-r)q^{\lfloor N/(v-1)\rfloor }$$ , where $$v=\lfloor (t/2+1)^2\rfloor $$ , $$0\le r\le v-2$$ and $$N\equiv r \mod (v-1)$$ . This new bound improves two previously known bounds of Blackburn, and Alon and Stav. On the other hand, $$M_{TA}(N,q,t)$$ is still not known for almost all t. In 2010, Blackburn, Etzion and Ng asked whether $$M_{TA}(N,q,t)\le cq^{\lceil N/t^2\rceil }$$ or not, where c is a constant depending only on N, and they have shown the only known validity of this bound for $$t=2$$ . By using some complicated combinatorial counting arguments, we prove this bound for $$t=3$$ . This is the first non-trivial upper bound in the literature for traceability codes with strength three.

Hankel determinants of linear combinations of consecutive Catalan-like numbers

Let (an)n≥0 be a sequence of the Catalan-like numbers. We evaluate Hankel determinants det[λai+j+μai+j+1]0≤i,j≤n and det[λai+j+1+μai+j+2]0≤i,j≤n for arbitrary coefficients λ and μ. Our results unify many known results of Hankel determinant evaluations for classic combinatorial counting coefficients, including the Catalan, Motzkin and Schröder numbers.

On The Improvement of Combinatorial Mathematics Teaching From Generating Function

Generating function is an effective method to solve combinatorial counting problem, but it most likely to be neglected in combinatorial mathematics teaching. In this paper, we provides a demonstration for combinatorial mathematics teaching improvement by using the generating function solving combination number and the sum of preceding terms among sequence of numbers.

The traveling salesman problem on grids with forbidden neighborhoods

We introduce the traveling salesman problem with forbidden neighborhoods (TSPFN). This is an extension of the Euclidean TSP in the plane where direct connections between points that are too close are forbidden. The TSPFN is motivated by an application in laser beam melting. In the production of a workpiece in several layers using this method one hopes to reduce the internal stresses of the workpiece by excluding the heating of positions that are too close. The points in this application are often arranged in some regular (grid) structure. In this paper we study optimal solutions of TSPFN instances where the points in the Euclidean plane are the points of a regular grid. Indeed, we explicitly determine the optimal values for the TSPFN and its associated path version on rectangular regular grids for different minimal distances of the points visited consecutively. For establishing lower bounds on the optimal values we use combinatorial counting arguments depending on the parities of the grid dimensions. Furthermore we provide construction schemes for optimal TSPFN tours for the considered cases.

A Dichotomy for Real Weighted Holant Problems

Holant is a framework of counting characterized by local constraints. It is closely related to other well-studied frameworks such as the counting constraint satisfaction problem (#CSP) and graph homomorphism. An effective dichotomy for such frameworks can immediately settle the complexity of all combinatorial problems expressible in that framework. Both #CSP and graph homomorphism can be viewed as subfamilies of Holant with the additional assumption that the equality constraints are always available. Other subfamilies of Holant such as Holant* and Holantc problems, in which we assume some specific sets of constraints to be freely available, were also studied. The Holant framework becomes more expressive and contains more interesting tractable cases with less or no freely available constraint functions, while, on the other hand, it also becomes more challenging to obtain a complete characterization of its time complexity. Recently, a complexity dichotomy for a variety of subfamilies of Holant such as #CSP, graph homomorphism, Holant*, and Holantc for Boolean domain was proved. In this paper, we prove a dichotomy for the general Holant framework where all the constraints are real symmetric functions on Boolean inputs. This setting already captures most of the interesting combinatorial counting problems defined by local constraints, such as (perfect) matching. This is the first time a dichotomy is obtained for general Holant problems without any auxiliary functions. One benefit of working with the Holant framework is some powerful new reduction technique such as the holographic reduction. Along the proof of our dichotomy, we introduce a new reduction technique, namely realizing a constraint function by approximating it. This new technique is employed in our proof in a situation where it seems that all previous reduction techniques fail; thus, this new idea of reduction might also be of independent interest. Besides proving a dichotomy and developing a new technique, we also obtained some interesting by-products. We prove a dichotomy for #CSP, restricting to instances where each variable appears a multiple of d times for any d. We also prove that counting the number of Eulerian orientations on 2k-regular graphs is #P-hard for any $${k \geq 2}$$k?2.

2D Toda τ-Functions as Combinatorial Generating Functions

Two methods of constructing 2D Toda τ-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group Sn by multiplication of the conjugacy class sums Cλ ∈C(Sn) in the group algebra by elements of an abelian group of central elements. Extending the characteristic map to the tensor product C(Sn )⊗ C(Sn) leads to double expansions in terms of power sum symmetric functions, in which the coefficients count the number of such paths. Applying the same map to sums over the orthogonal idempotents leads to diagonal double Schur function expansions that are iden- tified as τ-functions of hypergeometric type. The second method is the standard construc- tion of τ-functions as vacuum-state matrix elements of products of vertex operators in a fermionic Fock space with elements of the abelian group of convolution symmetries .A homomorphism between these two group actions is derived and shown to be intertwined by the characteristic map composed with fermionization. Applications include Okounkov's generating function for double Hurwitz numbers, which count branched coverings of the Riemann sphere with specified ramification profiles at two branch points, and only simple branching at all the others, and various analogous combinatorial counting functions.

A model counter for constraints over unbounded strings

Model counting is the problem of determining the number of solutions that satisfy a given set of constraints. Model counting has numerous applications in the quantitative analyses of program execution time, information flow, combinatorial circuit designs as well as probabilistic reasoning. We present a new approach to model counting for structured data types, specifically strings in this work. The key ingredient is a new technique that leverages generating functions as a basic primitive for combinatorial counting. Our tool SMC which embodies this approach can model count for constraints specified in an expressive string language efficiently and precisely, thereby outperforming previous finite-size analysis tools. SMC is expressive enough to model constraints arising in real-world JavaScript applications and UNIX C utilities. We demonstrate the practical feasibility of performing quantitative analyses arising in security applications, such as determining the comparative strengths of password strength meters and determining the information leakage via side channels.

Particle methods: An introduction with applications

Interacting particle methods are increasingly used to sample from complex high-dimensional distributions. They have found a wide range of applications in applied probability, Bayesian statistics and information engineering. Understanding rigorously these new Monte Carlo simulation tools leads to fascinating mathematics related to Feynman-Kac path integral theory and their interacting particle interpretations. In these lecture notes, we provide a pedagogical introduction to the stochastic modeling and the theoretical analysis of these particle algorithms. We also illustrate these methods through several applications including random walk confinements, particle absorption models, nonlinear filtering, stochastic optimization, combinatorial counting and directed polymer models.

Asymptotic distribution of integers with certain prime factorizations

Let p1<p2<⋯<pν<⋯ be the sequence of prime numbers and let m be a positive integer. We give a strong asymptotic formula for the distribution of the set of integers having prime factorizations of the form pmk1pmk2⋯pmkn with k1⩽k2⩽⋯⩽kn. Such integers originate in various combinatorial counting problems; when m=2, they arise as Matula numbers of certain rooted trees.

Holographic algorithms without matchgates

The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, surprised the complexity community by showing certain problems, very similar to #P complete problems, were in fact in the class P. In particular, the theory produces algebraic tests for a problem to be in P. In this article we describe the geometric basis of these algorithms by (i) replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix, and (ii) replacing the computation of weighted perfect matchings of an auxiliary graph by computing the Pfaffian of the directly constructed skew-symmetric matrix. This procedure indicates a more geometric approach to complexity classes. It also leads to more general constructions where one replaces the “Grassmann–Plücker identities” which test for admissibility by other algebraic tests. Natural problems treatable by these methods have been previously considered in a different context, and we present one such example.