Exact Enumeration Research Articles

Exact Enumeration Approach to Estimate the Theta Temperature of Interacting Self-Avoiding Walks on the Simple Cubic Lattice.

We compute the exact root-mean-square end-to-end distance of the interacting self-avoiding walk (ISAW) up to 27 steps on the simple cubic lattice. These data are used to construct a fixed point equation to estimate the theta temperature of the collapse transition of the ISAW. With the Bulirsch-Stoer extrapolation method, we obtain accurate results that can be compared with large-scale long-chain simulations. The free parameter ω in extrapolation is precisely determined using a parity property of the ISAW. The systematic improvement of this approach is feasible by adopting the combination of exact enumeration and multicanonical simulations.

On the hardness of quadratic unconstrained binary optimization problems

We use exact enumeration to characterize the solutions of quadratic unconstrained binary optimization problems of less than 21 variables in terms of their distributions of Hamming distances to close-by solutions. We also perform experiments with the D-Wave Advantage 5.1 quantum annealer, solving many instances of up to 170-variable, quadratic unconstrained binary optimization problems. Our results demonstrate that the exponents characterizing the success probability of a D-Wave annealer to solve a quadratic unconstrained binary optimization correlate very well with the predictions based on the Hamming distance distributions computed for small problem instances.

Edge-colored directed subgraph enumeration on the connectome

Following significant advances in image acquisition, synapse detection, and neuronal segmentation in connectomics, researchers have extracted an increasingly diverse set of wiring diagrams from brain tissue. Neuroscientists frequently represent these wiring diagrams as graphs with nodes corresponding to a single neuron and edges indicating synaptic connectivity. The edges can contain “colors” or “labels”, indicating excitatory versus inhibitory connections, among other things. By representing the wiring diagram as a graph, we can begin to identify motifs, the frequently occurring subgraphs that correspond to specific biological functions. Most analyses on these wiring diagrams have focused on hypothesized motifs—those we expect to find. However, one of the goals of connectomics is to identify biologically-significant motifs that we did not previously hypothesize. To identify these structures, we need large-scale subgraph enumeration to find the frequencies of all unique motifs. Exact subgraph enumeration is a computationally expensive task, particularly in the edge-dense wiring diagrams. Furthermore, most existing methods do not differentiate between types of edges which can significantly affect the function of a motif. We propose a parallel, general-purpose subgraph enumeration strategy to count motifs in the connectome. Next, we introduce a divide-and-conquer community-based subgraph enumeration strategy that allows for enumeration per brain region. Lastly, we allow for differentiation of edges by types to better reflect the underlying biological properties of the graph. We demonstrate our results on eleven connectomes and publish for future analyses extensive overviews for the 26 trillion subgraphs enumerated that required approximately 9.25 years of computation time.

Preimages under a popqueue-sorting algorithm

Abstract Following the footprints of what has been done with other sorting devices, we study a popqueue and define an optimal sorting algorithm, called Cons. Our results include a description of the set of all the preimages of a given permutation, an enumeration of the set of the preimages of permutations with some specific properties and, finally, the exact enumeration of permutations having 0, 1 and 2 preimages, respectively, with a characterization of permutations having 3 preimages.

The complexity of simple counting: ERP findings reveal early perceptual and late numerical processes in different arrangements

The counting process can only be fully understood when taking into account the visual characteristics of the sets counted. Comparing behavioral data as well as event-related brain potentials (ERPs) evoked by different task-irrelevant arrangements of dots during an exact enumeration task, we aimed to investigate the effect of illusory contour detection on the counting process while other grouping cues like proximity were controlled and dot sparsity did not provide a cue to the numerosity of sets. Adult participants (N = 37) enumerated dots (8–12) in irregular and two different types of regular arrangements which differed in the shape of their illusory dot lattices. Enumeration speed was affected by both arrangement and magnitude. The type of arrangement influenced an early ERP negativity peaking at about 270 ms after stimulus onset, whereas numerosity only affected later ERP components (> 300 ms). We also observed that without perceptual cues, magnitude was constructed at a later stage of cognitive processing. We suggest that chunking is a prerequisite for more fluent counting which influences automatic processing (< 300 ms) during enumeration. We conclude that the procedure of exact enumeration depends on the interaction of several perceptual and numerical processes that are influenced by magnitude and arrangement.

Effect of solvent gradient inside the entropic trap on polymer migration.

By employing the exact enumeration technique on the lattice model of a polymer, we study the migration of the polymer chain across an entropic trap in a quasiequilibrium condition and explore the effect of solvent gradient present in the entropic trap which acts both parallel and perpendicular to the direction of migration. The Fokker-Planck formalism utilizes the free energy landscape of a polymer chain across the channel in the presence of the entropic trap to calculate the migration time. It is revealed that the migration is fast when the solvent gradient acts along the migration axis (i.e., x axis) inside the channel in comparison to the channel having the entropic trap. We report here for the first time that the entropic trap makes the migration faster at a certain value of solvent gradient. We also study the effect of transverse solvent gradient (along the y axis) inside the trap and investigate the structural changes of the polymer during migration through the channel. We observe the nonmonotonic dependence of migration time on the solvent gradient.

Asymptotic behaviour of the containment of certain mesh patterns

We present some results on the proportion of permutations of length n containing certain mesh patterns as n grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between 0 and 1.

An investigation of -symmetry breaking in tight-binding chains

We consider non-Hermitian -symmetric tight-binding chains where gain/loss optical potentials of equal magnitudes ±iγ are arbitrarily distributed over all sites. The main focus is on the threshold γ c beyond which -symmetry is broken. This threshold generically falls off as a power of the chain length, whose exponent depends on the configuration of optical potentials, ranging between 1 (for balanced periodic chains) and 2 (for unbalanced periodic chains, where each half of the chain experiences a non-zero mean potential). For random sequences of optical potentials with zero average and finite variance, the threshold is itself a random variable, whose mean value decays with exponent 3/2 and whose fluctuations have a universal distribution. The chains yielding the most robust -symmetric phase, i.e. the highest threshold at fixed chain length, are obtained by exact enumeration up to 48 sites. This optimal threshold exhibits an irregular dependence on the chain length, presumably decaying asymptotically with exponent 1, up to logarithmic corrections.

DNA migration through semi-circular gradient channel

We study quasi-equilibrium migration of double stranded DNA (dsDNA) through semi-circular narrow channel by exact enumeration method. We employ Fokker–Planck formalism to analyze the slow migration process of dsDNA through this symmetric channel which resembles the Herpes Simplex Virus -1 (HSV-1) migration through the vestibular nuclei in brainstem. We explore all possibilities of varying solvent gradient inside the semi circular area to ensure slow migration which is essential in DNA nanopore sequencing techniques. We also shed light on configurational and orientational properties of DNA and sequential preference of the dsDNA while migrating through the channel. We report that GC rich homo-sequence DNA migrates more slowly through the channel than AT rich and AT-GC di-block DNA. In this paper, a simple lattice model is developed to explore various physical parameters dependence on slow migration process of a dsDNA.

Aerosol is the optimal route of respiratory tract infection to induce pathological lesions of colibacillosis by a lux-tagged avian pathogenic Escherichia coli in chickens

ABSTRACT Pathogenesis of colibacillosis caused by avian pathogenic Escherichia coli (APEC) in poultry is unclear and experimental studies reveal substantial inconsistency. In this study, the impact of three infection routes differing in the site of deposition of inoculum in the respiratory tract, were investigated. Two-weeks-old chickens were infected with a lux-tagged APEC strain via aerosol, intranasally or intratracheally, and sequentially sampled along with uninfected birds. At 1 and 3 days post infection (dpi), liver or spleen to body-weight ratios in all infected groups were significantly higher than in negative control, while at 7 dpi, such differences were significant in both organs in the aerosol-infected group. The infection-strain colonized tracheas and lungs in infected birds at 1 dpi and persisted until 7 dpi. Among infected groups, in lungs, bacterial load at 1 dpi was significantly lower in intranasally-inoculated birds. Histology revealed that, independent of infection route, lesions were mostly seen in the lower respiratory organs (lungs and air sacs) characterized by bronchitis/pneumonia and airsacculitis. Birds infected via aerosol showed the highest mean lesion score in lungs while intranasal application caused the mildest pathological changes, and difference between the two groups was significant at 1 dpi. In spleen, heterophilic infiltrations were prominent in affected birds. Interestingly, tracheas were pathologically unaffected. Altogether, the results demonstrated the importance of infection route, with aerosol being the most suitable to induce pathological lesions of colibacillosis without predisposing factors. Furthermore, the lux-tagged APEC strain was discriminated from native isolates enabling exact differentiation and enumeration. RESEARCH HIGHLIGHTS Lux-tagged APEC strain was used for infection to differentiate from native E. coli. Pathologically, lungs, air sacs and spleen but not trachea were affected. The route of infection strongly impacts the pathological outcome with APEC. The infection with APEC via aerosol caused the most severe lesions in chickens.

On the failure of effective-field theory in predicting a spurious spontaneous ordering and phase transition of Ising nanoparticles, nanoislands, nanotubes and nanowires

The present work clarifies a failure of the effective-field theory in predicting a false spontaneous long-range order and phase transition of Ising nanoparticles, nanoislands, nanotubes and nanowires with either zero- or one-dimensional magnetic dimensionality. It is conjectured that the standard formulation of the effective-field theory due to Honmura and Kaneyoshi generally predicts for the Ising spin systems a spurious spontaneous long-range order with nonzero critical temperature regardless of their magnetic dimensionality whenever at least one Ising spin has coordination number greater than two. The failure of the effective-field theory is exemplified on a few paradigmatic exactly solved examples of zero- and one-dimensional Ising nanosystems: star, cube, decorated hexagon, star of David, branched chain, sawtooth chain, two-leg and hexagonal ladders. The presented exact solutions illustrate eligibility of a few rigorous analytical methods for exact treatment of the Ising nanosystems: exact enumeration, graph-theoretical approach, transfer-matrix method and decoration-iteration transformation. The paper also provides a substantial survey of the scientific literature, in which the effective-field theory led to a false prediction of the spontaneous long-range ordering and phase transition.

Preimages under the Queuesort algorithm

Following the footprints of what has been done with the algorithm Stacksort, we investigate the preimages of the map associated with a slightly less well known algorithm, called Queuesort. After having described an equivalent version of Queuesort, we provide a recursive description of the set of all preimages of a given permutation, which can be also translated into a recursive procedure to effectively find such preimages. We then deal with some enumerative issues. More specifically, we investigate the cardinality of the set of preimages of a given permutation, showing that all cardinalities are possible, except for 3. We also give exact enumeration results for the number of permutations having 0,1 and 2 preimages. Finally, we consider the special case of those permutations π whose set of left-to-right maxima is the disjoint union of a prefix and a suffix of π: we determine a closed formula for the number of preimages of such permutations, which involves two different incarnations of ballot numbers, and we show that our formula can be expressed as a linear combination of Catalan numbers.

Standard and inverse site percolation of straight rigid rods on triangular lattices: Isotropic and perfectly oriented deposition and removal.

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, linear k-mers (particles occupying k consecutive sites along one of the lattice directions) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by randomly removing sets of k consecutive monomers (linear k-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with size k: (i) [(ii)] standard isotropic[oriented] percolation threshold θ_{c,k}[ϑ_{c,k}], which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k}[ϑ_{c,k}] is reached by isotropic[oriented] deposition of straight rigid k-mers on an initially empty lattice; and (iii) [(iv)] inverse isotropic[oriented] percolation threshold θ_{c,k}^{i}[ϑ_{c,k}^{i}], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i}[ϑ_{c,k}^{i}] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. θ_{c,k}, ϑ_{c,k}, θ_{c,k}^{i}, and ϑ_{c,k}^{i} are determined for a wide range of k (2≤k≤512). The obtained results indicate that (1)θ_{c,k}[θ_{c,k}^{i}] exhibits a nonmonotonous dependence on the size k. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around k=11, and finally increases and asymptotically converges towards a definite value for large segments θ_{c,k→∞}=0.500(2) [θ_{c,k→∞}^{i}=0.500(1)]; (2)ϑ_{c,k}[ϑ_{c,k}^{i}] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long k-mers ϑ_{c,k→∞}=0.5334(6) [ϑ_{c,k→∞}^{i}=0.4666(6)]; (3) for both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(ϑ)=0.5. Thus a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and ϑ_{c,k}+ϑ_{c,k}^{i}=1) which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems, and (4) in all cases, the critical concentration curves divide the θ space in a percolating region and a nonpercolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for the whole range of k.

Learning exact enumeration and approximate estimation in deep neural network models

A system for approximate number discrimination has been shown to arise in at least two types of hierarchical neural network models—a generative Deep Belief Network (DBN) and a Hierarchical Convolutional Neural Network (HCNN) trained to classify natural objects. Here, we investigate whether the same two network architectures can learn to recognise exact numerosity. A clear difference in performance could be traced to the specificity of the unit responses that emerged in the last hidden layer of each network. In the DBN, the emergence of a layer of monotonic ‘summation units’ was sufficient to produce classification behaviour consistent with the behavioural signature of the approximate number system. In the HCNN, a layer of units uniquely tuned to the transition between particular numerosities effectively encoded a thermometer-like ‘numerosity code’ that ensured near-perfect classification accuracy. The results support the notion that parallel pattern-recognition mechanisms may give rise to exact and approximate number concepts, both of which may contribute to the learning of symbolic numbers and arithmetic.

Number of Dominating Sets in Cylindric Square Grid Graphs

A dominating set of a graph is a subset D of the vertices such that every vertex not in D is adjacent to some vertex of D. In this paper, we introduce several variants of dominating sets in the square grid, periodic square grid and cylindric square grid by considering translation symmetry. We provide their exact enumerations in terms of domination polynomials. We also analyze the asymptotic behavior of the growth rates of their cardinality.

Dynamics of a polymer chain translocating through varying cone-shaped channels.

By employing the exact enumeration technique, we study consequences of different apex angles of a wedge-shaped channel on the mean first passage time and free-energy profile of a linear polymer chain translocating from the cis- to the trans-side through an interacting pore. We investigate effects of asymmetry arising in the free-energy profile due to the change in apex angles and its dependence on the first passage time. We report the combined effect of entropy (arising due to apex angles) and pore interaction on the nonmonotonic behavior of the translocation time. The effect of different solvent quality across the channel has also been explored. We show that the increase in monomer-monomer interaction leads to the formation of globules near the pore, which drives the process faster.

Bounded affine permutations I. Pattern avoidance and enumeration

We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations.

Parking function varieties for combinatorial tree models

We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted labelled trees of important combinatorial tree families including labelled ordered, unordered and binary trees. Using combinatorial decompositions of the underlying structures yields, after solving the resulting equations, implicit characterizations of suitable generating functions of the total number of such tree parking functions for trees of size n and n successful drivers, from which we obtain exact and asymptotic enumeration results. The approach can be extended to the general situation of tree parking functions for trees of size n and m<n drivers for which we are also able to characterize the generating functions solutions, which allow, by applying analytic combinatorics techniques, a study of the asymptotic behaviour of the total number of tree parking functions and distributions for n→∞ depending on the load factor 0<α=mn<1.

Impaired large numerosity estimation and intact subitizing in developmental dyscalculia.

It is believed that the approximate estimation of large sets and the exact quantification of small sets (subitizing) are supported by two different systems, the Approximate Number System (ANS) and Object Tracking System (OTS), respectively. It is a current matter of debate whether they are both impaired in developmental dyscalculia (DD), a specific learning disability in symbolic number processing and calculation. Here we tackled this question by asking 32 DD children and 32 controls to perform a series of tasks on visually presented sets, including exact enumeration of small sets as well as comparison of large, uncountable sets. In children with DD, we found poor sensitivity in processing large numerosities, but we failed to find impairments in the exact enumeration of sets within the subitizing range. We also observed deficits in visual short-term memory skills in children with dyscalculia that, however, did not account for their low ANS acuity. Taken together, these results point to a dissociation between quantification skills in dyscalculia, they highlight a link between DD and low ANS acuity and provide support for the notion that DD is a multifaceted disability that covers multiple cognitive skills.

Counting all unfolded self-avoiding walks on a finite lattice strip of width three

By extending the method of the author’s earlier paper, a purely recursive approach is devised for counting the total number of unfolded self-avoiding walks terminating along the line x=n, within the finite lattice strip {0,1,…,n}×{0,1,2,3} of width three. This approach yields a generating function for the sequence of numbers just described, without the need for any exact enumeration, usually entailed with an application of the transfer matrix algorithm. As a result of determining this generating function, the total number of unfolded walks within the lattice strip can then be easily computed.