On the growth rate of 1324-avoiding permutations

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity. Rather, the number of 1324-avoiding permutations of length $n$ behaves as $$B\cdot \mu^n \cdot \mu_1^{n^{\sigma}} \cdot n^g.$$ We estimate $\mu=11.60 \pm 0.01,$ $\sigma=1/2,$ $\mu_1 = 0.0398 \pm 0.0010,$ $g = -1.1 \pm 0.2$ and $B =9.5 \pm 1.0.$Comment: 20 pages, 10 figure

On consecutive pattern-avoiding permutations of length 4, 5 and beyond

We review and extend what is known about the generating functions for consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite. For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.Comment: 23 pages, 2 figures (update of references, plus web link to enumeration data). Minor update. Typos corrected. One additional referenc

Counting occurrences of patterns in permutations

We develop a new, powerful method for counting elements in a {\em multiset.} As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and the general behaviour of these is reasonably well-known. We slightly extend some of the known results in that case, and exhaustively study the case of patterns of length 4, about which there is little previous knowledge. For such patterns, there are seven Wilf classes, and based on extensive enumerations and careful series analysis, we have conjectured the asymptotic behaviour for all classes. Finally, we investigate a proposal of Blitvi\'c and Steingr\'imsson as to the range of a parameter for which a particular generating function formed from the occurrence sequences is itself a Stieltjes moment sequence

On the Photometric Accuracy of RHESSI Imaging and Spectrosocopy

We compare the photometric accuracy of spectra and images in flares observed with the Ramaty High Energy Solar Spectroscopic Imager (RHESSI)}spacecraft. We test the accuracy of the photometry by comparing the photon fluxes obtained in different energy ranges from the spectral-fitting software SPEX with those fluxes contained in the images reconstructed with the Clean, MEM, MEM-Vis, Pixon, and Forward-fit algorithms. We quantify also the background fluxes, the fidelity of source geometries, and spatial spectra reconstructed with the five image reconstruction algorithms. We investigate the effects of grid selection, pixel size, field-of-view, and time intervals on the quality of image reconstruction. The detailed parameters and statistics are provided in an accompanying CD-ROM and web page. We find that Forward-fit, Pixon, and Clean have a robust convergence behavior and a photometric accuracy in the order of a few percents, while MEM does not converge optimally for large degrees of freedom (for large field-of-views and/or small pixel sizes), and MEM-Vis suffers in the case of time-variable sources. This comparative study documents the current status of the RHESSI spectral and imaging software, one year after launch.Comment: 2 Figures, full version on http://www.lmsal.com/~aschwand/eprints/2003_photo/index.htm

Revealing the structure of language model capabilities

Building a theoretical understanding of the capabilities of large language models (LLMs) is vital for our ability to predict and explain the behavior of these systems. Here, we investigate the structure of LLM capabilities by extracting latent capabilities from patterns of individual differences across a varied population of LLMs. Using a combination of Bayesian and frequentist factor analysis, we analyzed data from 29 different LLMs across 27 cognitive tasks. We found evidence that LLM capabilities are not monolithic. Instead, they are better explained by three well-delineated factors that represent reasoning, comprehension and core language modeling. Moreover, we found that these three factors can explain a high proportion of the variance in model performance. These results reveal a consistent structure in the capabilities of different LLMs and demonstrate the multifaceted nature of these capabilities. We also found that the three abilities show different relationships to model properties such as model size and instruction tuning. These patterns help refine our understanding of scaling laws and indicate that changes to a model that improve one ability might simultaneously impair others. Based on these findings, we suggest that benchmarks could be streamlined by focusing on tasks that tap into each broad model ability.Comment: 10 pages, 3 figures + references and appendices, for data and analysis code see https://github.com/RyanBurnell/revealing-LLM-capabilitie

Luminous Intensity for Traffic Signals: A Scientific Basis for Performance Specifications

Humnan factors experiments on visual responses to simulated traffic signals using incandescent lamps and light-emitting diodes are described

Psychometric Network Analysis of the Hungarian WAIS

The positive manifold—the finding that cognitive ability measures demonstrate positive correlations with one another—has led to models of intelligence that include a general cognitive ability or general intelligence (g). This view has been reinforced using factor analysis and reflective, higher-order latent variable models. However, a new theory of intelligence, Process Overlap Theory (POT), posits that g is not a psychological attribute but an index of cognitive abilities that results from an interconnected network of cognitive processes. These competing theories of intelligence are compared using two different statistical modeling techniques: (a) latent variable modeling and (b) psychometric network analysis. Network models display partial correlations between pairs of observed variables that demonstrate direct relationships among observations. Secondary data analysis was conducted using the Hungarian Wechsler Adult Intelligence Scale Fourth Edition (H-WAIS-IV). The underlying structure of the H-WAIS-IV was first assessed using confirmatory factor analysis assuming a reflective, higher-order model and then reanalyzed using psychometric network analysis. The compatibility (or lack thereof) of these theoretical accounts of intelligence with the data are discussed

Working memory for social information: Chunking or domain-specific buffer?

Humans possess unique social abilities that set us apart from other species. These abilities may be partially supported by a large capacity for maintaining and manipulating social information. Efficient social working memory might arise from two different sources: chunking of social information or a domain-specific buffer. We test these hypotheses with functional magnetic resonance imaging (fMRI) by manipulating sociality and working memory load in an n-back paradigm. We observe (i) an effect of load in the frontoparietal control network, (ii) an effect of sociality in regions associated with social cognition and face processing, and (iii) an interaction within the frontoparietal network such that social load has a smaller effect than nonsocial load. These results support the hypothesis that working memory is more efficient for social information than for nonsocial information, and suggest that chunking, rather than a domain-specific buffer, is the mechanism of this greater efficiency