The large-parts formula for p(n)

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$ is a partition of $n$ with $a_1 \leq ... \leq a_k$ and $a_0 = 0$, then the sum of $\lfloor(a_k + a_{k-1}) / (a_{k-1} + 1)\rfloor$ over all partitions of $n$ is equal to $2p(n) - 1$.Comment: Four pages; 1 figur

Coalescent simulation in continuous space

Coalescent simulation has become an indispensable tool in population genetics and many complex evolutionary scenarios have been incorporated into the basic algorithm. Despite many years of intense interest in spatial structure, however, there are no available methods to simulate the ancestry of a sample of genes that occupy a spatial continuum. This is mainly due to the severe technical problems encountered by the classical model of isolation by distance. A recently introduced model solves these technical problems and provides a solid theoretical basis for the study of populations evolving in continuous space. We present a detailed algorithm to simulate the coalescent process in this model, and provide an efficient implementation of a generalised version of this algorithm as a freely available Python module

Bayesian inference of ancestral recombination graphs

We present a novel algorithm, implemented in the software ARGinfer, for probabilistic inference of the Ancestral Recombination Graph under the Coalescent with Recombination. Our Markov Chain Monte Carlo algorithm takes advantage of the Succinct Tree Sequence data structure that has allowed great advances in simulation and point estimation, but not yet probabilistic inference. Unlike previous methods, which employ the Sequentially Markov Coalescent approximation, ARGinfer uses the Coalescent with Recombination, allowing more accurate inference of key evolutionary parameters. We show using simulations that ARGinfer can accurately estimate many properties of the evolutionary history of the sample, including the topology and branch lengths of the genealogical tree at each sequence site, and the times and locations of mutation and recombination events. ARGinfer approximates posterior probability distributions for these and other quantities, providing interpretable assessments of uncertainty that we show to be well calibrated. ARGinfer is currently limited to tens of DNA sequences of several hundreds of kilobases, but has scope for further computational improvements to increase its applicability

A community-maintained standard library of population genetic models

The explosion in population genomic data demands ever more complex modes of analysis, and increasingly, these analyses depend on sophisticated simulations. Recent advances in population genetic simulation have made it possible to simulate large and complex models, but specifying such models for a particular simulation engine remains a difficult and error-prone task. Computational genetics researchers currently re-implement simulation models independently, leading to inconsistency and duplication of effort. This situation presents a major barrier to empirical researchers seeking to use simulations for power analyses of upcoming studies or sanity checks on existing genomic data. Population genetics, as a field, also lacks standard benchmarks by which new tools for inference might be measured. Here, we describe a new resource, stdpopsim, that attempts to rectify this situation. Stdpopsim is a community-driven open source project, which provides easy access to a growing catalog of published simulation models from a range of organisms and supports multiple simulation engine backends. This resource is available as a well-documented python library with a simple command-line interface. We share some examples demonstrating how stdpopsim can be used to systematically compare demographic inference methods, and we encourage a broader community of developers to contribute to this growing resource.Open access journalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Efficient ancestry and mutation simulation with msprime 1.0

Stochastic simulation is a key tool in population genetics, since the models involved are often analytically intractable and simulation is usually the only way of obtaining ground-truth data to evaluate inferences. Because of this, a large number of specialized simulation programs have been developed, each filling a particular niche, but with largely overlapping functionality and a substantial duplication of effort. Here, we introduce msprime version 1.0, which efficiently implements ancestry and mutation simulations based on the succinct tree sequence data structure and the tskit library. We summarize msprime’s many features, and show that its performance is excellent, often many times faster and more memory efficient than specialized alternatives. These high-performance features have been thoroughly tested and validated, and built using a collaborative, open source development model, which reduces duplication of effort and promotes software quality via community engagement

htsget: a protocol for securely streaming genomic data

Summary: Standardized interfaces for efficiently accessing high-throughput sequencing data are a fundamental requirement for large-scale genomic data sharing. We have developed htsget, a protocol for secure, efficient and reliable access to sequencing read and variation data. We demonstrate four independent client and server implementations, and the results of a comprehensive interoperability demonstration. Availability and implementation: http://samtools.github.io/hts-specs/htsget.html Supplementary information: Supplementary data are available at Bioinformatics online