The distribution of cycles in breakpoint graphs of signed permutations

Breakpoint graphs are ubiquitous structures in the field of genome rearrangements. Their cycle decomposition has proved useful in computing and bounding many measures of (dis)similarity between genomes, and studying the distribution of those cycles is therefore critical to gaining insight on the distributions of the genomic distances that rely on it. We extend here the work initiated by Doignon and Labarre, who enumerated unsigned permutations whose breakpoint graph contains $k$ cycles, to signed permutations, and prove explicit formulas for computing the expected value and the variance of the corresponding distributions, both in the unsigned case and in the signed case. We also compare these distributions to those of several well-studied distances, emphasising the cases where approximations obtained in this way stand out. Finally, we show how our results can be used to derive simpler proofs of other previously known results

On the distribution of the number of cycles in the breakpoint graph of a random signed permutation

International audienceWe use the finite Markov chain embedding technique to obtain the distribution of the number of cycles in the breakpoint graph of a random uniform signed permutation. This further gives a very good approximation of the distribution of the reversal distance between two random genomes

Constitutionalism Outside the Courts

This essay is a chapter to be included in the forthcoming Oxford Handbook on the U.S. Constitution. Using the actions of Arkansas Governor Orville Faubus during the Little Rock crisis of 1957 and the U.S. Supreme Court’s subsequent decision in Cooper v. Aaron as a lens, it explores constitutional interpretation and enforcement by extrajudicial institutions. I explore the critique of Cooper’s notion of judicial supremacy by departmentalists like Walter Murphy, empirical scholars skeptical of judicial efficacy like Gerald Rosenberg, and popular constitutionalists like Larry Kramer and Mark Tushnet. I also consider four distinct institutional forms of extrajudicial constitutional interpretation and enforcement: protection of constitutional values through political processes and checks and balances; the role of social movements in shaping constitutional meaning; resolution of particular constitutional controversies in the political branches through processes of “constitutional construction”; and the role of “administrative constitutionalism.” The critique of judicial supremacy and the analysis of extrajudicial interpretation and enforcement have had a salutary impact in broadening the horizons of constitutional law. To the extent that theories of constitutionalism outside the courts are used to go further and attack judicial review, however, I find them less persuasive. One need not believe in judicial supremacy to value the courts’ ultimate settlement function in litigated constitutional controversies. If the Constitution is to continue to act as an external constraint on political action, then constitutionalism outside the courts can never be wholly autonomous of constitutionalism inside

Average-case analysis of perfect sorting by reversals (Journal Version)

Perfect sorting by reversals, a problem originating in computational genomics, is the process of sorting a signed permutation to either the identity or to the reversed identity permutation, by a sequence of reversals that do not break any common interval. B\'erard et al. (2007) make use of strong interval trees to describe an algorithm for sorting signed permutations by reversals. Combinatorial properties of this family of trees are essential to the algorithm analysis. Here, we use the expected value of certain tree parameters to prove that the average run-time of the algorithm is at worst, polynomial, and additionally, for sufficiently long permutations, the sorting algorithm runs in polynomial time with probability one. Furthermore, our analysis of the subclass of commuting scenarios yields precise results on the average length of a reversal, and the average number of reversals.Comment: A preliminary version of this work appeared in the proceedings of Combinatorial Pattern Matching (CPM) 2009. See arXiv:0901.2847; Discrete Mathematics, Algorithms and Applications, vol. 3(3), 201

The Wooster Voice (Wooster, OH), 1963-04-19

This edition of the Wooster Voice, published on April 19 of 1963, is four pages long. The far left column on the first page is an article about the revolution in Iraq on February 8 which overthrew the Kassem dictatorship. The new Voice staff for the next school year is announced on the first page. The Spring Concert for the Wooster Symphony Orchestra will take place on Monday evening and will be conducted by guest conductor Michael Charry. The weekly athletics updates can be found on the third page.https://openworks.wooster.edu/voice1961-1970/1057/thumbnail.jp

Efficient Sampling of Parsimonious Inversion Histories with Application to Genome Rearrangement in Yersinia

Inversions are among the most common mutations acting on the order and orientation of genes in a genome, and polynomial-time algorithms exist to obtain a minimal length series of inversions that transform one genome arrangement to another. However, the minimum length series of inversions (the optimal sorting path) is often not unique as many such optimal sorting paths exist. If we assume that all optimal sorting paths are equally likely, then statistical inference on genome arrangement history must account for all such sorting paths and not just a single estimate. No deterministic polynomial algorithm is known to count the number of optimal sorting paths nor sample from the uniform distribution of optimal sorting paths