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

An Algorithm for Distributing Coalitional Value Calculations among Cooperating Agents

The process of forming coalitions of software agents generally requires calculating a value for every possible coalition which indicates how beneficial that coalition would be if it was formed. Now, instead of having a single agent calculate all these values (as is typically the case), it is more efficient to distribute this calculation among the agents, thus using all the computational resources available to the system and avoiding the existence of a single point of failure. Given this, we present a novel algorithm for distributing this calculation among agents in cooperative environments. Specifically, by using our algorithm, each agent is assigned some part of the calculation such that the agents’ shares are exhaustive and disjoint. Moreover, the algorithm is decentralized, requires no communication between the agents, has minimal memory requirements, and can reflect variations in the computational speeds of the agents. To evaluate the effectiveness of our algorithm, we compare it with the only other algorithm available in the literature for distributing the coalitional value calculations (due to Shehory and Kraus). This shows that for the case of 25 agents, the distribution process of our algorithm took less than 0.02% of the time, the values were calculated using 0.000006% of the memory, the calculation redundancy was reduced from 383229848 to 0, and the total number of bytes sent between the agents dropped from 1146989648 to 0 (note that for larger numbers of agents, these improvements become exponentially better)

Rush to Justice: The Disappearing Civil Trial

I address an issue largely ignored by political science: the role of litigation procedure in influencing policy outcomes. The specific aim of my dissertation is to explain the phenomenon of diminishing civil trials and the concomitant policy effects caused by this diminution. I begin with the following research puzzle: United States federal civil case filings have increased steadily over time to reach the highest levels ever seen. Federal civil case filings have increased over 400% between 1964-2014. Yet, despite the increased number of case filings, the percentage of cases reaching trial is 1/12th what it was in 1964. Presently, only 1% of all cases filed reach trial. The puzzle is why have the number of trials diminished even though case filings have increased? And, does the diminution of trials have a substantive impact on national policies? My dissertation theorizes that the decrease in trials has resulted from a modification in litigation incentives created by changes to the Federal Rules of Civil Procedure, namely, the summary judgment process. In 1986, the U.S. Supreme Court issued a trilogy of decisions which dramatically altered the usage of summary judgment, and in so doing, the economic incentives for litigants to go to trial. To address the aforementioned questions, I utilize a multi-level research design. First, in a macro-level study of aggregate data, I utilize advanced time series methods and find structural breaks in the data consistent with these changes to the summary judgment process. Additionally, continuing the macro-level analysis, I examine the consequences of this procedural change by analyzing the effects of the change to summary judgment on federal employment discrimination cases. I find diminishing employment discrimination trials cause increased disparity in income ratios between men and women, as well as African-Americans and whites. The findings suggest that changes to legal procedure by the federal judiciary have hurt legislative endeavors to promote income equality, despite Congress’ efforts to incentivize civil rights litigation. Finally, I perform a micro-level analysis evaluating case-level data. Through creation of a unique dataset, I evaluate the causes of motions for summary judgment and the determinants of judicial grants of such motions. The data suggest that several factors, including litigant resources and the type of case brought (such as civil rights cases), impact a court’s decision to dismiss claims using this procedural device. My results suggest policymakers must rethink their efforts at legislating equality through private statutory enforcement

Confidence and Constraint: Public Opinion, Judicial Independence, and the Roberts Court

This Article uses statistical models to show the relationship between public opinion of the Supreme Court and the Court’s propensity to invalidate federal laws on constitutional grounds. Merril, Conway, and Ura analyze this connection to underscore the loss of judicial independence as a result of declining public opinion. The authors note this decline in public opinion allows the President and Congress to leverage public opinion against the Court in order to influence whether a federal law will be invalidated