Brian Nakamura

Using functional equations to enumerate 1324-avoiding permutations

We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r=0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n=31. We also extend those functional equations to account for the number of inversions and derive analogous algorithms.

Automatic Generation of Theorems and Proofs on Enumerating Consecutive-Wilf Classes

This article, describes two complementary approaches to enumeration, the positive and the negative, each with its advantages and disadvantages. Both approaches are amenable to automation, and we apply it to the currently active subarea, initiated in 2003 by Sergi Elizalde and Marc Noy, of enumerating consecutive-Wilf classes (i.e. consecutive pattern-avoidance) in permutations.

Experimental designs for testing metal detectors at a large sports stadium

When utilizing metal detectors at a large venue such as a sports stadium, there are the competing objectives of accuracy of the patron screening and the speed of throughput. This research, carried out in collaboration with the security staff at MetLife Stadium in New Jersey as well as other stadiums, analyzed two patron screening methods: handheld metal detectors (“wands”) and walk-through metal detectors (“walk-throughs”). An initial experimental design was created to understand the effectiveness of wanding. This design was used with MetLife Stadium security during three training sessions. The data collected was used to understand (a) if the prohibited item was found and (b) in how many attempts. Various prohibited as well as allowable metal items were hidden at random in various locations on the body of individuals, who were then scored based on the importance weight of the item (guns were given more weight than keys for example). Trainees were then assigned a performance score based on speed and accuracy and were tested until they reached a minimum required score. Building on this initial experiment, a second more formal experiment was created to help MetLife Stadium staff understand how walk-throughs would perform outdoors at different security settings. This experiment focused less on training security staff and more on understanding the performance of walk-throughs in real situations (as opposed to idealized lab situations). This experiment was created to understand the walk-through performance at each setting in the outdoor environment; e.g., does a walk-through catch each of the pre-specified prohibited items, and is this consistent across machines on the same setting? Because of the number of factors to be considered (type of item, location, orientation, walk-through setting, etc.), designing the experiment required a sophisticated approach called Combinatorial Experimental Design. The experiment was part of two DHS-supported projects on best practices for stadium security.

The ACCAM model: simulating aviation mission readiness for U.S. coast guard stations

We present a model and discrete event simulation of USCG Air Stations, accounting for the mission demands and maintenance procedures pertaining to USCG aircraft. The simulation provides aircraft availability distributions and mission performance metrics based on varying input scenarios, including changes in the number of stationed aircraft and maintenance targets. The Air Station model is novel in its relatively simple, easily tunable, renewal process treatment of maintenance procedures, mitigating the need for the modeling of complex maintenance subprocesses and the resulting statistical estimation of numerous parameters. The simulation also models mission requirements such as Search and Rescue that are stochastic in time and space. Simulations are consistent with historical data and offer insights into hypothetical scenarios.