Home Page, William J. Keith

Welcome to my home page at Drexel University. I am a mathematician, a number theorist, a partition theorist, and a combinatorial partition theorist, in increasing order of specificity. The photo to the right was taken on a hiking trip with a Drexel student group for which I was the faculty advisor; it links to a full image of the whole group with the view from the summit.

Teaching

After receiving my Ph.D. from Penn State in the summer of 2007, I joined the faculty of the Math Department at Drexel University that Fall, in the non-tenure track position of Senior Lecturer. In the Fall of 2009 I became a Teaching Assistant Professor. As of Fall 2010, I have entered a postdoctoral position at the University of Lisbon, in the Centro Estruturas Lineares e Combinatórias. I would like to kindly thank Drexel for continuing to host this webpage.

Ongoing Research

My current duties are postdoctoral research, exploring the properties of partitions and related objects such as tableaux and lattice paths. Some of the recent questions that have interested me:

The "first differences" of a partition are simply the differences between parts, e.g. the first part minus the second part, the second part minus the third, etc. The second differences are the difference of the first differences, and so forth. These are analogous, for a discrete sequence, to the derivatives of a function, and they are much explored as statistics for partitions. What, then, of the analogue to an integral: the first part, the first part plus the sum of the first and second part, etc.? Little explored, these "progressive sums" of partitions look to early investigation as if they have some curious properties; if some observations and conjectures work out, this could be an interesting new statistic for partitions.

Here is a very simple statement: The hooksets of partitions are the complements of numerical semigroups. Some use of this is made in this paper (joint w/ Rishi Nath, CUNY-York), where the map that relates the two sets is described. But I think there is a great deal of unexploited potential here, especially in open questions for semigroups, such as the Wilf question or the Fibonacci-like number of semigroups with a given genus.

My current group projects, somewhat more broadly, involve a study of vexillary involutions here at CELC, and a long-distance collaboation with Robert Boyer at Drexel on extremal behavior of generating functions related to Herb Wilf's prefabs.

Past Papers

George Andrews introduced the notion of a "signed partition" in a recent article. Once you've seen this definition, the things start popping up everywhere. How does a combinatorialist approach identities for these objects? How should we modify the standard partition toolkit to approach it? I have now produced a paper on this question, "A bijective toolkit for signed partitions," which has been accepted for publication in annals of Combinatorics. You are welcome to download a preprint here.
Another idea of Andrews is the k-marked Durfee symbol, and the associated full rank of such a symbol, which encapsulates information on the moments of the rank function for ordinary partitions. My thesis work included a section deriving identities for the distribution of the populations of Durfee symbols with full ranks congruent to the various residues for odd moduli. This work has been accepted for publication in Discrete Mathematics; it is available at http://dx.doi.org/10.1016/j.disc.2009.02.032.
In a casual meeting of local mathematicians Herb Wilf mentioned to us the fact that the sum of the values zeta(k) - 1, where zeta is the Riemann zeta function and k ranges from 2 to infinity, is 1. When a sequence of nonzero positive numbers is so polite as to do this, he suggested, it ought to be the probabilities of something, and asked what it might be. I recently answered the question in a short note, which can be found here.
I have recently become interested in the area of tableaux combinatorics and the related behavior of symmetric functions, especially the k-Schur functions defined by Jen Morse and her collaborators. In addition to the standard open conjectures of the field, I am wondering if there exist signed variants of the related questions that could be explored.
Along those lines, I have answered an open conjecture in a paper of Chunwei Song relating the maj statistics of Catalan words, or Dyck paths, to those of tableaux. This paper is in submission; a preprint can be downloaded from the arXiv.
My recent presentations at the Georgia Southern q-series 2011 conference, and at the Combintorics seminar at the University of Coimbra, were on a project to write down a form of the generating function for simultaneous s/t-cores. The slides for the extended version of that presentation may be found here. They are in beamer format.
A collection of several other papers of mine can be found on the arXiv.

I am very interested in landing a permanent position at a good university, especially one with a research group active in number theory and/or combinatorics. I have plenty of experience teaching, and can make research progress under a full teaching load, so I would like to think I would make a valuable contribution to the scholarly community at any high-quality institution.

If you'd like to learn more about my background, education, and experience, please take a look at my CV and my publication list (pdfs). If you would really like to delve into my research, here's the 731K pdf of my thesis! I would be happy to answer any questions by email. I have an application available with AMS standard cover sheet and reference letters, including teaching commentary, at mathjobs, and am always eager to hear from academic employers or fellow mathematicians who might be interested in my work.