William J. Keith
Michigan Tech University Math Department
Tenure-track Assistant Professor



Teaching

My courses for the fall of 2014 are:

MA3210, Section 1  Introduction to Combinatorics  8:05am - 9:20am, T Th  Fisher 325  Course page will be linked when assigned
MA2320, Section 2  Elementary Linear Algebra  Online  N/A  Course page will be linked when assigned

Outside of my office hours, students should get in touch with me using my MTU email, wjkeith [at] mtu.edu (no spaces).
My office hours are T Th 12-1pm. My office is Fisher 316. I am generally available at other times if a student emails me well in advance. This and other class information is available on the syllabus for each class, which is also on the course webpages linked above.


Research

My research centers on partition theory and related q-series and symmetric function identities. Over the course of my postdoctorate at the University of Lisbon, I expanded my areas of interest to other enumerative combinatorics questions, though I remain primarily interested in partitions.

For the standard outline of my research, please help yourself to a copy of my CV and publication list, both in pdf format. For more detail, please browse below.

Selected Publications and Preprints

A few of my more recent papers (and my thesis). I link to the published versions of these papers when possible, but preprints of all my work are available on the arXiv.

  • My thesis, generalizing two theorems in the literature on congruences for the full rank and on a theorem of Fine, both to more general modulus.

  • Proof of a conjectured q,t-Schröder identity. Discrete Mathematics, Volume 310, Issue 19, 6 October 2010, Pages 2489 - 2494. The k=2 case (one with interesting combinatorial interpretation) of a larger open conjecture by Chunwei Song related to the q,q limit of the q,t-Schröder theorem.

  • A Bijection for Partitions with initial repetitions. The Ramanujan Journal, February 2012, Volume 27, Issue 2, pp 163-167. A short paper with a bijective proof of another theorem of Andrews, proved with q-series techniques.

  • The 2-adic Valuation of Plane Partitions and Totally Symmetric Plane Partitions. Elec. Journ. of Comb., Volume 19 (2012), paper 48. Answering a question of Amdeberhan, Manna and Moll.

  • (Joint w/ Rishi Nath, CUNY-York) Partitions with prescribed hooksets. Journal of Combinatorics and Number theory, Volume 3 (1) 2011. The link goes to the preprint, since the journal is not as widely accessible.

  • A Ramanujan congruence analogue for Han's hook-length formula mod 5, and other symmetries. In submission; the link is the arXiv preprint. Raise the partition function to the power 1-b (or its reciprocal to b-1), and expand it as a power series in q with coefficients in the indeterminate b. The resulting polynomials have a large number of very pleasing symmetries.

  • Congruences for 9-regular partitions modulo 3. There seem to be a surprising number of congruences modulo 2 and 3 for partitions in which parts are not divisible by various values, and a topic of my current research is understanding why.


    Ongoing Research

    These are a few of the ongoing research questions which interest me. I am always happy to receive comments from interested colleagues, and would be pleased to collaborate with someone who has useful ideas in these directions. Graduate students considering combinatorics who find some of these questions interesting are encouraged to contact me as well.

    1.) I continue with great interest to study more about the polynomials described in the "Ramanujan congruence analogue" paper, which I think are fascinating combinatorial objects with properties that deserve exploration.

    2.) In non-partition work, I am presently interested in another conjecture of Amdeberhan, Manna and Moll, this one their open conjecture on the 2-adic valuation of the Stirling numbers of the second kind.

    3.) As mentioned above, I have recently become interested in m-regular partitions, especially their low-modulus congruences.

    4.) Of the famous problems that interest me, I would name the Borwein Conjecture, on the positivity of coefficients of a series defining a certain weighted sum over specialized partitions, and Lehmer's Conjecture on the non-zeroness of the coefficients of Ramanujan's tau function.