|William J. Keith|
Michigan Tech University Math Department
Tenure-track Assistant Professor
My course for the spring of 2017 is:
|MA 5360, Section 1
|| ||3:35pm - 4:50pm, TR
||I am available for office isits most times with some notice. My office is Fisher 316.
||Outside of my office hours, students should get in touch with me using my MTU email, wjkeith [at] mtu.edu (no spaces). I am generally available at other times MWF 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. On Tuesdays and Thursdays I am typically involved in research activities and unavailable for student meetings.
I currently serve on the Undergraduate Committee and help organize the Combinatorics Seminar.
Here is the Combinatorics Seminar Schedule for the current semester, with previous schedules back to the Spring of 2013. Seminars this semester run Thursdays 1-2pm in Fisher 130.
|My research is in combinatorics, specializing in partition theory and related q-series and identities.|
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, I list below a few of my papers (and my thesis). Preprints of all my work are available on the arXiv.
Graduate students who find the topics of these papers interesting are warmly encouraged to discuss potential thesis work with me!
Selected Publications and Preprints
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.
Restricted k-color partitions. Overpartitions are a hot current topic in partition research, and colored partitions are an old standby in the literature. Here I unify the two topics and ask about the combinatorial properties of the resulting objects.
(Joint with Fabrizio Zanello and his graduate student Samuel Judge.) On the density of the odd values of the partition function. One of the oldest and hardest questions in partition theory is whether it is true, as it seems to be, that half of all partition numbers are odd and half even. This paper conjectures and proves a family of formulas that relate the density of odd values of various powers of the partition function, and derives some conditional bounding results on those densities.
The part-frequency matrices of a partition. A generalization of Glaisher's map to all partitions suggests a new statistic for working with partition bijections and congruences.
(Joint with Donald Kreher and Dalibor Fronček.) A note on nearly platonic graphs. One of the fun things about teaching lower-division courses is that sometimes a challenging question can still pop up. When I was teaching an undergraduate combinatorics course, I found myself asking whether it was possible to draw a graph that was platonic except that one face was a different degree. It turns out this is impossible, and not trivial to show!
Proof of Xiong's conjectured refinement of Euler's partition theorem. One of the first theorems a student of partitions learns is that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. This theorem has been refined in many ways over the centuries; this paper proves a conjecture in that direction which someone else advanced recently.
Major index over descent for pattern-avoiding permutations. An open question of bruce Sagan and collaborators is whether the major index of 321-avoiding permutations is distributed unimodally among permutations of a given length. In examining this question I noticed a fascinating formula which refines Stanley's q-analogue of the Frame-Robinson-Thrall formula for the major index of standard Young Tableaux by adding descents, at least for two-rowed permutations. Analysis of this formula could be an excellent route to proof of this conjecture.
In 2016-2017 I supervised the master's thesis of J. T. Davies in research permutation statistics. Mr. Davies is currently in doctoral study at the University of Waterloo in Canada. Our motivating question: the major index is symmetric over some sets of pattern-avoiding permutations in Sn with fixed descent number, and (maj, des) form a Mahonian pair. Are there conditions analogous to pattern avoidance (and hopefully equally interesting) for other pairs such as (den, exc) which are known to be Mahonian but are not distributed symmetrically over pattern-avoidance classes?
If graduate students are interested in research with me they should let me know. I am presently open to becoming a graduate advisor for a new student.
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.) My most immediate current project is extending the refinement of Stanley's formula listed given in the last listed paper. I think it would be an exciting result if this could be generalized to standard Young tableaux of any shape.
2.) I am interested in m-regular partitions, especially their low-modulus congruences. Related to this, I would like to show properties of singular overpartitions related to known theorems such as the Pak-Postnikov (m,c) theorem.
3.) I have recently been studying Kanade and Russell's very curious conjectures on asymmetric versions of the Göllnitz-Gordon theorem.
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.