William J. Keith
Michigan Tech University Math Department
Associate Professor


Teaching

My courses for the Fall of 2024 are:

MA 2330, Section R02    Introduction to Linear Algebra   9:00am - 10:00am, M W F    Fisher Hall 138    Canvas page
MA 4209, Section R01    Combinatorics and Graph Theory   12:00pm - 1:00pm, M W F    EERC 314    Canvas page

My regular office hours are Monday and Wednesday 3pm - 4pm. My office is Fisher 314. Should students have class during these hours theyy are welcome to make an appointment for a more convenient time to meet. I also have a Zoom office if students prefer, https://michigantech.zoom.us/j/7738766078, where I can take appointments.
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 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.

Service

I currently serve on the Undergraduate Committee and coordinate assessment activities for the Department. I am also the alternate faculty Senator for the Math Department.

Here is the Partitions Seminar schedule and archival page for the current and previous semesters, with schedules, and videos and slides, of talks back to its start in Spring 2021. This Seminar is currently active.

Here is the Algebra and Combinatorics Seminar schedule for previous semesters, with schedules back to the Spring of 2013. The Seminar is no longer running.


Research

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 which includes a full publication list. 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, although for the moment I consider myself fully-engaged with advisees.

Selected Publications and Preprints

  • (Joint w/ George Andrews.) Schmidt-type theorems for partitions with uncounted parts. Schmidt's theorem states that the number of partitions into distinct parts, wherein we only count partitions in odd-indexed places, has the same generating function as the normally-weighted partitions. This theorem can be substantially generalized, to periodic but otherwise arbitrary subsets of parts being counted, being identified with sets of colored partitions or multipartitions. This may be a useful technique for establishing sum-product q-series identities.

  • Families of major index distributions: closed forms and unimodality. Proving that the major index of 321-avoiding permutations is distributed unimodally among permutations of a given length spurs observations on potential refinements of Stanley's q-analogue of the Frame-Robinson-Thrall formula for the major index of standard Young Tableaux by adding descents. These turn out to be connected to Schur polynomials, possibly in deeper ways than have currently been discovered.

  • 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.

  • 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.

  • A Ramanujan congruence analogue for Han's hook-length formula mod 5, and other symmetries. 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!

  • 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.


    Graduate Students

    In 2016-2017 I supervised the master's thesis of J. T. Davies in permutation statistics. 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?

    In 2020-2021 I supervised the master's thesis of Emily Anible in the distribution of major index over tableaux with fixed descent number. Our motivating question: if we fix the descent number and ask for the distribution of the major index over all standard Young tableaux of a given shape, it seems that in many cases some beautiful and combinatorially interesting formulas arise. A full restriction giving such a formula would refine Stanley's q-Frame-Robinson-Thrall formula.

    I am currently overseeing the doctoral work of Hunter Waldron, who has just submitted his first paper on Schmidt-type theorems.

    I am currently overseeing the doctoral work of Philip Cuthbertson.

    I presently consider myself fully-engaged for graduate student advising.


    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.) Schmidt-type theorems, such as those discussed by Mr. Waldron and in my recent submission with George Andrews.

    2.) Extending the refinement of Stanley's formula listed given in the maj-descent paper. I think it would be an exciting result if this could be generalized to standard Young tableaux of any shape.

    3.) 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.

    4.) Dousse and Kim's conjectures on unimodality for the overpartition analogue of the Gaussian coefficients seems quite interesting to me.

    5.) Bergeron's ad - bc conjecture.