Specialty Seminar in Partition Theory, q-Series and Related Topics
Host: Michigan Technological University
Department of Mathematical Sciences



This is the schedule and archival page for this multi-University Specialty Seminar. We will host the schedule of upcoming talks for the current semester, videos of previous talks, and slides when available. Previous semesters are found lower on the page.

In the Fall Semester of 2021, the Seminar is running at Zoom address https://michigantech.zoom.us/j/81840782304 on Thursdays from 12:20 - 1:10pm US Eastern Daylight Time (-4 GMT), with time before and after for chat and questions.

The current organizer of the Seminar is William J. Keith of Michigan Tech. If you would like to volunteer to speak, or contact him for any other purpose, you may reach him at wjkeith @ mtu . edu (without spaces).

Date: Sep 16 2021Title: Combinatorial Perspectives on Dyson's Crank and the Mex of Partitions
Speaker: Brian HopkinsAbstract: Recently there have been several connections established between Dyson's storied crank and a newer partition statistic, now called the mex: the least positive integer that is not a part of the partition. We will revisit and extend these results with an emphasis on combinatorial proofs. One highlight is a generating function expression for the number of partitions with a bounded crank that does not include an alternating sum, which leads to a combinatorial interpretation involving types of Durfee rectangles. This includes joint work with James Sellers, Dennis Stanton, and Ae Ja Yee.

Affiliation: Saint Peters University
Video: Video
Slides: Brian Hopkins slides
Date: Sep 23 2021Title: Schmidt Type partitions and modular forms (joint work with Peter Paule)
Speaker: George Andrews Abstract: In 1999, Frank Schmidt noted that the number of partitions of integers into distinct parts in which the first, third, fifth, etc. summands add to n is equal to p(n), the number of ordinary partitions of n. By invoking MacMahon's theory of Partition Analysis, we provide a context for this result which leads directly to many other theorems of this nature.

Affiliation: Penn State
Video: Video
Slides: George Andrews slides
Date: Sep 30 2021Title: Weight 3/2 moonshine for the Thompson group
Speaker: Maryan KhaqanAbstract: In recent work, we characterize all infinite-dimensional graded modules for the Thompson group whose graded traces are certain weight 3/2 weakly holomorphic modular forms satisfying special properties. This is an instance of moonshine for the Thompson group. In this talk, I will demonstrate how we can use one such module to study the ranks of certain families of elliptic curves. In particular, while number theory's contribution to moonshine is plentiful, this talk demonstrates an example of how moonshine can be used to answer questions in number theory.

SPECIAL TIME: Dr. Khaqan is connecting from Europe; this Seminar will be held 1 hour EARLIER than the usual time.

Affiliation: Stockholm University
Video: Video
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Date: Oct 7 2021Title:
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Date: Oct 14 2021Title: On new modulo 8 cylindric partition identities
Speaker: Ali UncuAbstract: We will discuss new sum-product identities that emerged from the study of cylindric partitions. Cylindric partitions were defined by Gessel and Krattenthaler in 1997 in the context of non-intersecting lattice paths. These combinatorial objects later appeared naturally in many different contexts. Most recently, Corteel and Welsh re-derived the A2 Rogers-Ramanujan identities originally proven by Andrews, Schilling and Warnaar using cylindric partitions. In their paper, they presented a general recurrence relation for cylindric partitions which can be applied to any class of such partitions. In a joint effort, the speaker, Corteel and Dousse studied a different class of cylindric partitions. This study led to the discovery of many intriguing multisum-product identities.

Affiliations: Univ. of Bath & OEAW RICAM
Video: Video
Slides: Ali Uncu slides
Date: Oct 21 2021Title: Product-sum identities from certain restricted plane partitions
Speaker: Walter BridgesAbstract: A cylindric partition is a sort of restricted plane partition that can be thought of as wrapping around the surface of a cylinder. A profile describes their shape, and Borodin proved that for every profile, the generating function is a (modular) infinite product. Recently, Corteel and Welsh proved systems of recurrences for cylindric partitions of all profiles and found sum generating functions that solve these in several cases, thus establishing new connections between product-sum identities and an important combinatorial object.
We discuss an extension of Corteel-Welsh's work to symmetric cylindric and double skew shifted plane partitions, which, thanks to recent work of Han and Xiong, also come with infinite product generating functions. This is joint work with Ali Uncu.

Affiliation: Universität zu Köln
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Date: Oct 28 2021Title: Generating Functions for Certain Weighted Cranks(joint work with Ae Ja Yee)
Speaker: Shreejit BandyopadhyayAbstract: Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this talk, we first consider the crank generating function weighted by the number of ones and obtain analytical formulas for this weighted crank function under conditions of the crank being less than or equal to some specific integer. We connect these functions to the generating functions of partitions with certain sizes of Durfee rectangles. We then consider a generalization of the crank for k-colored partitions, and investigate the generating function for this generalized crank weighted by the number of parts in the first subpartition of a k-colored partition. We show that the cumulative generating functions in this case are the same as the generating functions for certain unimodal sequences.

Affiliation: Penn State
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Date: Nov 11 2021Title:
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Date: Nov 18 2021Title:
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Date: Dec 2 2021Title:
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Date: Dec 9 2021Title:
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Spring 2021
Date: Feb 18 2021Title: Identities of Hecke type and Rogers-Ramanujan type
Speaker: Shane ChernAbstract: In this talk, I will present several basic hypergeometric transformations from which dozens of identities of Hecke type and Rogers-Ramanujan type would be deduced. The results come from my joint papers with Chun Wang.

Affiliation: Penn State
Video: Video
Slides: Shane Chern slides
Date: Feb 25 2021Title: 5-adic Convergence Over Modular Curves of Genus 1: The Andrews--Sellers Congruences and Beyond
Speaker: Nicolas SmootAbstract: A major topic of interest in the theory of partitions is the study of infinite families of congruences---regular patterns of divisibility of a given partition function by arbitrarily large powers of a given prime. In the century since Ramanujan's groundbreaking work on this subject, our understanding has grown substantially. A notable feature of this subject is the considerable range in the difficulty of proving a given congruence family. Possible complications include the genus of the underlying modular curve, issues in the representation of the associated family of modular functions, and more subtle matters involving piecewise p-adic convergence. In this talk we will discuss these complications, together with some recent techniques in proving congruence families, as well as the prospect of an algorithmic framework for studying them.

Affiliation: RISC
Video: Video
Slides: Nicolas Smoot slides
Date: Mar 4 2021Title: Under construction: a multiplicative theory of integer partitions
Speaker: Robert SchneiderAbstract: Pdf link

Affiliation: Univ. of Georgia
Video: Video
Slides: Robert Schneider slides, extended notes
Date: Mar 18 2021Title: The Unimodularity of Gaussian polynomials $N+m\brack m$ for a few small values of $m$
Speaker: Brandt KronholmAbstract: In this talk I will give a proof of the unimodularity of Gaussian polynomials $N+4\brack 4$, $N+3\brack 3$, and $N+2\brack 2$. The proof will come from an examination of a collection of generating functions for these Gaussian polynomials.

(TeX pdf abstract)

Affiliation: Univ. of Texas-RGV
Video: Video
Slides: Brandt Kronholm slides
Date: Mar 25 2021Title: Distribution of partition statistics in arithmetic progressions
Speaker: Ken OnoAbstract: Recent works at the interface of algebraic geometry, number theory, representation theory, and topology have provided decompositions of the set of size n integer partitions. The examples we consider arise from extensions of the Nekrasov-Okounkov and Han hook product formulas, and the study of Betti numbers of various n point Hilbert schemes. In analogy with Dirichlet's Theorem on primes in arithmetic progressions, we sort the size n partitions by congruence conditions on the relevant partition invariants. For the Hilbert schemes, we prove that homology is equidistributed for large n. For t-hooks in Ferrers-Young diagrams, we prove distributions which are often non-uniform. The cases where t ∈ {2, 3} stand out, as there are congruence classes where such counts are identically zero!

This is joint work with Kathrin Bringmann, Will Craig, and Joshua Males.

Affiliation: Univ. of Virginia
Video: Video
Slides: Ken Ono slides
Date: Apr 1 2021Title: Sequentially Congruent Partitions and Partitions into Squares
Speaker: James SellersAbstract: In recent work, Max Schneider and Robert Schneider studied a curious class of integer partitions called "sequentially congruent" partitions wherein the mth part is congruent to the (m+1)st part modulo m, and the smallest part is congruent to zero modulo the number of parts in the partition. After sharing a number of results that Schneider and Schneider proved regarding these sequentially congruent partitions, we will then prove that the number of sequentially congruent partitions of weight n is equal to the number of partitions of weight n where all parts are squares. We then show how our proof naturally generalizes to other classes of partitions where the parts are all kth powers for some fixed k. This is joint work with Robert Schneider and Ian Wagner.

Affiliation: Univ. of Minn.-Duluth
Video: Video
Slides: James Sellers slides
Date: Apr 8 2021Title: Partition Eisenstein series and semi-modular forms
Speaker: Matt JustAbstract: We identify a class of "semi-modular" forms invariant on special subgroups of GL2(Z), which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms. We ask whether other examples exist, and what properties they all share.

Affiliation: UGA
Video: Video
Slides: Matthew Just slides
Date: Apr 15 2021Title: A poset of generating functions of partitions of n
Speaker: Tim WagnerAbstract: In this talk, we present our work on a novel poset: Pn ={Gλ | λ ⊢ n}, the poset of generating functions of partitions of n. The motivation behind this poset is an intriguing and beautiful conjecture by François Bergeron. Our strongest result on this poset is that two "balancing" operations on the principal hooks of a partition λ produce generating functions at least as large as Gλ (in the ordering of Pn). This imposes a strong necessary condition on the maxima of Pn. We conjecture an asymptotic value of |Pn|, and show that determining |Pn| exactly appears to be nontrivial. This we demonstrate by providing an infinite family of non-conjugate pairs of partitions that have the same generating function. Finally, we discuss asymptotic results on the number of maxima in this poset.

Affiliation: MTU
Video: Video
Slides: Tim Wagner slides
Date: Apr 22 2021Title:
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Date: Apr 29 2021Title: A bijection for partitions into parts simultaneously s-regular and t-distinct
Speaker: William J. KeithAbstract: We consider partitions simultaneously satisfying two different restrictions from the classic theory. We construct a bijection which realizes the main symmetry of the class and ask about its characteristics, and whether the bijection can be improved. We are particularly interested in the place of Glaisher's map in the problem. This is work in progress and ideas are welcome.

Affiliation: Michigan Tech
Video: Video
Slides: William Keith slides
Date: May 6 2021Title: Gupta, Ramanujan, Dyson, and Ehrhart: Formulas for Partition Functions, Congruences, Cranks, and Polyhedral Geometry
Speaker: Joselyne RodriguezAbstract: We will revisit Gupta's 1975 result regarding properties of formulas for restricted partitions with the goal of generalizing the result. We will then use this result to prove an infinite family of Ramanujan-style partition congruences. We find and prove combinatorial witnesses, also known as cranks, for the congruences using polyhedral geometry.

Affiliation: Univ. of Texas-RGV
Video: Video
Slides: Joselyne Rodriguez slides