Specialty Seminar in Partition Theory, q-Series and Related TopicsHost: 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 Spring Semester of 2022, the Seminar is running at Zoom address https://michigantech.zoom.us/j/87405452604 on Thursdays from 2:00 - 2:50pm US Eastern Daylight Time (-4 UTC), 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).

Spring 2022

Date: Jan 20 2022 | Title: Statistical distributions on integer partitions |

Speaker: Ken Ono | Abstract: In this talk we examine three types of distributions on integer partitions.
(1) Generalizing a classical theorem of Erdos and Lehner, we determine the distribution of parts in partitions that are multiples of a fixed integer
(2) For a fixed positive integer
(3) For a fixed integer
This is joint work with Michael Griffin, Larry Rolen, and Wei-Lun Tsai. |

Affiliation: Univ. of Virginia | |

Video: | Video |

Slides: | Ken Ono slides |

Date: Jan 27 2022 | Title: A Conjecture on Seaweed Algebras and Partitions |

Speaker: William Craig | Abstract: In 2018 Coll, Mayers and Mayers initiated the study of the index statistic of seaweed algebras from a partition-theoretic point of view. Within their framework, they conjecture that the difference between the number of partitions into odd parts having odd index and those with even index has the peculiar generating function (q, -q. We prove this conjecture using a variation of the circle method and Euler-Maclaurin summation.^{3}; q^{4})_{∞}^{-1} |

Affiliation: Univ. of Virginia | |

Video: | Video |

Slides: | William Craig slides |

Date: Feb 10 2022 | Title: Partition identities with eventually periodic residue conditions |

Speaker: Kağan Kurşungöz | Abstract: The majority of integer partition identities relate gap conditions to residue conditions. Gap conditions control how many times a part can appear in the partitions, and residue conditions stipulate which parts can appear without a bound on the number of repetitions. Kanade and Russell in 2015 developed a computerized method to discover partition identities, and made a fundamental change in our paradigm. In a PURE project in the summer of 2021, students implemented a variant of Kanade and Russell's IdentityFinder to discover partition identities with a novel theme, namely "eventually periodic residue conditions". Most of these identities can be proven by elementary means, some of them require transformations to be recognized as corollaries of known results. There is one new construction of an evidently positive generating function due to a lucky mistake. This is the preliminary report on joint work with Salih Numan Büyükbaş, Vahit Alp Hıdıroğlu, and Ömer Surhay Kocakaya, our graduate students. |

Affiliation: Sabancı Üniversitesi | |

Video: | Video |

Slides: | Kağan Kurşungöz slides |

Date: Feb 17 2022 | Title: Construction of Evidently Positive Series and An Alternative Construction for a Family of Partition Generating Functions due to Kanade and Russell |

Speaker: Halime Ömrüuzun Seyrek | Abstract: We give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our alternative construction, we use ordinary partitions instead of jagged partitions. We also present new generating functions which are evidently positive series. To obtain those generating functions, we first construct an evidently positive series for a key infinite product. In that construction, a series of combinatorial moves is used to decompose an arbitrary partition into a base partition together with some auxiliary partitions that bijectively record the moves.
This is a joint work with Prof. Kağan Kurşungöz. |

Affiliation: Sabancı Üniversitesi | |

Video: | Video |

Slides: | Halime Ömrüuzun Seyrek slides |

Date: Mar 3 2022 | Title: Generalizations and refinements of Schmidt's Theorem |

Speaker: William J. Keith | Abstract: Recently, George Andrews spoke in this Seminar on the theorem of Schmidt that the statistic summing only odd parts of partitions into distinct parts has the same distribution as the ordinary weight on all partitions. We show that this theorem can be generalized to partitions into parts appearing fewer than m times, and counting parts in various places modulo m, yielding tidy generating functions for colored partitions. This count can then be further refined by the number of parts of each color appearing. Additional questions present themselves for further investigation.This is joint work with George Andrews. |

Affiliation: Michigan Tech | |

Video: | Video |

Slides: | William Keith slides |

Date: Mar 10 2022 | Title: Linked partition ideals and Schur's 1926 partition theorem |

Speaker: Shane Chern | Abstract: Issai Schur's famous 1926 partition theorem states that the number of partitions of n into distinct parts congruent to ± 1 modulo 3 is the same as the number of partitions of n such that every two consecutive parts have difference at least 3 and that no two consecutive multiples of 3 occur as parts. In this talk, we consider some variants of Schur's theorem, especially their Andrews-Gordon type generating functions, from the perspective of span one linked partition ideals introduced by George Andrews. Our investigation has interesting connections with basic hypergeometric series, q-difference equations, computer algebra, and so on. |

Affiliation: Dalhousie University | |

Video: | Video |

Slides: | Shane Chern slides |

Date: Mar 24 2022 | Title: Completing the A Andrews–Schilling–Warnaar identities_{2} |

Speaker: Matthew Russell | Abstract: We study the Andrews–Schilling–Warnaar sum-sides for the principal characters of standard (i.e., integrable, highest weight) modules of A. These characters have been studied recently by various subsets of Corteel, Dousse, Foda, Uncu, Warnaar and Welsh. We prove complete sets of identities for moduli 5 through 8 and 10, in Andrews–Schilling–Warnaar form. The cases of moduli 6 and 10 are new. Our methods depend on the Corteel–Welsh recursions governing the cylindric partitions and on certain relations satisfied by the Andrews–Schilling–Warnaar sum-sides. We speculate on the role of the latter in the proofs of higher modulus identities. Further, we provide a complete set of conjectures for modulus 9. We show that at any given modulus, a complete set of conjectures may be deduced using a subset of "seed" conjectures. These seed conjectures are obtained by appropriately truncating conjectures for the "infinite" level. Additionally, for moduli _{2}^{(1)}3k, we use an identity of Weierstraß to deduce new sum-product identities starting from the results of Andrews–Schilling–Warnaar. Joint work with Shashank Kanade. |

Affiliation: UIUC | |

Video: | Video |

Slides: | Matthew Russell slides |

Date: Apr 14 2022 | Title: Copartitions and Their Connections to Classical Partition-Theoretic Objects |

Speaker: Dennis Eichhorn | Abstract: Copartitions are a generalization of partitions with connections to many classical topics in partition theory, including two-colored partitions, Rogers-Ramanujan partitions, partitions with parts separated by parity, and crank statistics. In this talk, we introduce copartitions and focus on these connections. This talk is joint work with Hannah Burson. |

Affiliation: UC-Irvine | |

Video: | Video |

Slides: | Dennis Eichhorn slides |

Date: Apr 21 2022 | Title: Copartitions: Parity and Positivity |

Speaker: Hannah Burson | Abstract: Copartitions are a generalization of partitions with connections to a wide range of partition-theoretic topics. Additionally, they provide combinatorial interpretations for the coefficients in the q-series expansions of a family of infinite products. In this talk, we discuss properties of those coefficients, with a special focus on what we know about parity and positivity. Though this talk can be thought of as a sequel to the talk last week, all needed definitions will be restated and no previous exposure to copartitions is necessary. This talk is joint work with Dennis Eichhorn. |

Affiliation: Univ. of Minn. | |

Video: | Video |

Slides: | Hannah Burson slides |

Date: Apr 28 2022 | Title: A bijective proof and generalization of the non-negative crank-odd mex identity |

Speaker: Isaac Konan | Abstract: Recent works of Andrews--Newman and Hopkins--Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer n, there are as many partitions of n with non-negative crank as partitions of n with odd mex. In this talk, we provide a generalization of this identity and prove it bijectively. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee. |

Affiliation: Université Claude Bernard Lyon 1 | |

Video: | Video |

Slides: | Isaac Konan slides |

Fall 2021

Date: Sep 16 2021 | Title: Combinatorial Perspectives on Dyson's Crank and the Mex of Partitions |

Speaker: Brian Hopkins | Abstract: 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 2021 | Title: 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 2021 | Title: Weight 3/2 moonshine for the Thompson group |

Speaker: Maryan Khaqan | Abstract: 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. |

Affiliation: Stockholm University | |

Video: | Video |

Slides: | |

Date: Oct 7 2021 | Title: |

Speaker: | Abstract: |

Affiliation: | |

Video: | |

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Date: Oct 14 2021 | Title: On new modulo 8 cylindric partition identities |

Speaker: Ali Uncu | Abstract: 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 2021 | Title: Product-sum identities from certain restricted plane partitions |

Speaker: Walter Bridges | Abstract: 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 | |

Video: | Video |

Slides: | Walter Bridges slides |

Date: Oct 28 2021 | Title: Generating Functions for Certain Weighted Cranks(joint work with Ae Ja Yee) |

Speaker: Shreejit Bandyopadhyay | Abstract: 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 | |

Video: | Video |

Slides: | Shreejit Bandopadhyay slides |

Date: Nov 4 2021 | Title: Proofs of Berkovich and Uncu's Conjectures on Integer Partitions using Frobenius Numbers |

Speaker: Damanvir Singh Binner | Abstract: We use techniques from elementary number theory (such as Frobenius numbers) to combinatorially prove four recent conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) regarding inequalities between the sizes of two closely related sets consisting of integer partitions whose parts lie in the interval {s, ..., L + s}. Further restrictions are placed on the sets by specifying impermissible parts as well as a minimum part. |

Affiliation: IISER Mohali | |

Video: | Video |

Slides: | Damanvir Singh Binner slides |

Date: Nov 11 2021 | Title: Plane Partition Congruences and Localization |

Speaker: Nicolas Smoot | Abstract: George Andrews and Peter Paule recently conjectured an infinite family of congruences mod powers of 3 for the 2-elongated plane partition function d. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$. This is a superb example of the utility of localization techniques to proving partition congruences._{2}(n) |

Affiliation: RISC-Linz | |

Video: | Video |

Slides: | Nicolas Smoot slides |

Date: Nov 18 2021 | Title: Compositions from Ferrers Diagrams |

Speaker: George Beck | Abstract: Think of the dots of the Ferrers diagram of an integer partition of n as a subset F of the lattice points in the plane. Let L be the set of lines l of rational slope s passing through the convex hull of F and a lattice point. Count the points of F that lie on each line of L (starting with the line with greatest y intercept) to form a composition of n. We present conjectures about the enumeration of various statistics for sets of such compositions. |

Affiliation: Wolfram Research | |

Video: | Video |

Slides: | Pdf of talk notebook, Executable talk notebook, CORE 6 notebook with auxiliary functions |

Date: Dec 2 2021 | Title: Semi-Modular Forms |

Speaker: Robert Schneider | Abstract: We introduce our developing theory of a new class of functions that complement modular forms, that are "half-modular" in a specific sense. Using variations on classical Eisenstein series involving integer partitions and also Lucas sequences, we construct first examples of semi-modular forms, and conjecture what other examples might look like. (Joint work with A. P. Akande and M. Just) |

Affiliation: Univ. of Georgia | |

Video: | Video |

Slides: | Robert Schneider slides |

Date: Dec 9 2021 | Title: Congruences for k-elongated partition diamonds |

Speaker: James Sellers | Abstract: In 2007, George Andrews and Peter Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken k-diamond partitions. On the way to broken k-diamond partitions, Andrews and Paule introduced the idea of k-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of k-elongated partition diamonds in a paper that is to appear in the Journal of Number Theory. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers d produced by summing the links of _{k}(n)k-elongated plane partition diamonds of length n is given by (q for each ^{2};q^{2})_{∞}^{k}/(q;q)_{∞}^{3k+1}k ≥ 1. A significant portion of their recent paper involves proving several congruence properties satisfied by d, _{1}d and _{2}d, using modular forms as their primary proof tool. Since then, Nicolas Smoot has extended the work of Andrews and Paule, refining one of their conjectures and proving an infinite family of congruences modulo arbitrarily large powers of 3 for the function _{3}d.
_{2}In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions k. The proof techniques employed are all elementary, relying on generating function manipulations and classical q-series results.
This is joint work with Robson da Silva of Universidade Federal de Sao Paulo (Brazil). |

Affiliation: UMN-Duluth | |

Video: | Video |

Slides: | James Sellers slides |

Spring 2021

Date: Feb 18 2021 | Title: Identities of Hecke type and Rogers-Ramanujan type |

Speaker: Shane Chern | Abstract: 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 2021 | Title: 5-adic Convergence Over Modular Curves of Genus 1: The Andrews--Sellers Congruences and Beyond |

Speaker: Nicolas Smoot | Abstract: 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 2021 | Title: Under construction: a multiplicative theory of integer partitions |

Speaker: Robert Schneider | Abstract: Pdf link |

Affiliation: Univ. of Georgia | |

Video: | Video |

Slides: | Robert Schneider slides, extended notes |

Date: Mar 18 2021 | Title: The Unimodularity of Gaussian polynomials $N+m\brack m$ for a few small values of $m$ |

Speaker: Brandt Kronholm | Abstract: 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. |

Affiliation: Univ. of Texas-RGV | |

Video: | Video |

Slides: | Brandt Kronholm slides |

Date: Mar 25 2021 | Title: Distribution of partition statistics in arithmetic progressions |

Speaker: Ken Ono | Abstract: 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 2021 | Title: Sequentially Congruent Partitions and Partitions into Squares |

Speaker: James Sellers | Abstract: 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 2021 | Title: Partition Eisenstein series and semi-modular forms |

Speaker: Matt Just | Abstract: We identify a class of "semi-modular" forms invariant on special subgroups of GL, 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._{2}(Z) |

Affiliation: UGA | |

Video: | Video |

Slides: | Matthew Just slides |

Date: Apr 15 2021 | Title: A poset of generating functions of partitions of n |

Speaker: Tim Wagner | Abstract: In this talk, we present our work on a novel poset: P, the poset of generating functions of partitions of _{n} ={G_{λ} | λ ⊢ n}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 G (in the ordering of _{λ}P). This imposes a strong necessary condition on the maxima of _{n}P. We conjecture an asymptotic value of |_{n}P|, and show that determining |_{n}P| 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._{n} |

Affiliation: MTU | |

Video: | Video |

Slides: | Tim Wagner slides |

Date: Apr 22 2021 | Title: |

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Date: Apr 29 2021 | Title: A bijection for partitions into parts simultaneously s-regular and t-distinct |

Speaker: William J. Keith | Abstract: 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 2021 | Title: Gupta, Ramanujan, Dyson, and Ehrhart: Formulas for Partition Functions, Congruences, Cranks, and Polyhedral Geometry |

Speaker: Joselyne Rodriguez | Abstract: 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 |