Seminar in Partition Theory, qSeries and Related Topics Host: Michigan Technological University Department of Mathematical Sciences 
Date: September 7 2023  Title: Partition Fixed Points: Connections, Generalizations, and Refinements 
Speaker: Brian Hopkins  Abstract: Last year, Blecher and Knopfmacher introduced the idea of a fixed point to integer partitions, such as those with second part 2. For partitions presented in nonincreasing order, they made a conjecture about the number with or without a fixed point. We confirmed their conjecture by connecting fixed points to a growing network of related partition statistics: the Frobenius symbol, Dyson's crank, and the mex (minimal excludant). For generalizations, shifting the fixed point criterion to, e.g., second part 4, gives a family of results connected to the crank and related to a generalized mex defined by the speaker, Sellers, and Stanton. For refinements, the integer triangle determined by counting partitions with a particular fixed point has many interesting patterns including analogues of Pascal's lemma and identities for both diagonal and antidiagonal sums. Much of this is joint work with James Sellers. 
Affiliation: St. Peter's  
Video:  Video 
Slides:  Brian Hopkins slides 
Date: September 14 2023  Title: A Decomposition of Cylindric Partitions and Cylindric Partitions into Distinct Parts 
Speaker: Kağan Kurşungöz  Abstract: We show that cylindric partitions are in onetoone correspondence with a pair which has an ordinary partition and a colored partition into distinct parts. Then, we show the general form of the generating function for cylindric partitions into distinct parts and give some examples. We prove part of a conjecture by Corteel, Dousse, and Uncu. The approaches and proofs are elementary and combinatorial.
This is joint work with Halime Ömrüuzun Seyrek. The arxiv preprint is at https://arxiv.org/abs/2308.14514 , some updates due to updates in Warnaar's work will be incorporated in the near future. 
Affiliation: Sabancı Üniversitesi  
Video:  Video 
Slides:  Kağan Kurşungöz slides Kağan Kurşungöz slides, annotated 
Date: September 21 2023  Title: Cultivating Maple and Sage in Ramanujan's Garden 
Speaker: Frank Garvan  Abstract: This talk is on experimental math and how we can use Maple and Sage to gain insight into various rank and crank type functions. Includes some joint work with Rishabh Sarma. 
Affiliation: Univ. of Florida  
Video:  Video 
Slides:  Frank Garvan slides (pdf) Frank Garvan slides, raw Maple workbook 
Date: September 28 2023  Title: Infinite families of congruences modulo powers of 2 for partitions into odd parts with designated summands 
Speaker: James Sellers  Abstract: In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd. Recently, Herden, Sepanski, Stanfill, Hammon, Henningsen, Ickes, and Ruiz proved a number of Ramanujanlike congruences for the function PDO(n) which counts the number of partitions of weight n with designated summands wherein all parts must be odd. In the first part of this talk, we will use truly elementary techniques to prove some of the results conjectured by Herden, et. al. by proving the following two infinite families of congruences satisfied by PDO(n): For all k ≥ 0 and n ≥ 0, PDO(2^{k}(4n+3)) ≡ 0 (mod 4) and PDO(2^{k}(8n+7)) ≡ 0 (mod 8). We will then transition to a discussion of the following unexpected infinite family of internal congruences: For all k ≥ 0 and all n ≥ 0, PDO(2^{2k+3}n) ≡ PDO(2^{2k+1}n) (mod 2^{2k+3}). This family of results is proven via generating function dissections, various modular relations and recurrences involving a Hauptmodul on the classical modular curve X_{0}(6), and an induction argument which provides the final step in proving the necessary divisibilities. This latter result is joint work with Shane Chern, Dalhousie University. 
Affiliation: UMNDuluth  
Video:  Video 
Slides:  James Sellers slides 
Date: October 5 2023  Title: New Infinite Hierarchies Of Polynomial Identities Related To The Capparelli Partition Theorems 
Speaker: Ali Uncu  Abstract: We will talk about the polynomial refinements of Capparelli's identities and the beautiful infinite families of identities they give rise to. This is joint work with Alexander Berkovich. 
Affiliation: Austrian Acad. of Sciences Univ. of Bath  
Video:  Video 
Slides:  Ali Uncu slides 
Date: October 12 2023  Title: Methods in asymptotic statistics for partitions 
Speaker: Walter Bridges  Abstract: Choose a partition of a large number uniformly at random. How big do we expect the largest part to be? How many 1's are there? What does the Young diagram look like? For that matter, how can we generate large partitions efficiently, to collect data and make conjectures? In the first part of this talk, we will give an overview of the socalled Boltzmann model for partitions, a technique developed in the 90s by Fristedt, Pittel, Vershik and others for answering the above questions. Time permitting, we will sketch proofs of the distribution of small parts, the largest part, limit shapes for Young diagrams, and even the asymptotic formula for p(n), the number of partitions of n. In the second part of this talk, we discuss the speaker's adaptation of these techniques to unimodal sequences and partitions with gap conditions. This is joint work with Kathrin Bringmann. 
Affiliation: Universität zu Köln  
Video:  Video 
Slides:  Walter Bridges slides 
Date: October 19 2023 8pm US EDT  Title: Bressoud's Conjecture on the RogersRamanujan Identities 
Speaker: Kathy K. Q. Ji  Abstract: In 1980, Bressoud put forward a conjecture for a more general partition identity that implies many classical results in the theory of partitions. The general case for j=1 was recently resolved by Kim. In this talk, we present an answer to Bressoud's conjecture for the case j=0, resulting in a complete solution to the conjecture. It is somewhat unexpected that overpartitions play a crucial role in this regards.

Affiliation: Tianjin University  
Video:  Video 
Slides:  Kathy Q. Ji slides 
Date: October 26 2023  Title: Neighborly partitions, hypergraphs and Gordon's identities 
Speaker: Pooneh Afsharijoo  Abstract: The family of Gordon's identities is an important family of partition identities which generalizes the famous RogersRamanujan identities. We prove a family of partition identities which is "dual" in a some sense (that we will explain) to Gordon's identities. These new identities make intervene a new type of partitions (the (r, i)Neighborly partitions) and the hypergraphs (generalization of graphs) associated with them. This is a joint work with H. Mourtada; it generalizes a recent work by Z. Mohsen and H. Mourtada in which they give new partition identities which are dual to those of RogersRamanujan. 
Affiliation: Univ. Complutense Madrid  
Video:  Video 
Slides:  Pooneh Afsharijoo slides 
Date: November 2 2023 10am US EDT  Title: Asymptotic and exact formulas in the theory of partitions 
Speaker: Kathrin Bringmann  Abstract: In this talk I will explain asymptotics of various partition statistic and how modularity helps to show such behaviors. Besides the classical modular forms I will also speak about mock theta functions and false theta functions.

Affiliation: Universität zu Köln  
Video:  Video 
Slides:  Kathrin Bringmann slides 
Date: November 9 2023  Title: Ncolored generalized Frobenius partitions: Generalized Kolitsch identities 
Speaker: Zafer Selcuk Aygin  Abstract: We establish an asymptotical formula for the number of Ncolored generalized Frobenius partitions of n in terms of the partition function. These results are from a joint work with Professor Khoa Dang Nguyen of the University of Calgary.

Affiliation: Northwestern Polytechnic  
Video:  Video 
Slides:  Zafer Selcuk Aygin slides 
Date: November 30 2023  Title: dFold Partition Diamonds: Generating Functions and Partition Analysis 
Speaker: Dalen Dockery  Abstract: This is the first in a sequence of two talks concerning dfold partition diamonds. These combinatorial objects generalize both classical partitions and the plane diamond partitions introduced by Andrews, Paule, and Riese in 2001, as well as their Schmidt type counterparts (obtained by only using parts of certain indices). We use MacMahon's theory of partition analysis to find generating functions for this family of partitions. Along the way, we prove a result that extends several of MacMahon's elimination formulae to an arbitrary number of terms. To conclude this talk, we simplify the generating functions in the case of Schmidt type partitions by appealing to a result of Euler on Eulerian polynomials. This is joint work with Marie Jameson, James A. Sellers, and Samuel Wilson. Twopart series with James Sellers. 
Affiliation: UTKnoxville  
Video:  Video 
Slides:  Dalen Dockery slides 
Date: December 7 2023  Title: Arithmetic Properties of dfold Partition Diamonds 
Speaker: James Sellers  Abstract: This is the second in a sequence of two talks concerning dfold partition diamonds and serves as a continuation of Dalen Dockery's talk from the previous week. We will begin with a brief reminder of the topic, including the generating functions involved. Once that review is complete, we will use the generating functions in question to provide elementary proofs of infinitely many Ramanujanlike congruences satisfied by dfold partition diamonds. This is joint work with Dalen Dockery, Marie Jameson, and Samuel Wilson. Twopart series with Dalen dockery. 
Affiliation: UMNDuluth  
Video:  
Slides:  
Date: January 19 2023  Title: Selfconjugate 6cores and quadratic forms 
Speaker: Marie Jameson  Abstract: We will analyze the behavior of the selfconjugate 6core partition numbers sc_{6}(n) by utilizing the theory of quadratic and modular forms. In particular, we explore when sc_{6}(n) > 0. Positivity of sc_{t}(n) has been studied in the past, and Hanusa and Nath conjectured that sc_{6}(n) > 0 except when n ∈ {2, 12, 13, 73}. Assuming the Generalized Riemann Hypothesis, we are able to settle Hanusa and Nath's conjecture. 
Affiliation: UTKnoxville  
Video:  Video 
Slides:  Marie Jameson slides 
Date: January 26 2023  Title: Forgotten conjectures of Andrews for Nahmtype sums 
Speaker: Joshua Males  Abstract: In his famous '86 paper, Andrews made several conjectures on the function σ(q) of Ramanujan, including that it has coefficients (which count certain partitiontheoretic objects) whose sup grows in absolute value, and that it has infinitely many Fourier coefficients that vanish. These conjectures were famously proved by AndrewsDysonHickerson in their '88 Invent. paper, and the function σ has been related to the arithmetic of ℤ[√6] by Cohen (and extensions by Zwegers), and is an important first example of quantum modular forms introduced by Zagier.
A closer inspection of Andrews' '86 paper reveals several more functions that have been a little left in the shadow of their sibling σ, but which also exhibit extraordinary behaviour. In an ongoing project with Folsom, Rolen, and Storzer, we study the function v_{1}(q) which is given by a Nahmtype sum and whose coefficients count certain differences of partitiontheoretic objects. We give explanations of four conjectures made by Andrews on v_{1}, which require a blend of novel and wellknown techniques, and reveal that v_{1} should be intimately linked to the arithmetic of the imaginary quadratic field ℤ[√(3)]. 
Affiliation: Univ. of Manitoba  
Video:  Video 
Slides:  Joshua Males slides 
Date: February 2 2023  Title: RamanujanKolberg identities, regular partitions, and multipartitions 
Speaker: William J. Keith  Abstract: RamanujanKolberg identities relate subprogressions of the partition numbers and linear combinations of etaquotients, so named after Ramanujan's "most beautiful identity" ∑_{m ≥ 0} p(5m+4) q^{n} = 5 ∏_{n ≥ 1} (1q^{5n})^{5} / (1q^{n})^{6}. Descending from equality to congruence mod 2, recent work of the speaker and Fabrizio Zanello has produced a large number of these with implications for the study of the parity of the partition function, and ShiChao Chen has shown that these are part of an infinite family where the etaquotients required have a very small basis. Analyses of particular cases from that family yield many pleasing patterns: older work of the speaker and Zanello gave congruences for the mregular partitions for m odd, and this talk will be on more recent joint work which exhibits the very different behavior for m even which gives connections to multipartitions. All of these results in turn illuminate different aspects of the longstanding partition parity problem and hopefully provide some useful insight therein. This is joint work with Fabrizio Zanello. 
Affiliation: Michigan Technological University  
Video:  Video 
Slides:  William Keith slides 
Date: February 9 2023  Title: Some Legendre theorems for a class of partitions with early conditions 
Speaker: Darlison Nyirenda  Abstract: In this talk, we consider a subclass of partitions with early conditions, introduced by George E. Andrews. We present Legendre theorems associated with these partitions. The theorems provide combinatorial interpretation of some RogersRamanujan identities due to Lucy J. Slater. 
Affiliation: Univ. of the Witwatersrand  
Video:  Video 
Slides:  Darlison Nyirenda slides 
Date: February 23 2023  Title: A Unified Approach to Unimodality of Gaussian Polynomials 
Speaker: Ali K. Uncu  Abstract: In 2013, Pak and Panova proved the strict unimodality property of Gaussian Polynomials (a.k.a. qbinomial coefficients) , as polynomials in q. They showed it to be true for all ℓ, m ≥ 8 and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.
This is a joint work with Christoph Koutschan and Elaine Wong. For more data, see https://wongey.github.io/unimodality/. 
Affiliation: RICAMAAS  
Video:  Video 
Slides:  Ali Uncu slides 
Date: March 2 2023  Title: On some new "multisum = product" identities 
Speaker: Shashank Kanade  Abstract: Recently, in a joint work with Baker, Sadowski and Russell, motivated by vertexalgebraic considerations, certain new "multisum = product" identities emerged. Significantly, we found and proved a quadruplesum representation for Nandi's identities. Additionally, we found and proved a set of new identities where the mod10 products are the same as in level 4 principal characters of the affine Lie algebra D_{4}^{(3)}. (Recall that some of the earlier mod9 conjectures of the speaker jointly with Russell were about level 3 principal characters of the affine Lie algebra D_{4}^{(3)}.) I will explain these developments and give a sketch of the computational technique behind the proofs. 
Affiliation: Univ. of Denver  
Video:  Video 
Slides:  Shashank Kanade slides 
Date: March 16 2023  Title: Localization Applied to a Genus 1 Congruence Family. 
Speaker: Nicolas Smoot  Abstract: Over the last century, a large variety of infinite congruence families have been discovered and studied for different integer partition functions. These families exhibit a great variety with respect to the difficulty of their proof: some have been understood for decades, while others are still resistant to proof today. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0; as either number grows, the difficulty of the congruence family grows. In recent years we have developed a systematic method for handling families in which the genus is 0 but the cusp count is greater than 2. We will report on the successful extension of localization to a congruence family in the case of genus 1, for the 2elongated plane partition diamond counting function d_{2}(n) by powers of 7. We compare our method with other techniques for proving genus 1 congruence families, and discuss the implications regarding the overall classification of the subject. We finally conjecture a second congruence family for d_{3}(n) by powers of 7 which should be amenable to similar techniques. This is joint work with Koustav Banerjee. 
Affiliation: RISCLinz  
Video:  Video 
Slides:  Nicolas Smoot slides 
Date: March 23 2023  Title: Counting matrix points on curves and surfaces with partitions 
Speaker: Hasan Saad  Abstract: Partitions are ubiquitous in mathematics appearing in combinatorics, number theory, and representation theory. Here we discuss how partitions make it possible to count "matrix" points on curves and surfaces over finite fields. Using partition generating function formulas of Euler, Feit and Fine counted the number of n × n commuting matrices over finite fields. Building on this classical result, it is natural to count the number of "matrix" points over finite fields on curves and surfaces. In this talk we show how to employ partitions and Gaussian hypergeometric functions to count such n × n matrix points on elliptic curves and certain K3 surfaces. We then use these results to determine the SatoTate distribution of these points counts. 
Affiliation: Univ. of Virginia  
Video:  Video 
Slides:  Hasan Saad slides 
Date: March 30 2023  Title: General coefficientvanishing results associated with theta series 
Speaker: Shane Chern  Abstract: There are a number of sporadic coefficientvanishing results associated with theta series, which suggest certain underlying patterns. By expanding theta powers as a linear combination of products of theta functions, we present two strategies that will provide a unified treatment. Our approaches rely on studying the behavior of a product of two theta series under the action of the huffing operator. For this purpose, some explicit criteria are given. We may use the presented methods to not only verify experimentally discovered coefficientvanishing results, but also to produce a series of general phenomena. 
Affiliation: Dalhousie Univ.  
Video:  Video 
Slides:  Shane Chern slides 
Date: April 13 2023  Title: Combinatorial proofs and refinements of three partition theorems of Andrews 
Speaker: Shishuo Fu  Abstract: In his recent work, Andrews revisited twocolor partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these twocolor partitions with more familiar kinds of partitions, such as strict partitions and basis partitions. In this talk, we present bijective proofs as well as refinements of those three theorems of Andrews. Our refinements take into account the numbers of parts in each of the two colors. NOTE SPECIAL TIME: This talk is scheduled for 7pm US EDT, 5 hours later than usual, to accommodate a speaker from China. 
Affiliation: Chongqing Univ.  
Video:  Video 
Slides:  Shishuo Fu slides 
Date: Sep 1 2022  Title: Biases in parts among kregular and kindivisible partitions 
Speaker: Faye Jackson and Misheel Otgonbayar  Abstract: For integers k, t ≥ 2 and 1 ≤ r ≤ t, let D_{k}(r, t, n) and D_{k}^{×}(r, t, n) be the number of parts that are congruent to r mod t among partitions of n that are kregular (no parts repeating k or more times) and kindivisible (no parts divisible by k) respectively. Using the circle method, we find the asymptotic for each quantity as n tends to infinity. From this asymptotic, we find that in both kregular and kindivisible cases, the parts are weakly equidistributed among equivalence classes mod t. However, inspection of the lower order terms implies a bias towards the lower congruence classes for kregular partitions; that is, for 1 ≤ r < s ≤ t, we have D_{k}(r, t, n) ≥ D_{k}(s, t, n) for large n. We make this inequality explicit. In the kindivisible case however, the bias is much more chaotic due to the interaction of the multiplicative structures mod k and t. Most notably, while the kregular case matches with a simple combinatorial heuristic, there appears to be no such heuristic for the kindivisible partitions. 
Affiliation: Univ. of Virginia  
Video:  Video 
Slides:  Jackson and Otgonbayar slides 
Date: Sep 15 2022  Title: Elementary Proofs of Infinite Families of Congruences for Merca's Cubic Partitions 
Speaker: James Sellers  Abstract: In the September 2022 issue of the Ramanujan Journal, Mircea Merca published a paper whose focus was on the function A(n) which is defined to be the difference between the number of cubic partitions of n into an even number of parts and the number of cubic partitions of n into an odd number of parts. (Recall that a cubic partition is a partition wherein the even parts are allowed to appear in two different colors.)
Recently, using modular forms and Smoot's Mathematica implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all n ≥ 0,
A(9n+5) ≡ 0 mod 3, A(27n+26) ≡ 0 mod 3.
In this talk, we will provide elementary proofs of these two congruences via classical generating function manipulations. We then prove two infinite families of nonnested Ramanujanlike congruences modulo 3 satisfied by A(n) wherein Merca's original two congruences serve as the initial members of each family. This is joint work with Robson da Silva, Universidade Federal de Sao Paulo, Brazil. 
Affiliation: UMNDuluth  
Video:  Video 
Slides:  James Sellers slides 
Date: Sep 22 2022  Title: Bijective approaches for Schmidt type theorems 
Speaker: Hunter Waldron  Abstract: We study two bijections to find new Schmidt type theorems, which concern the equinumerosity of partitions with parts counted at only given indices, and other more well studied families of partitions. First, we show that Mork's bijection, originally given as a proof of Schmidt's theorem, is identical to an older bijection of Bessenrodt when applied to a 2modular diagram, which implies a Schmidt type refinement of Euler's theorem. We then develop an idea from a recent paper of Andrews and Keith to construct a bijection between partitions counted at the indices r, t+r, 2t+r, ... and tcolored partitions. 
Affiliation: MTU  
Video:  Video 
Slides:  Hunter Waldron slides 
Date: Sep 29 2022  Title: Hook Length and Symplectic Content in Partitions 
Speaker: Cristina Ballantine  Abstract: The irreducible polynomial representations of the general linear group are indexed by partitions. The dimension of a representation is given by a formula involving the hook lengths and the contents of cells in the Young diagram of the indexing partition. There are also analogous formulas for irreducible polynomial representations of symplectic and orthogonal groups. I will discuss the combinatorial nature of identities involving hooks and the symplectic (and orthogonal) contents. These have been conjectured by T. Amdeberhan and are inspired by the NekrasovOkounkov hooklength formula. Time permitting, I will also introduce some auxiliary results. The representation theory is mentioned only as motivation and I will not discuss it further in the talk. (Joint work with Tewodros Amdeberhan and George Andrews.) 
Affiliation: Holy Cross  
Video:  Video is available upon request; please email host or speaker. 
Slides:  Cristina Ballantine slides 
Date: Oct 6 2022  Title: Partitions, Kernels, and Localization 
Speaker: Nicolas Smoot  Abstract: Since Ramanujan's groundbreaking work, there has been a large variety of infinite congruence families for partition functions modulo prime powers. These families vary enormously with respect to the difficulty of proving them. We will discuss the application of the localization method to proving congruence families by walking through the proof of one recently discovered congruence family for the counting function for 5elongated plane partitions. In particular, we will discuss a critical aspect of such proofs, in which the associated generating functions of a given congruence family are members of the kernel of a certain linear mapping to a vector space over a finite field. We believe that this approach holds the key to properly classifying congruence families. 
Affiliation: RISCLinz  
Video:  Video 
Slides:  Nicolas Smoot slides 
Date: Oct 13 2022  Title: Combinatorial constructions of generating functions of cylindric partitions with small profiles into unrestricted or distinct parts 
Speaker: Halime Ömrüuzun Seyrek  Abstract: Cylindric partitions into profiles c=(1,1) and c=(2,0) are considered. The generating functions into unrestricted cylindric partitions and cylindric partitions into distinct parts with these profiles are constructed. The constructions are combinatorial and they connect the cylindric partitions with ordinary partitions. The generating function of cylindric partitions with the said profiles turn out to be combinations of two infinite products. This is a joint work with Kağan Kurşungöz. 
Affiliation: Sabancı Üniversitesi  
Video:  Video 
Slides:  Halime Ömrüuzun Seyrek slides 
Date: Nov 3 2022  Title: Finding odd values of p(n) 
Speaker: Catherine Cossaboom and Sharon Zhou  Abstract: One of the deepest open problems in the theory of partitions is the assertion that 50% of the values of p(n) are even (resp. odd). Very little is known in the direction of this conjecture. The best known lower bounds for the number of even (resp. odd) values relies on variants of Euler's classical recurrence for p(n), typically souped up with the theory of modular forms. In this talk, we will introduce a new framework that explicitly provides infinitely many such relations. These results follow from the phenomenon of Hecke nilpotency. 
Affiliation: Univ. of Virginia  
Video:  Video 
Slides:  Cossaboom and Zhou slides 
Date: Nov 10 2022  Title: Basic hypergeometric summation theorems with symmetry in four variables 
Speaker: Jonathan BradleyThrush  Abstract: The _{6}φ_{5} summation theorem of L.J. Rogers has a formulation, due to F.H. Jackson, which is symmetrical with respect to three of its parameters. I will reexamine the problem, first considered by Jackson, of extending Rogers's identity to one which possesses a fourfold symmetry. Analogous to Jackson's formulation of the _{6}φ_{5} summation theorem, there is a form of Bailey's _{6}ψ_{6} identity which is symmetrical in three parameters. Motivated by this, I will present a bilateral summation theorem with symmetry in four variables, giving an outline proof and briefly examining a few interesting special cases. The theory of WZ pairs will be employed to obtain further results; some partial fraction expansions will also be considered. I will conclude with the statements of a few open problems. 
Affiliation: Univ. of Florida  
Video:  Video 
Slides:  Jonathan BradleyThrush slides 
Date: Nov 17 2022  Title: Compositiontheoretic series in partition theory 
Speaker: Robert Schneider and Drew Sills  Abstract: We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are kgonal numbers; our proofs employ Ramanujan's theta functions. We explore applications to lacunary qseries, and to a new class of compositiontheoretic Dirichlet series. 
Affiliation: MTU and GSU  
Video:  Video 
Slides:  Sills slides and Schneider slides 
Date: Dec 1 2022  Title: Modulo d extension of parity results in RogersRamanujanGordon type overpartition identities 
Speaker: Kağan Kurşungöz  Abstract: Sang, Shi and Yee, in 2020, found overpartition analogs of Andrews' results involving parity in RogersRamanujanGordon identities. Their result partially answered an open question of Andrews'. The open question was to involve parity in overpartition identities. We extend Sang, Shi, and Yee's work to arbitrary moduli, and also provide a missing case in their identities. We also unify proofs of RogersRamanujanGordon identities for overpartitions due to Lovejoy and Chen et. al.; Sang, Shi, and Yee's results; and ours. Although verification type proofs are given for brevity, a construction of series as solutions of functional equations between partition generating functions is sketched. This is joint work with Mohammad Zadehdabbagh. 
Affiliation: Sabancı Üniversitesi  
Video:  Video 
Slides:  Kurşungöz slides and slides with written notes 
Date: Dec 8 2022  Title: Recent problems in partitions and other combinatorial functions 
Speaker: Larry Rolen  Abstract: In this talk, I will discuss recent work, joint with a number of collaborators, on analytic and combinatorial properties of the partition and related functions. This includes work on recent conjectures of Stanton, which aim to give a deeper understanding into the "rank" and "crank" functions which "explain" the famous partition congruences of Ramanujan. I will describe progress in producing such functions for other combinatorial functions using the theory of modular and Jacobi forms and recent connections with Lietheoretic objects due to GritsenkoSkoruppaZagier. I will also discuss how analytic questions about partitions can be used to study Stanton's conjectures, as well as recent conjectures on partition inequalities due to ChernFuTang and HeimNeuhauser, which are related to the NekrasovOkounkov formula. 
Affiliation: Vanderbilt University  
Video:  Video 
Slides:  Larry Rolen slides 
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 A. For example, the case of A=2 is the case of even parts in partitions of a large integer n. These limiting distributions are of Gumbel type, which are used to predict earthquakes! These results have implications in the algebraic geometry of n point Hilbert schemes.
(2) For a fixed positive integer t, we determine the distribution of the number of hook lengths of size t among the partitions of n. As n tends to infinity, the distributions are asymptotically normal.
(3) For a fixed integer t > 3, we determine the distribution of the number of hook lengths that are multiples of t among the partitions of n. As n tends to infinity, the distributions are asymptotically shifted Gamma distributions.
This is joint work with Michael Griffin, Larry Rolen, and WeiLun 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 partitiontheoretic 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^{3}; q^{4})_{∞}^{1}. We prove this conjecture using a variation of the circle method and EulerMaclaurin summation. 
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 AndrewsGordon 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, qdifference equations, computer algebra, and so on. 
Affiliation: Dalhousie University  
Video:  Video 
Slides:  Shane Chern slides 
Date: Mar 24 2022  Title: Completing the A_{2} AndrewsSchillingWarnaar identities 
Speaker: Matthew Russell  Abstract: We study the AndrewsSchillingWarnaar sumsides for the principal characters of standard (i.e., integrable, highest weight) modules of A_{2}^{(1)}. 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 AndrewsSchillingWarnaar 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 sumsides. 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 3k, we use an identity of Weierstraß to deduce new sumproduct identities starting from the results of AndrewsSchillingWarnaar. Joint work with Shashank Kanade. 
Affiliation: UIUC  
Video:  Video 
Slides:  Matthew Russell slides 
Date: Apr 14 2022  Title: Copartitions and Their Connections to Classical PartitionTheoretic Objects 
Speaker: Dennis Eichhorn  Abstract: Copartitions are a generalization of partitions with connections to many classical topics in partition theory, including twocolored partitions, RogersRamanujan 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: UCIrvine  
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 partitiontheoretic topics. Additionally, they provide combinatorial interpretations for the coefficients in the qseries 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 nonnegative crankodd mex identity 
Speaker: Isaac Konan  Abstract: Recent works of AndrewsNewman and HopkinsSellers 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 nonnegative 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 
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 infinitedimensional 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 
Slides:  
Date: Oct 14 2021  Title: On new modulo 8 cylindric partition identities 
Speaker: Ali Uncu  Abstract: We will discuss new sumproduct identities that emerged from the study of cylindric partitions. Cylindric partitions were defined by Gessel and Krattenthaler in 1997 in the context of nonintersecting lattice paths. These combinatorial objects later appeared naturally in many different contexts. Most recently, Corteel and Welsh rederived the A2 RogersRamanujan 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 multisumproduct identities. 
Affiliations: Univ. of Bath & OEAW RICAM  
Video:  Video 
Slides:  Ali Uncu slides 
Date: Oct 21 2021  Title: Productsum 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 productsum identities and an important combinatorial object.
We discuss an extension of CorteelWelsh'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 kcolored partitions, and investigate the generating function for this generalized crank weighted by the number of parts in the first subpartition of a kcolored 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) 263284) 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 2elongated plane partition function d_{2}(n). 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. 
Affiliation: RISCLinz  
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: SemiModular Forms 
Speaker: Robert Schneider  Abstract: We introduce our developing theory of a new class of functions that complement modular forms, that are "halfmodular" in a specific sense. Using variations on classical Eisenstein series involving integer partitions and also Lucas sequences, we construct first examples of semimodular 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 kelongated 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 kdiamond partitions. On the way to broken kdiamond partitions, Andrews and Paule introduced the idea of kelongated partition diamonds. Recently, Andrews and Paule revisited the topic of kelongated 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_{k}(n) produced by summing the links of kelongated plane partition diamonds of length n is given by (q^{2};q^{2})_{∞}^{k}/(q;q)_{∞}^{3k+1} for each k ≥ 1. A significant portion of their recent paper involves proving several congruence properties satisfied by d_{1}, d_{2} and d_{3}, 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 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 d_{k} for an infinite set of values of k. The proof techniques employed are all elementary, relying on generating function manipulations and classical qseries results. This is joint work with Robson da Silva of Universidade Federal de Sao Paulo (Brazil). 
Affiliation: UMNDuluth  
Video:  Video 
Slides:  James Sellers slides 
Date: Feb 18 2021  Title: Identities of Hecke type and RogersRamanujan type 
Speaker: Shane Chern  Abstract: In this talk, I will present several basic hypergeometric transformations from which dozens of identities of Hecke type and RogersRamanujan 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: 5adic Convergence Over Modular Curves of Genus 1: The AndrewsSellers Congruences and Beyond 
Speaker: Nicolas Smoot  Abstract: A major topic of interest in the theory of partitions is the study of infinite families of congruencesregular 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 padic 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 TexasRGV  
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 NekrasovOkounkov 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 thooks in FerrersYoung diagrams, we prove distributions which are often nonuniform. 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 semimodular forms 
Speaker: Matt Just  Abstract: We identify a class of "semimodular" forms invariant on special subgroups of GL_{2}(ℤ), which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisensteinlike series summed over integer partitions, and use it to construct families of semimodular forms. We ask whether other examples exist, and what properties they all share. 
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_{n} ={G_{λ}  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 G_{λ} (in the ordering of P_{n}). This imposes a strong necessary condition on the maxima of P_{n}. We conjecture an asymptotic value of P_{n}, and show that determining P_{n} exactly appears to be nontrivial. This we demonstrate by providing an infinite family of nonconjugate 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 29 2021  Title: A bijection for partitions into parts simultaneously sregular and tdistinct 
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 Ramanujanstyle partition congruences. We find and prove combinatorial witnesses, also known as cranks, for the congruences using polyhedral geometry. 
Affiliation: Univ. of TexasRGV  
Video:  Video 
Slides:  Joselyne Rodriguez slides 