Number Theory Seminar

George Andrews

Posted on in Number Theory Seminar

Partition Identities for k-Regular Partitions with Distinct Parts

When/Where:

April 12, 2022, 1:55 — 2:45 pm at LIT 368.

Abstract:

We start with a little-known Euler type theorem (due to Alladi) which is the following:

The number of partitions of n into distinct parts not divisible by k (i.e. k-regular partitions with distinct parts) equals the number of partitions of n into odd parts none repeated more than k-1 times.

k=1 and 2 are tautologies.  k=3 plays a prominent role in Schur’s 1926 partition theorem.  Both Alladi and Schur have further partition identities related to k=2 which we will discuss.  Obviously, k = infinity is Euler’s theorem.  We then proceed to k=4 where an empirical investigation leads to a result for overpartitions.  We conclude with a proof of the k=4 case and look at results and possibilities for k>4.

 

 

 

 

Ae Ja Yee

Posted on in Number Theory Seminar

Overpartition analogues of generalized Rogers-Ramanujan type partitions

When/Where:

April 5, 2022, 1:55 — 2:45 pm on Zoom

Abstract:

In the past two decades, there have been a lot of research centered around overpartitions, some of which concern overpartition analogues of Rogers–Ramanujan type identities. In this talk, I will present Rogers–Ramanujan type overpartition identities by considering Bressoud’s even moduli generalization of the Rogers–Ramanujan identity and its overpartition analogue given by Chen, Sang and Shi in 2015. This talk is based on joint work with Shreejit Bandyopadhyay.

Ali Kemal Uncu

Posted on in Number Theory Seminar

Some Recent Progress in Cylindric Partitions II

When/Where:

March 29, 2022, 1:55 — 2:45 pm on Zoom

Abstract:

This week we will look at symmetric cylindric partitions as well as skew double shifted plane partitions. We will later shift our focus on weighted treatment of these objects and prove Schmidt’s partition theorem as well as a recent result of Andrews and Paule. This is a joint work with Walter Bridges.

Ali Kemal Uncu

Posted on in Number Theory Seminar

Some Recent Progress in Cylindric Partitions

When/Where:

March 22, 2022, 1:55 — 2:45 pm on Zoom

Abstract:

The cylindric partitions defined by Gessel and Krattenthaler has been of recent interest after a paper by Corteel and Welsh. In this talk, we will present the necessary background to the topic and my (joint with Corteel and Dousse) recent contribution to this subject. We will also mention some newer developments and talk about prospects.

Jonathan Bradley-Thrush

Posted on in Number Theory Seminar

Unilateral and bilateral summation theorems in the theory of basic hypergeometric series

When/Where:

March 15, 2022, 1:55 — 2:45 pm at LIT 368.

Abstract:

 The 6φ5 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 re-examine the problem, first considered by Jackson, of extending Rogers’s identity to one which possesses a fourfold symmetry. I will then provide some examples to demonstrate how the theory of elliptic functions may be used to convert certain unilateral summation theorems into bilateral summation theorems. The bilateral series considered will include those which feature in Ramanujan’s 1ψ1 identity and Bailey’s 6ψ6 identity, along with a few others which are similarly expressible as infinite products. Combinatorial interpretations of several q-series identities will also be given in terms of Ferrers diagrams.

 

Aritram Dhar

Posted on in Number Theory Seminar

New Relations of the mex with other partition statistics

When/Where:

March 1, 2022, 1:55 — 2:45 pm at LIT 368.

Abstract:

In a recent pioneering work, Andrews and Newman defined an extended function \(p_{A,a}(n)\) of their minimal excludant or “mex” of a partition function and by considering the cases \(p_{k,k}(n)\) and \(p_{2k,k}(n)\), they unearthed connections to the rank and crank of partitions and some restricted partitions. In this work, we generalize their results and associate the extended mex function to the number of partitions of an integer with arbitrary bound on the rank and crank. We also derive a new result expressing the smallest parts function of Andrews as a finite sum of this extended mex function. We obtain some more restricted partition identities as well.

This is joint work with Avi Mukhopadhyay and Rishabh Sarma.

James Sellers

Posted on in Number Theory Seminar

Congruences for k-elongated partition diamonds

When/Where:

February 22, 2022, 1:55 — 2:45 pm on Zoom

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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 recently appeared 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 k-elongated plane partition diamonds of length n is given by \(\frac{(q^2;q^2)_\infty^k}{(q;q)_\infty^{3k+1}\)  for each \(k\geq 1\).  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 [late]d_k[/late] for an infinite set of values of 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 and Mike Hirschhorn of the University of New South Wales.

 

 

 

 

 

Alexander Berkovich

Posted on in Number Theory Seminar

The Capparelli partition theorems and related infinite hierarchies of q-hypergeometric identities

When/Where:

February 8, 2022, 1:55 — 2:45 pm at LIT 368.

Abstract:

I review my recent work joint with Ali Kemal Uncu

Krishnaswami Alladi

Posted on in Number Theory Seminar

Weighted partition identities and links with Schmidt type theorems

When/Where:

February 1, 2022, 1:55 — 2:45 pm at LIT 368.

Abstract:

I will revisit my 1994 paper in TAMS in which I first discussed weighted partition identities. I will provide alternate proofs of two Schmidt type theorems using my weighted approach. Some open problems will be discussed.

 

 

 

 

Ali Uncu

Posted on in Number Theory Seminar

Reflecting (on) the modulo 9 Kanade-Russell (conjectural) identities

When/Where:

January 25, 2022, 1:55 — 2:45 pm on Zoom

Abstract:

I will be reflecting on the five modulo 9 Kanade-Russell conjectures and our recent efforts towards proving them. I will also present some new modulo 45 conjectures we found while we were trying to prove the modulo 9 conjectures.

This is a joint work with Wadim Zudilin.