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Algebra & Combinatorics Seminar

Title: Recent developments in the theory of the restricted partition function $p(n,N)$
Speaker: Atul Dixit (IIT, Gandhinagar)
Date: 09 October 2019
Time: 3:30 pm
Venue: LH-1, Mathematics Department

A beautiful $q$-series identity found in the unorganized portion of Ramanujan’s second and third notebooks was recently generalized by Maji and I. This identity gives, as a special case, a three-parameter identity which is a rich source of partition-theoretic information allowing us to prove, for example, Andrews’ famous identity on the smallest parts function $\mathrm{spt}(n)$, a recent identity of Garvan, and identities on divisor generating functions, to name a few. Guo and Zeng recently derived a finite analogue of Uchimura’s identity on the generating function for the divisor function $d(n)$. This motivated us to look for a finite analogue of my generalization of Ramanujan’s aforementioned identity with Maji. Upon obtaining such a finite version, our quest to look for a finite version of Andrews’ $\mathrm{spt}$-identity necessitated finding finite analogues of rank, crank and their moments. We could obtain finite versions of rank and crank for vector partitions. We were also able to obtain a finite analogue of a partition identity recently conjectured by George Beck and proven by Shane Chern. I will discuss these and some related results. This is joint work with Pramod Eyyunni, Bibekananda Maji and Garima Sood.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 21 Jun 2024