## Thursday, February 18, 2021

### Gaurav Bhatnagar (Ashoka) February 18, 2021 - 3:55 PM-5:00 PM

Talk announcement

Title: The Partition-Frequency Enumeration Matrix

Speaker: Gaurav Bhatnagar (Ashoka University)

When: February 18, 2021 - 3:55 PM - 5:00 PM (IST)

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan's $\tau$ function, sums of squares and triangular numbers, and for $\zeta(2n)$, where $n$ is a positive integer. These include classical results due to Euler, Ramanujan, and others.  As one application, we embed Ramanujan's famous congruences $p(5n+4)\equiv 0\;$ (mod $5)$ and $\tau(5n+5)\equiv 0\;$ (mod $5)$ into an infinite family of such congruences.