Riordan Arrays, Pseudo-Involutions, and AI-Assisted Mathematical Discovery

Abstract: The Riordan group is a group of infinite lower triangular matrices that are defined by two generating functions, $g$ and $f$. The $k^{\mathrm{th}}$ column of each matrix has a generating function related to $gf^k$. There are many applications of Riordan arrays, among other things they can be used to count combinatorial objects and to prove combinatorial identities

In this talk, I will introduce Riordan arrays and discuss recent joint work with Dewayne Dixon and Moussa Doumbia on pseudo-involutions. In the Appell subgroup, the pseudo-involution condition reduces to the simple identity \[ g(z)g(-z)=1, \] which leads to a useful classification and several interesting integer sequences. I will also discuss how artificial intelligence can assist mathematical discovery by helping generate examples, identify patterns, and suggest conjectures, while exact computation and proof remain essential.

computation and languagemachine learningcombinatoricsnumber theory

Audience: researchers in the topic

AI, Combinatorics and Number Theory Seminar

Series comments: This seminar is devoted to the responsible and constructive use of AI in number theory and combinatorics. It is a space to think carefully about what these tools can and cannot do, and where the limits actually lie. Enthusiasm and skepticism are both welcome here, and we ask only that they be voiced with generosity. Questions from students and from people new to these areas are especially encouraged.

The seminar meets Online via Zoom from 2:00 PM – 3:00 PM Eastern Time.

Please email the organizers to join the mailing list and receive the Zoom links.