IMath Seminar: Dr. Jordan Tirrell

You are all invited to a seminar featuring Dr. Jordan Tirrell (Department of Mathematics and Computer Science, Washington College, USA). This is an in-person event and it will be held on Wednesday, 26 June 2024, 10:00 AM at the Math Bldg Conference Room.

Dr. Jordan Tirrell (Department of Mathematics and Computer Science, Washington College, USA)
Title: Unimodality of \(q\)-Twotorials via Alternating Gamma Vectors
Abstract: We explore the unimodality of the polynomials \((1+q)(1+q^2)…(1+q^n)\), which we call \(q\)-twotorials in relation to \(q\)-factorials, and seek a combinatorial proof of their unimodality. While there are simple proofs of unimodality for \(q\)-factorials, a combinatorial proof of unimodality for \(q\)-twotorials has been elusive. We follow a suggestion by Brittenham, Carroll, Petersen, and Thomas (2016) to leverage the combinatorics of the alternating gamma coefficients through domino tilings and ballot paths. We introduce a technique that provides a combinatorial proof for the unimodality of \(q\)-twotorials for all \(n < 25\) with specific exceptions. Our approach can also be applied to a larger class of related polynomials, and we are able to give combinatorial proofs of unimodality and non-unmodality for certain infinite families, potentially laying the groundwork for future investigations. This is joint work with undergraduates.