Elise Lockwood, Oregon State University

*Subtleties of the multiplication principle*

The multiplication principle is fundamental to combinatorics, underpinning many standard formulas and providing justification for counting strategies.
Given its importance, the way it is presented in textbooks is surprisingly varied.
In this talk, I identify key elements of the principle and present a categorization of statement types found in a textbook analysis.
I also incorporate excerpts from a reinvention study that shed light on how students reason through these key elements.
Findings from both the textbook analysis and the reinvention study reveal surprisingly subtle aspects of the multiplication principle.
I conclude with a number of potential mathematical and pedagogical implications for the teaching and learning of the principle.

Edward Hanson, SUNY New Paltz

*𝔰𝔩 _{2}(ℂ)-modules and orthogonal polynomials*

The Lie algebra 𝔰𝔩_{2}(ℂ) is the vector space of 2×2 matrices with trace zero over the complex numbers.
In this talk, we will explore some connections between the structure of certain 𝔰𝔩_{2}(ℂ)-modules, sequences of orthogonal polynomials, and algebraic graph theory.