Speaker: Jia Huang, University of Nebraska, Kearney

Title: Nonassociativity of some binary operations

Abstract: Let $*$ be a binary operation on a set $X$ and let $x_0,x_1,\ldots,x_n$ be $X$-valued indeterminates. Define two parenthesizations of $x_0*x_1*\cdots*x_n$ to be equivalent if they give the same function from $X^{n+1}$ to $X$. Under this equivalence relation, we study the number $C_{*,n}$ of equivalence classes and the largest size $\widetilde C_{*,n}$ of an equivalence class. It is well known that $1\le C_{*,n}\le C_n$ and $1\le \widetilde C_{*,n}\le C_n$, where $C_n := \frac{1}{n+1}{2n\choose n}$ is the ubiquitous Catalan number. Moreover, $C_{*,n}=1 \Leftrightarrow$ $*$ is associative $\Leftrightarrow \widetilde C_{*,n}=C_n$. Thus $C_{*,n}$ and $\widetilde C_{*,n}$ measure how far the operation $*$ is away from being associative. In this talk we will present various results on $C_{*,n}$ and $\widetilde C_{*,n}$ when $*$ satisfies a multiparameter generalization of associativity.