** 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.