Maximally Asymmetric Multiple-Valued Functions

Abstract

The asymmetry of a function $f(x_{1},\ x_{2},\ \ldots,\ x_{n})$ is the fewest elements of the range of $f$ that must be changed so that $f$ becomes a symmetric function. The functions with maximal asymmetry for the case of r-valued n-variable functions have been characterized and counted for $r=2$ in two previous papers. In this paper, we extend these results to $r > 2$. We do this for two types of symmetry, functions whose value is unchanged by 1) any permutation of the variable labels and by 2) any permutation of variable labels and variable values. We also derive the maximum possible asymmetry. We show that, as $n\rightarrow\infty$ and $r$ is fixed, the maximum asymmetry approaches $(r-1)r^{n-1}$.