Paint cost spectrum of perfect $k$-ary trees

We determine the paint cost spectrum for perfect $k$-ary trees. A coloring of the vertices of a graph $G$ with $d$ colors is said to be \emph{$d$-distinguishing} if only the trivial automorphism preserves the color classes. The smallest such $d$ is the distinguishing number of $G$ and is denoted $dist(G)\mathrm{.}$ The \emph{paint cost of $d$-distinguishing $G$}, denoted ${\rho }^d(G)$, is the minimum size of the complement of a color class over all $d$-distinguishing colorings. A subset $S$ of the vertices of $G$ is said to be a \emph{fixing set} for $G$ if the only automorphsim that fixes the vertices in $S$ pointwise is the trivial automorphism. The cardinality of a smallest fixing set is denoted $fix(G)$. In this paper, we explore the breaking of symmetry in perfect $k$-ary trees by investigating what we define as the \emph{paint cost spectrum} of a graph $G$: $(dist(G);{\rho }^{dist(G)}(G),{\rho }^{dist(G)+1}(G),\dots ,{\rho }^{fix(G)+1}(G))$ and the \emph{paint cost ratio} of $G$, which is defined to be the fraction of paint costs in the paint cost spectrum equal to $fix(G)$. We determine both the paint cost spectrum and the paint cost ratio completely for perfect $k$-ary trees. We also prove a lemma that is of interest in its own right: given an $n$-tuple, $n\ge 2$ of distinct elements of an ordered abelian group and $1\le k\le n!-1$, there exists a $k\times n$ row permuted matrix with distinct column sums.