  There is a growing interest in studying the distribution of certain labels in
products of permutations since the work of Stanley addressing a conjecture of
B\'{o}na. This paper is concerned with a problem in that direction. Let $D$ be
a permutation on the set $[n]=\{1,2,\ldots, n\}$ and $E\subset [n]$. Suppose
the maximum possible number of cycles uncontaminated by the $E$-labels in the
product of $D$ and a cyclic permutation on $[n]$ is $\theta$ (depending on $D$
and $E$). We prove that for arbitrary $D$ and $E$ with few exceptions, the
number of cyclic permutations $\gamma$ such that $D\circ \gamma$ has exactly
$\theta-1$ $E$-label free cycles is at least $1/2$ that of $\gamma$ for $D\circ
\gamma$ to have $\theta$ $E$-label free cycles, where $1/2$ is best possible.
An even more general result is also conjectured.
