  In this paper, we study the distribution of the number of occurrences of the
simplest frame pattern, called the $\mu$ pattern, in $n$-cycles. Given an
$n$-cycle $C$, we say that a pair $\langle i,j \rangle$ matches the $\mu$
pattern if $i < j$ and as we traverse around $C$ in a clockwise direction
starting at $i$ and ending at $j$, we never encounter a $k$ with $i < k < j$.
We say that $ \langle i,j \rangle$ is a nontrivial $\mu$-match if $i+1 < j$.
Also, an $n$-cycle $C$ is incontractible if there is no $i$ such that $i+1$
immediately follows $i$ in $C$.
  We show that the number of incontractible $n$-cycles in the symmetric group
$S_n$ is $D_{n-1}$, where $D_n$ is the number of derangements in $S_n$.
Further, we prove that the number of $n$-cycles in $S_n$ with exactly $k$
$\mu$-matches can be expressed as a linear combination of binomial coefficients
of the form $\binom{n-1}{i}$ where $i \leq 2k+1$. We also show that the
generating function $NTI_{n,\mu}(q)$ of $q$ raised to the number of nontrivial
$\mu$-matches in $C$ over all incontractible $n$-cycles in $S_n$ is a new
$q$-analogue of $D_{n-1}$, which is different from the $q$-analogues of the
derangement numbers that have been studied by Garsia and Remmel and by Wachs.
We show that there is a rather surprising connection between the charge
statistic on permutations due to Lascoux and Sch\"uzenberger and our
polynomials in that the coefficient of the smallest power of $q$ in
$NTI_{2k+1,\mu}(q)$ is the number of permutations in $S_{2k+1}$ whose charge
path is a Dyck path. Finally, we show that $NTI_{n,\mu}(q)|_{q^{\binom{n-1}{2}
-k}}$ and $NT_{n,\mu}(q)|_{q^{\binom{n-1}{2} -k}}$ are the number of partitions
of $k$ for sufficiently large $n$.
