  The Cayley continuants are referred to the determinants of tridiagonal
matrices in connection with the Sylvester continuants. Munarini-Torri found a
striking combinatorial interpretation of the Cayley continuants in terms of the
joint distribution of the number of odd cycles and the number of even cycles of
permutations of $[n]=\{1,2,\ldots, n\}$. In view of a general setting,
$r$-regular cycles (with length not divisible by $r$) and $r$-singular cycles
(with length divisible by $r$) have been extensively studied largely related to
roots of permutations. We introduce the wide band Cayley continuants as an
extension of the original Cayley continuants, and we show that they can be
interpreted in terms of the joint distribution of the number of $r$-regular
cycles and the number of $r$-singular cycles over permutations of $[n]$.
