The Wide Band Cayley Continuants

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 $[n]$.