Butler  gave recurrence relations for computing the number Nr(n) of r-valued (r ≥ 2) switching functions of n variables realizable by a cascaded network of n - 1 r-valued cells. The cascades are assumed to have a fixed-input variable assignment. In this correspondence similar recurrence relations are presented for the number Mr(n) of such cascade realizable functions that depend on all n input variables. In particular, for the ternary valued case, explicit formulas for N3(n) and M3(n) are given. Some asymptotic properties of Nr(n) and Mr(n) are also derived, which show that their growth is asymptotically exponential as n increases.