  For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck
path of length $(k+1)n$ is a lattice path that starts at $(0, 0)$, ends at
$((k+1)n, 0)$, stays weakly above the line $y = -t$, and consists of steps from
the step-set $\{(1, 1), (1, -k)\}$. We enumerate the family of $k_t$-Dyck paths
by considering the number of down-steps at a height of $i$ modulo $k$. Given a
tuple $(a_1, a_2, \ldots, a_k)$ we find an exact enumeration formula for the
number of $k_t$-Dyck paths of length $(k+1)n$ with $a_i$ down-steps at a height
of $i$ modulo $k$, $1 \leq i \leq k$. The proofs given are done via bijective
means or with generating functions.
