Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo $k$

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