Enumeration of generalized Dyck paths based on the height of down-steps modulo \(k\) (Q2684899)

Summary: 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 \(k, 1 \leqslant i \leqslant k\). The proofs given are done via bijective means or with generating functions.