On a generalisation of \(k\)-Dyck paths

We consider a family of non-negative lattice paths consisting of the step set \(\{(1, 1), (1, -k)\}\) called \(k\)-Dyck paths, which are enumerated by the generalised Catalan numbers. By removing the non-negativity condition but restricting the path to above the line \(y = -t\) we obtain a family of lattice paths called \(k_t\)-Dyck paths which have interesting enumerative properties. We enumerate these paths using various combinatorial methods, and perform analysis of associated parameters.