Down-step statistics in generalized Dyck paths

In this talk we will discuss the enumeration of the number of down-steps between pairs of up-steps in $k_t$-Dyck paths, which are a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$. These down-step statistics have applications in other combinatorial enumeration problems. We will also establish a connection between these $k_t$-Dyck paths and binary matrices used in coding theory, connecting the two areas of study and leading to a new interpretation for the Catalan numbers.