Enumeration of $S$-omino towers and row-convex $k$-omino towers

We first enumerate a generalization of domino towers that was proposed by Tricia M. Brown (J. Integer Seq. 20 (2017)), which we call S-omino towers. We establish equations that the generating function must satisfy and then apply the Lagrange inversion formula to find a closed formula for the number of towers. We also show a connection to generalized Dyck paths and provide an explicit bijection. Finally, we consider the set of row-convex k-omino towers, introduced by Brown, and calculate an exact generating function.