Notes on Riordan arrays and lattice paths

In this note, we explore links between Riordan arrays and lattice paths. We begin by describing Riordan arrays, and some of their generalizations, including rectifications and triangulations. We the consider Riordan array links to lattice paths with steps of type $(a,b)$, where $a$ and $b$ are nonnegative. We consider common Riordan arrays that are linked to lattice paths, as well as showing links between almost Riordan arrays and lattice paths. We then consider lattice paths with step sets that include downward steps, and show how the $A$-matrix characterization of Riordan arrays plays a key role in analysing corresponding Riordan arrays.