Pascal's Formulas and Vector Fields

We consider four examples $T=(T(n,k))_{0\le k\le n}$ of combinatorial triangles (Pascal, Stirling of both types, Euler) : through saddle-point asymptotics, their Pascal's formulas define four vector fields, together with their field lines that turn out to be the conjectured limit of sample paths of four well known Markov chains. We prove this asymptotic behaviour in three of the four cases. Our results lead to a new proof of Koršunov's formula for the enumeration of accessible complete deterministic automata, and to the design of an efficient rejection method for the random generation of this class of automata.