Pascal's formulas and vector fields
Résumé
We consider four examples of combinatorial triangles $\left(T(n,k)\right)_{0\le k\le n}$ (Pascal, Stirling of both types, Euler) : their Pascal's formulas define four vector fields, together with their field lines that turn out to be the limit of sample paths of four well known Markov chains. Using saddle-point asymptotics, we prove fluid approximation results in some of the four cases.
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