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.
@article{10_37236_12591,
author = {Philippe Chassaing and Jules Flin and Alexis Zevio},
title = {Pascal's formulas and vector fields},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {1},
doi = {10.37236/12591},
zbl = {1564.05008},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12591/}
}
TY - JOUR
AU - Philippe Chassaing
AU - Jules Flin
AU - Alexis Zevio
TI - Pascal's formulas and vector fields
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/12591/
DO - 10.37236/12591
ID - 10_37236_12591
ER -
%0 Journal Article
%A Philippe Chassaing
%A Jules Flin
%A Alexis Zevio
%T Pascal's formulas and vector fields
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/12591/
%R 10.37236/12591
%F 10_37236_12591
Philippe Chassaing; Jules Flin; Alexis Zevio. Pascal's formulas and vector fields. The electronic journal of combinatorics, Tome 32 (2025) no. 1. doi: 10.37236/12591
