Fractal Patterns in Gaussian and Stirling Number Tables

Geometrical representations of certain classical number tables modulo a given prime power (binomials, Gaussian \(g\)-binomials and Stirling numbers of \(1st\) and \(2nd\) kind) generate patterns with self-similarity features. Moreover, these patterns appear to be strongly related for all number tables under consideration, when a prime power is fixed.

These experimental observations are made precise by interpreting the recursively defined number tables as the output of certain cellular automata \((CA)\). For a broad class of \(CA\) it has been proven \([11]\) that the long time evolution can generate fractal sets, whose properties can be understood by means of hierarchical iterated function systems. We use these results to show that the mentioned number tables (mod \(p^v\)) induce fractal sets which are homeomorphic to a universal fractal set denoted by \(\mathcal{S}_{p^v}\) which we call Sierpinski triangle (mod \(p^v\)).