Two Types of Rotations in Associated Stirling Numbers

In a previous article [1], we have found that associated Stirling numbers of first and second kind can be compacted, at any order and using a linear transformation, in a structure of arithmetical triangle. We show then a strong link between the congruence of such numbers and this common geometrical layout. It leads to non-trivial combinatorial and modular properties which would be much more difficult to find and to prove without such a structure. Amongst other things, the binomial coefficient reveals a new surprising ability to build modular areas, angles and rotations, geared to the order r and independent of their homonyms in terms of classical space transformations. Probably for the first time in arithmetic science, two rotations of deeply different natures, one geometrical and the other modular, are highlighted for the same Stirling numbers and their associated.