Hermite subdivision schemes are iterative refinement algorithms which produce a function and its derivatives
in the limit.
In this talk we are interested in Hermite subdivision schemes which reproduce polynomials of high degree.
We show that this can be characterized by operator factorizations involving Taylor operators and difference factorizations of a rank one vector scheme.
Explicit expressions for these operators are derived, which are based on an interplay between Stirling numbers and p-Cauchy numbers. Furthermore, we discuss how this framework can be used to prove high smoothness of the limit functions.