I've recently found myself doing some work on local rings, and I found the following quantity keeps popping up-
Let $A$ be a local commutative unital ring, with maximal ideal $\newcommand{\mfr}{\mathbf} \mfr{m}$. For any $n\in\mathbb{N}$, put $\beta(n)=\dim_{A/\mfr{m}}\mfr{m}^n/\mfr{m}^{n+1}$.
It seems quite plausible that one might consider this quantity an important invariant of the ring $A$. I was only wondering if this quantity has a commonly used name, and possibly notation.
Thank you, shai