Let $(W,S)$ be a Coxeter system with simple reflections $S=\{s_i : i \in I\}$, let $\Phi=\Phi^+ \sqcup (-\Phi^+)$ be its associated real root system with simple roots $\Delta=\{\alpha_i : i \in I\}$, and let $\ell:W\to \mathbb{N}\cup\{0\}$ denote the Coxeter length function with respect to $S$. For $w\in W$, define the inversion set

\begin{align*} N(w):=\{\alpha\in \Phi^+ : w\alpha\in -\Phi^+\}. \end{align*}

Then, for every $w\in W$,

\begin{align*} \ell(w)=|N(w)|. \end{align*}

Moreover, for every $w\in W$ and every $i\in I$,

\begin{align*} \ell(ws_i)=\ell(w)+1 \quad \Longleftrightarrow \quad w\alpha_i\in \Phi^+, \end{align*}

and

\begin{align*} \ell(ws_i)=\ell(w)-1 \quad \Longleftrightarrow \quad w\alpha_i\in -\Phi^+. \end{align*}