  We decompose the adjoint representation of $\mathfrak{sl}_{r+1}=\mathfrak
{sl}_{r+1}(\mathbb C)$ by a purely combinatorial approach based on the
introduction of a certain subset of the Weyl group called the \emph{Weyl
alternation set} associated to a pair of dominant integral weights. The
cardinality of the Weyl alternation set associated to the highest root and zero
weight of $\mathfrak {sl}_{r+1}$ is given by the $r^{th}$ Fibonacci number. We
then obtain the exponents of $\mathfrak {sl}_{r+1}$ from this point of view.
