$(k+1)$-potent Matrices in triangular matrix Groups and Incidence Algebras of Finite Posets

Let $\mathbb{K}$ be a field such that $char(\mathbb{K})\nmid k$ and $char(\mathbb{K})\nmid k+1$. We describe all $(k+1)$-potent matrices over the group of upper triangular matrix. In the case that $\mathbb{K}$ is a finite field we show how to compute the number of these elements in triangular matrix groups and use this formula to compute the number of $(k+1)$-potent elements in the Incidence Algebra $\mathcal{I}(X,\mathbb{K})$ where $X$ is a finite poset.