A cluster algebra approach to presentations of the monoid of uniform block permutations

We describe the mutation class of a certain quiver with a frozen vertex and associate these quivers with potentials appearing in our mutation class to presentations of the monoid \({\mathscr {P}}_{n+1}\) ${\mathcal{P}}_{n+1}$ of uniform block permutations on the set \(\{1,2,\ldots , n+1\}\) $\{1,2,\dots ,n+1\}$ . Some classical and known presentations of \({\mathscr {P}}_{n+1}\) ${\mathcal{P}}_{n+1}$ , including FitzGerald’s presentation and Everitt and Fountain’s presentation, are recovered.