When , the vertices of the permutoassociahedron can be represented by bracketing all the permutations of three terms , , and . There are six such permutations, , , , , , and , and each of them admits two bracketings (obtained from one another by associativity). For instance, can be bracketed as or as . Hence, the -dimensional permutoassociahedron is the dodecagon with vertices , , , , , , , , , , , and .
When , the vertex is adjacent to exactly three other vertices of the permutoassociahedron: , , and . The first two vertices are reached from via associativity and the third via a transposition. The vertex is adjacent to four vertices. Two of them, and , are reached via associativity, and the other two, and , via a transposition. This illustrates that, in dimension and above, the permutoassociahedron is not a simple polytope.[3]