Maximal elements in the Hurwitz graph

This is the authors' abstract. We don't add key points for this paper.

The Hurwitz graph has as its vertices the maximal chains in the lattice of non-crossing partitions, with an edge between two chains which differ at exactly one place. Adin and Roichman introduced an order on this graph by choosing a base element and showed that the maximal elements are counted by the Catalan numbers. We show that this set of maximal elements is in natural bijection with non-crossing partitions so that the graph structure corresponds to that of the permutahedron, restricted to non-crossing partitions. We use this to compute the diameter of the graph of maximal elements. We also briefly consider the graph obtained by identifying maximal chains with minimal factorizations of a full cycle into transpositions in the symmetric groups and quotienting by commutation relations.