Toward enumerating the chains of maximum length of Cambrian and m-eralized Cambrian lattices

The Cambrian lattice corresponding to a Coxeter element c of An, denoted Camb(c),

is the subposet of An induced by the c-sortable elements, and the m-eralized Cambrian

lattice corresponding to c, denoted Cambm(c), is dened as a subposet of the

braid group accompanied with the right weak ordering induced by the c-sortable elements

under certain conditions. Both of these families generalize the well-studied

Tamari lattice Tn rst introduced by D. Tamari in 1962. S. Fishel and L. Nelson

enumerated the chains of maximum length of Tamari lattices.

In this dissertation, I study the chains of maximum length of the Cambrian and

m-eralized Cambrian lattices, precisely, I enumerate these chains in terms of other

objects, and then nd formulas for the number of these chains for all m-eralized

Cambrian lattices of A1, A2, A3, and A4. Furthermore, I give an alternative proof

for the number of chains of maximum length of the Tamari lattice Tn, and provide

conjectures and corollaries for the number of these chains for all m-eralized Cambrian

lattices of A5.