Toward enumerating the chains of maximum length of Cambrian and m-eralized Cambrian lattices
Description
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.
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.
Date Created
The date the item was original created (prior to any relationship with the ASU Digital Repositories.)
2017
Agent
- Author (aut): Al-Suleiman, Sultan
- Thesis advisor (ths): Fishel, Susanna
- Committee member: Childress, Nancy
- Committee member: Czygrinow, Andrzej
- Committee member: Jones, John
- Committee member: Spielberg, John
- Publisher (pbl): Arizona State University