Publication Date



Open access

Embargo Period


Degree Type


Degree Name

Doctor of Philosophy (PHD)


Mathematics (Arts and Sciences)

Date of Defense


First Committee Member

Michelle Wachs Galloway

Second Committee Member

Drew Armstrong

Third Committee Member

Marvin Mielke

Fourth Committee Member

Orlando Alvarez


The lattice B_n of subsets of the set {1, 2, ..., n} ordered by inclusion and the lattice \Pi_n of partitions of {1, 2, ..., n} ordered by refinement are two of the most fundamental examples in the theory of partially ordered sets (posets). A natural well-studied q–analogue of the subset lattice is the lattice B_n(q) of subspaces of the n–dimensional vector space F^n_q over the field F_q with q elements ordered by inclusion. There are many justifications for viewing this as a q–analogue. One comes from the fact that the number of maximal chains of B_n is n!, while the number of maximal chains of B_n(q) equals the q–analogue of n! which is defined by [n]_q! := [n]_q[n − 1]_q . . . [1]_q, where [n]_q := 1 + q + · · · + q^{n−1}. Another justification comes from studying the topology of a certain simplicial complex associated with the poset, called the order complex. The order complex associated with B_n is homeomorphic to a single sphere of dimension n−2, while the order complex associated with B_n(q) has the homotopy type of a wedge of q^{\binom{n}{ 2}} spheres of dimension n − 2. It is well-known that the order complex associated with \Pi_n has the homotopy type of a wedge of (n−1)! spheres of dimension n−3. Various q–analogues of the partition lattice \Pi_n have been proposed over the years, starting with the Dowling lattices introduced in a 1973 paper of Dowling. Posets studied by Welker and by Hanlon, Hersh, and Shareshian involve direct sum decompositions of vector spaces over F_q. While these posets have interesting properties analogous to those of \Pi_n, such as having the homotopy type of a wedge of spheres, none have the desirable property that the number of spheres is a q–analogue of (n−1)!. The q–analogue proposed in this thesis is the poset \Pi_n(q) of direct sum decompositions of subspaces of F^n_q whose summands all have dimension at least 2, ordered by inclusion of summands. This is actually a q–analogue of a poset that is isomorphic to \Pi_n, namely the poset of partitions of subsets of {1, 2, ..., n} in which each block has size at least 2. We show that the order complex associated with \Pi_n(q) has the homotopy type of a wedge of f(q)[n−1]_q! spheres of dimension n − 3 where f(q) is a polynomial in q that is equal to 1 when q is set equal to 1. In order to prove this result, we initiate a study of a much more general class of posets, which includes \Pi_n, \Pi_n(q), and the k–equal partition lattices introduced by Björner, Lovász, and Yao in 1992. The k–equal partition lattice \Pi^{=k}_ n is the subposet of \Pi_n consisting of partitions for which each block has size at least k or 1. In this general class, the roles of B_n and B_n(q) in the definitions of \Pi_n and \Pi_n(q) are played by an arbitrary geometric lattice L. We use shellability theory to prove that the order complex associated with a general k–equal decomposition lattice \Pi^{=k}_ L has the homotopy type of a wedge of spheres in varying dimensions when k > 2 and just in dimension n − 3 when k = 2. Shellability theory also enables us to derive a complicated formula for the number of spheres in each dimension. The nontrivial step of reducing the complicated formula in the case of \Pi^{=2}_{ Bn(q)} = \Pi_n(q) to the desired f_n(q)[n − 1]_q! formula uses Stanley’s theory of exponential structures.


q-analogue; partition lattice; k-equal; EL-shellability