Publication Date



Open access

Embargo Period


Degree Type


Degree Name

Doctor of Philosophy (PHD)


Mathematics (Arts and Sciences)

Date of Defense


First Committee Member

Drew Armstrong

Second Committee Member

Ken Baker

Third Committee Member

Alexander Dvorsky

Fourth Committee Member

Victor Milenkovic


The idea of the lattice of non-crossing partitions, NC(n), is inspired by early work of Kreweras. In this thesis we study the action of dihedral group D_2n on NC(n), especially the sublattice NC(n)^F in which all the elements are fixed by a reflection F, and then we extend our work to the characters of the dihedral group acting on NC(n). We start from enumerative properties of the lattice NC(n)^F . Next we investigate the recursive structure on the lattice NC(n)^F related to central binomial coefficients and the Catalan numbers. We proceed to look into combinatorial structure of a graded sublattice which is named "the pruned sublattice". Two characters Alpha_S and Beta_S introduced by Stanley of dihedral groups acting on NC(n) are computed with respect to certain rank-selected subposet NC(n)_S in NC(n). We first recall Montenegro's computation of Beta_[n-2] from his unpublished manuscript. Based on the cyclic sieving phenomenon of Reiner, Stanton and White, we obtain a general result for all Alpha_S's, where S is a subset of [n] of size 1 or n-2.


non-crossing partition; dihedral group; characters; group action