A Heyting Algebra on Dyck Paths of Type AA and BB

In this article we investigate the lattices of Dyck paths of type $A$ and $B$ under dominance order, and explicitly describe their Heyting algebra structure. This means that each Dyck path of either type has a relative pseudocomplement with respect to some other Dyck path of the same type. While the proof that this lattice forms a Heyting algebra is quite straightforward, the explicit computation of the relative pseudocomplements using the lattice-theoretic definition is quite tedious. We give a combinatorial description of the Heyting algebra operations join, meet, and relative pseudocomplement in terms of height sequences, and we use these results to derive formulas for pseudocomplements and to characterize the regular elements in these lattices.Comment: Final version. 21 pages, 5 figure

Algebraic Methods for Reducibility in Nowhere-Zero Flows

We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of reducibility. We define variables which in some sense count the number of nowhere-zero flows of certain type in a graph and then deduce equalities and inequalities that must hold for all graphs. We then show how to use these algebraic expressions to prove reducibility. In our case, these inequalities and equalities are linear. We can thus use the well developed theory of linear programming to obtain certificates of these proof. We make publicly available computer programs we wrote to generate the algebraic expressions and obtain the certificates

Path Tableaux and the Combinatorics of the Immanant Function

Immanants are a generalization of the well-studied determinant and permanent. Although the combinatorial interpretations for the determinant and permanent have been studied in excess, there remain few combinatorial interpretations for the immanant. The main objective of this thesis is to consider the immanant, and its possible combinatorial interpretations, in terms of recursive structures on the character. This thesis presents a comprehensive view of previous interpretations of immanants. Furthermore, it discusses algebraic techniques that may be used to investigate further into the combinatorial aspects of the immanant. We consider the Temperley-Lieb algebra and the class of immanants over the elements of this algebra. Combinatorial tools including the Temperley-Lieb algebra and Kauffman diagrams will be used in a number of interpretations. In particular, we extend some results for the permanent and determinant based on the $R$-weighted planar network construction, where $R$ is a convenient ring, by Clearman, Shelton, and Skandera. This thesis also presents some cases in which this construction cannot be extended. Finally, we present some extensions to combinatorial interpretations on certain classes of tableaux, as well as certain classes of matrices