# Discrete Mathematics and Combinatorics Commons™

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## Recent Articles in Discrete Mathematics and Combinatorics

Discrete Structures In Finite Type Cluster Algebras, Salvatore Stella Northeastern University

#### Discrete Structures In Finite Type Cluster Algebras, Salvatore Stella

##### Mathematics Dissertations

Due to their recursive definition, manipulating cluster algebras in an efficient way can be hard. Several combinatorial models have been developed in order to overcome this difficulty; here we investigate some of them in the finite type case.

In the first part of this thesis, using the parametrization of cluster variables by their g-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we extend the original construction of generalized associahedra by F. Chapoton, S. Fomin and A. Zelevinsky to any choice of acyclic initial cluster, and compare it to the one given by C. Hohlweg, C. Lange, and H ...

On The Parallelization Of A Search For Counterexamples To A Conjecture Of Erd\H{O}S, ShengWei Shen McMaster University

#### On The Parallelization Of A Search For Counterexamples To A Conjecture Of Erd\H{O}S, Shengwei Shen

##### Open Access Dissertations and Theses

Denote by $k_t(G)$ the number of cliques of order $t$ in a graph $G$ having $n$ vertices. Let $k_t(n) = \min\{k_t(G)+k_t(\overline{G}) \}$ where $\overline{G}$ denotes the complement of $G$. Let $c_t(n) = {k_t(n)}/{\tbinom{n}{t}}$ and $c_t$ be the limit of $c_t(n)$ for $n$ going to infinity. A 1962 conjecture of Erd\H{o}s stating that $c_t = 2^{1-\tbinom{t}{2}}$ was disproved by Thomason in 1989 for all $t\geq 4$. Tighter counterexamples have been constructed by Jagger, {\v S}{\v t}ov{\' \i}{\v c}ek and ...

The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson University of Nebraska - Lincoln

#### The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson

##### Dissertations, Theses, and Student Research Papers in Mathematics

A linear extension of a partially ordered set is simply a total ordering of the poset that is consistent with the original ordering. The linear extension diameter is a measure of how different two linear extensions could be, that is, the number of pairs of elements that are ordered differently by the two extensions. In this dissertation, we calculate the linear extension diameter of grids. This also gives us a nice characterization of the linear extensions that are the farthest from each other, and allows us to conclude that grids are diametrally reversing.

A linear extension of a poset might ...