A Partial Order Structure on the Shellings of Lexicographically Shellable Posets.
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Date
2024-01-09
Authors
Lacina, Stephen
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Publisher
University of Oregon
Abstract
This dissertation has two main topics. The first is the introduction and in-depth study of a new poset theoretic structure designed to help us better understand the notion of lexicographic shellability of partially ordered sets (posets). Lexicographic shelling of posets was introduced by Bj{\"o}rner via a type of poset labeling known as an EL-labeling and was generalized by Bj{\"o}rner and Wachs to the notion of CL-labeling. We introduce and study a partial order structure on the maximal chains of any finite bounded poset $P$ which has a CL-labeling $\lambda$. We call this partial order the maximal chain descent order induced by $\lambda$, denoted $P_{\lambda}(2)$. We show that this new partial order can be thought of as the structure of the set of shellings of $P$ ``derived from $\lambda$". A motivating example is the weak order of type A. Another especially interesting class of examples produces natural partial orders on standard Young tableaux. We prove several results about the cover relations of maximal chain descent orders in general. We characterize the EL-labelings whose maximal chain descent orders have the expected cover relations, and we prove that this is the case for many important families of EL-labelings.
The second main topic of this dissertation is that of determining the poset topology of two families of lattices known as $s$-weak order and the $s$-Tamari lattice.
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Keywords
Lexicographic Shellability, Poset Topology, Topological Combinatorics