DMS Combinatorics Seminar

Speaker: Josey Graves (Auburn University)

Title: Widths of Finite Posets under the Majorization Ordering

Abstract: We focus on the structural properties of two types of posets, $P(n,m)$ and $P'(n,m)$, both of which are ordered by the majorization ordering. Specifically, we consider those cases where $1 \leq n \leq 4$. The poset $P(n,m)$ consists of sequences of non-negative integers of length $n$ that sum to $m$. The second poset, $P'(n,m)$, is a subposet of $P(n,m)$ where we restrict to the decreasing sequences.  We demonstrate that these posets exhibit Sperner-like properties. In particular, we show that the largest antichain in $P(n,m)$ and $P'(n,m)$ for $1 \leq n \leq 4$ is realized by a "middle" "level," similar to that of the classical Sperner theorem. We use the term "middle" loosely here, as there may be many levels or induced levels which are maximal, and they all generally occur in the middle section of these posets. In the case of $P(n,m)$, we provide explicit chain decompositions, while for $P'(n,m)$ we give explicit chain decompositions for $n \in \{1,2\}$. For $P'(n,m)$ when $n \in \{3,4\}$, we give an inductive proof of the existence of a minimal chain decomposition on the outer layer(s), with induction handling the smaller poset.