Is the Symmetric Group Sperner?

2019/01/01 by Larry H. Harper, Harper, Larry H., Gene B. Kim +1
Computer Science · Mathematics · #05D05 #05E99 #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1901.00197

openalex publication_date 2019/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An antichain A in a poset P is a subset of P in which no two elements are comparable. Sperner showed that the maximal antichain in the Boolean lattice, B_n = \ 0 < 1 \ⁿ, is the largest rank (of size \binomn\lfloor n/2 \rfloor). This type of problem has been since generalized, and a graded poset P is said to be Sperner if the largest rank of P is its maximal antichain. In this paper, we will show that the symmetric group S_n, partially ordered by refinement (or equivalently by absolute order), is Sperner.

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