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arXiv · 1901.00197

Is the Symmetric Group Sperner?

Larry H. Harper·Gene B. Kim
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Abstract

An antichain $\mathcal{A}$ in a poset $\mathcal{P}$ is a subset of $\mathcal{P}$ in which no two elements are comparable. Sperner showed that the maximal antichain in the Boolean lattice, $\mathcal{B}_n = \left\{ 0 < 1 \right\}^n$, is the largest rank (of size $\binom{n}{\lfloor n/2 \rfloor}$). This type of problem has been since generalized, and a graded poset $\mathcal{P}$ is said to be Sperner if the largest rank of $\mathcal{P}$ 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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Larry H. Harper, Gene B. Kim. 2019-01-03. Is the Symmetric Group Sperner?. https://arxiv.org/abs/1901.00197

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