Tags: #theorem

Statement

Let $P$ be a finite ranked poset with rank numbers $r_1,\dots ,r_l\ge 1$. If $P$ is a symmetric chain decomposition exists, then $P$ is

• rank symmetric poset
• rank unimodal poset
• Sperner poset

Note that the converse is not true (if $P$ satisfies these properties, it's not necessarily all of those.)

Proof

Note that the chains in a symmetric chain decomposition are rank symmetric and unimodal, and the union of these still satisfy that property. Now, let $n$ be the number of chains in the SCD. Any antichain must contain at most 1 from each chain, so it is upper bounded by $n$. But since it's unimodal, the number of chains is equal to the number of elements in the middle rank, $r_{\lfloor \frac{l}{2}\rfloor }$.