Notation Let $Q$ be a partially ordered set. We let $\operatorname{Chain}(Q)$ denote the collection of all nonempty, finite, linearly ordered subsets of $Q$. We regard $\operatorname{Chain}(Q)$ as a partially ordered set, where the partial order is given by inclusion. In the special case where $Q = [n] = \{ 0 < 1 < \ldots < n \} $ for some nonnegative integer $n$, we denote the partially ordered set $\operatorname{Chain}(Q)$ by $\operatorname{Chain}[n]$.