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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 $\operatorname{Chain}[n]$.