Maximal chain in a Boolean lattice

A maximal chain in $\mathcal{P}([n])$ contains one set of every cardinality from zero to $n$, with each set obtained from the preceding one by adding a single element.

A uniformly random maximal chain is obtained from a uniformly random ordering of the ground set by taking its successive initial segments. A fixed $h$-element set occurs with probability ${\left(\genfrac{}{}{0ex}{}{n}{h}\right)}^{-1}$.