Given a nonempty finite multiset $S$ of positive integers, we wish to find a partially ordered set $P$ of minimal cardinality such that the multiset of cardinalities of all maximal chains in $P$ equals $S$. This paper establishes upper and lower bounds on the size of $P$: $\max(S) + \lceil \log_2 |S| \rceil <= |P| <= \max(S) + |S| - 1$, and both bounds are tight.
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