Minimal Posets with Prescribed Maximal Chain Cardinalities

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 $P$: $\max (S)+\lceil {\log }_2\mathrm{\mid }S\mathrm{\mid }\rceil <=\mathrm{\mid }P\mathrm{\mid }<=\max (S)+\mathrm{\mid }S\mathrm{\mid }-1$, and both bounds are tight.