A note on linear extensions and incomparable pairs (Q807652)

For \(2\leq m\leq n\), let P(m,n) denote the family of (strictly) ordered sets on \(\{\) 1,2,...,n\(\}\) in which \(\{\) 1,...,m\(\}\) is an antichain. A characterization is then provided of those members of P(m,n) that maximize the proportion of linear extensions in which 1 lies above 2. The final section of the paper indicates that this suggests a whole class of problems relating to linear extensions and antichains. For example, one could ask to identify those ordered sets in P(m,n) that maximize the proportion of all linear extensions in which \(1>^*2>^*...>^*m\).