Antichains and linear extensions (Q1300333)

For any poset \(P\), let \(L(P)\) be the set of linear extensions of \(P\). Let \(m\geq 2\) be an integer and \(A\) be a finite set disjoint from \(\{1,2,\dots,m\}\). Denote by \(S(m,A)\) the family of posets \(P\) with the underlying set \(\{1,2,\dots,m\}\cup A\) in which \(\{1,2,\dots,m\}\) is an antichain. For \(P\in S(m,A)\) denote by \(L_m(P)\) the subset of \(L(P)\) whose linear extensions have \(1>2>\cdots>m\). The author deals with the problem of naximizing the number \(r(P)=|L_m(P)|/ |L(P)|\) over \(S(m,A)\) for fixed \(m\) and \(|A|\). He gives related results for \(|A|\in \{1,2\}\) and \(m\geq 2\).