On intervals in some posets of forests (Q1395823)

If \(I\) is a finite label set and \(\text{For}(I)\) is the set of leaf labeled forests of rooted binary trees, then the objects of investigation are intervals of \(\text{For}(I)\) equipped with a special order \(\leq\) which from its definition via a nice class of maps allows for a detailed investigation of these intervals via several interesting propositions leading to the observation that \((\text{For}(I),\leq)\) is ranked (from \(\widehat o\) up, there is no \(\widehat 1\)) whose combinatorial properties are established or quoted and permit detailed information on the roots of the M-polynomial, the Z-polynomial and the characteristic polynomial for intervals of \((\text{For}(I),\leq)\) to be obtained, including the main result that the characteristic polynomial of any such interval has only nonnegative integer roots, a stronger version of usual results along these lines in view of ``the poset conjecture'' for example.