Barrier strengthening beyond forest enumeration

Identify further natural nonsymmetric combinatorial rank sequences that are barrier log-concave but not log-concave, and determine structural conditions under which a known unimodality theorem extends to every uniform downward tilt.

Background

The paper gives examples of nonsymmetric rank sequences that are strictly barrier log-concave while failing ordinary log-concavity, including rank sequences of order ideals and clique counts with pendant vertices. These examples show that barrier log-concavity is a meaningful intermediate property between unimodality and full log-concavity.

The open problem seeks additional natural combinatorial settings and general structural preservation principles, beyond the specific mechanism of appending a nonincreasing terminal tail.