  This paper introduces a new and simple statistic on noncrossing partitions
that expresses each coordinate of the toric $h$-vector of a cubical complex,
written in the basis of the Adin $h$-vector entries, as the total weight of all
noncrossing partitions. The same model may also be used to obtain a very simple
combinatorial interpretation of the contribution of a cubical shelling
component to the toric $h$-vector. In this model, a strengthening of the
symmetry expressed by the Dehn-Sommerville equations may be derived from the
self-duality of the noncrossing partition lattice, exhibited by the involution
of Simion and Ullman.
