  Define an expansion poset to be the poset of monomials of a cluster variable
attached to an arc in a polygon, where each monomial is represented by the
corresponding combinatorial object from some fixed combinatorial cluster
expansion formula. We introduce an involution on several of the interrelated
combinatorial objects and constructions associated to type $A$ surface cluster
algebras, including certain classes of arcs, triangulations, and distributive
lattices. We use these involutions to formulate a dual version of skein
relations for arcs, and dual versions of three existing expansion posets. In
particular, this leads to two new cluster expansion formulas, and recovers the
lattice path expansion of Propp et al. We provide an explicit,
structure-preserving poset isomorphism between an expansion poset and its dual
version from the dual arc. We also show that an expansion poset and its dual
version constructed from the same arc are dual in the sense of distributive
lattices.
  We show that any expansion poset is isomorphic to a closed interval in one of
the lattices $L(m,n)$ of Young diagrams contained in an $m \times n$ grid, and
that any $L(m,n)$ has a covering by such intervals. We give two formulas for
the rank function of any lattice path expansion poset, and prove that this rank
function is unimodal whenever the underlying snake graph is built from at most
four maximal straight segments. We show that the support of any type $A$
cluster variable is the orbit of a groupoid. Finally, in work joint with
Nicholas Ovenhouse, we partially generalize $T$-paths to configurations of
affine flags, and prove that a $T$-path expansion analogous to the type $A$
case holds when the initial seed is from a fan triangulation.
