An Expansion Formula for Decorated Super-Teichmüller Spaces

Motivated by the definition of super-Teichm\"uller spaces, and Penner-Zeitlin's recent extension this to decorated space, as examples super Riemann surfaces, we use Ptolemy relations obtain formulas for $\lambda$-lengths associated arcs in a bordered surface. In special case disk, are able give combinatorial expansion diagonals polygon spirit Ralf Schiffler's $T$-path type $A$ cluster algebras. We further connect our super-friezes Morier-Genoud, Ovsienko, Tabachnikov, partial progress towards defining algebras $A_n$. particular, following Penner-Zeitlin, get (up signs) $\mu$-invariants triangles triangulated polygon, explain how these provide step understanding odd variables algebra.