  The $A_\infty$ T-system, also called the octahedron recurrence, is a
dynamical recurrence relation. It can be realized as mutation in a
coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We
define T-systems with principal coefficients from cluster algebra aspect, and
give combinatorial solutions with respect to any valid initial condition in
terms of partition functions of perfect matchings, non-intersecting paths and
networks. This also provides a solution to other systems with various choices
of coefficients on T-systems including Speyer's octahedron recurrence (Speyer
2007), generalized lambda-determinants (Di Francesco 2013) and (higher)
pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et
al. 2014).
