  Circuit topology refers to the arrangement of interactions between objects
belonging to a linearly ordered object set. Linearly ordered set of objects are
common in nature and occur in a wide range of applications in economics,
computer science, social science and chemical synthesis. Examples include
linear bio-polymers, linear signaling pathways in cells as well as topological
sorts appearing in project management. Using a statistical mechanical
treatment, we study circuit topology landscapes of linear polymer chains with
intra-chain contacts as a prototype of linearly sorted objects with
interactions. We find generic features of the topological space and study the
statistical properties of the space under the most basic constraints on the
occupancy of arrangements and topological interactions. We observe that a set
of correlated contact sites (a sector) could nontrivially influence the entropy
of circuits as the number of involved sites increases. Finally, we discuss how
constraints can be inferred from the information provided by local contact
distributions in presence of a sector.
