Abstract In previous publications [J. Geom. Phys. 38 (2001) 81 and references therein] the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface. With the help of this function the dynamical transition from pseudo-Anosov to periodic (Seifert-fibered) regime was studied. In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich [Commun. Math. Phys. 147 (1992) 1] inspired by earlier work of Witten on topological two-dimensional quantum gravity. To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some nontraditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics.