Bijections of plane Husimi graphs and certain combinatorial structures

Abstract

Plane Husimi graphs are combinatorial structures obtained when we replace edges in plane trees with complete graphs such that the resultant structures are connected and cycle-free. The formula that counts these structures is known to enumerate other combinatorial structures. In this paper, we construct bijections between the set of plane Husimi graphs and the sets of plane trees, dissections of convex polygons, sequences satisfying certain properties, standard Young tableaux, Deutsch paths and restricted lattice paths.