Maximal Chains in Bond Lattices

Let $G$ be a graph with vertex set $\{1,2,\ldots,n\}$. Its bond lattice, $BL(G)$, is sublattice of the partition lattice. The elements $BL(G)$ are partitions whose blocks induce connected subgraphs $G$.
 In this article, we consider graphs lattice consists only noncrossing partitions. We define family graphs, called triangulation property and show that any two produce isomorphic lattices. then look at enumeration maximal chains in lattices graphs. Stanley's map from to parking functions was our motivation. find restriction his certain Finally, number ordered cycle decompositions.