  We study the set of networks, which consist of sources, sinks and neutral
points, bijective to the permutations. The set of directed edges, which
characterizes a network, is constructed from a polyomino or a Rothe diagram of
a permutation through a Dyck tiling on a ribbon. We introduce a new
combinatorial object similar to a tree-like tableau, which we call a forest. A
forest is shown to give a permutation, and be bijective to a network
corresponding to the inverse of the permutation. We show that the poset of
networks is a finite graded lattice and admits an $EL$-labeling. By use of this
$EL$-labeling, we show the lattice is supersolvable and compute the M\"obius
function of an interval of the poset.
