  We revisit the concepts of acyclic orderings and number of acyclic orderings
of acyclic digraphs in terms of dispositions and counters for arbitrary
multidigraphs. We prove that when we add a sequence of nested directed paths to
a directed graph there is a unique polynomial such that the generatrix function
of the family of counters is the product of the polynomial and the exponential
function. We give an application, by considering a kind of digraphs arranged in
rows introduced by the authors in a previous paper, called dispositional
digraphs, in the particular case in which the digraph has two rows, to obtain
new families of linear differential equations of small order whose coefficients
are polynomials of small degree which admit polynomial solutions. In
particular, we obtain a new differential equation associated to Catalan
numbers, and the corresponding associated polynomials, which are solution of
this differential equation; we term them Catalan differencial equation and
Catalan polynomials, respectively. We prove that the Catalan polynomials
obtained when we connect the directed path to the second vertex of the lower
row of the digraph are orthogonal polynomials for an appropriate weight
function. We characterize the digraphs that maximize the counter of connected
dispositional digraphs and we find a new differential equation associated to
these digraphs. We introduce also dispositions and counters in any multidigraph
with non-strict inequalities in the dispositions, and we find new differential
equations associated to some of them.
