  Counting pattern avoiding ballot paths begins with a careful analysis of the
pattern. Not the length, but the characteristics of the pattern are responsible
for the difficulties in finding explicit solutions. Certain features, like
overlap and difference in number of right and up steps determine the solution
of the recursion formula. If the recursion can be solved by a polynomial
sequence, we apply the Finite Operator Calculus to find an explicit form of the
solution in terms of binomial coefficients. Keywords: Pattern avoidance, ballot
path, Dyck path, Finite Operator Calculus, Umbral Calculus
