Pattern Avoiding Ballot Paths and Finite Operator Calculus

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