Pattern Avoiding Ballot Paths and Finite Operator Calculus

TL;DR

This paper investigates pattern avoiding ballot paths, analyzing how pattern features influence recursive solutions, and applies Finite Operator Calculus to derive explicit formulas when solutions are polynomial sequences.

Contribution

It introduces a method to determine explicit solutions for pattern avoiding ballot paths using Finite Operator Calculus based on pattern characteristics.

Findings

Recursive formulas depend on pattern overlap and step differences.

Finite Operator Calculus yields explicit binomial coefficient formulas.

Method applies to polynomial sequence solutions in pattern avoidance problems.

Abstract

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