A Translation Method for Finding Combinatorial Bijections

Abstract

Consider a combinatorial identity that can be proved by induction. In this paper, we describe a general method for translating the inductive proof into a recursive bijection. Furthermore, we will demonstrate that the resulting recursive bijection can often be defined in a direct, non-recursive way. Thus, the translation method often results in a bijective proof of the identity that helps illuminate the underlying combinatorial structures. This paper has two main parts: First, we describe the translation method and the accompanying Maple code; and second, we give a few examples of how the method has been used to discover new bijections.