Mathematics Department, Saint Peter's University, New Jersey, USA
Bijective proofs are a popular means for verifying combinatorial theorems without much algebra; the goal is to make the results visually apparent. We will survey several of these proofs before focusing on a particular case: the mathematical objects n-color compositions, introduced by Agarwal in 2000, and a tool to visualize them, spotted tilings, introduced by the speaker in 2013. These allow us to give combinatorial interpretations for many known integer sequences and see new results about them. You may be surprised with what can be done just adding up positive integers and putting dots in rectangles.
This is joint work with Hua Wang of Georgia Southern University.