-
To appear in Memoirs of the American Mathematical Society.
Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions.
-
Rutgers Experimental Mathematics Seminar (March 10, 2022)
Combinatorial Exploration is a framework that unifies the often ad-hoc methods used in enumerative combinatorics. In this talk we will explain how Combinatorial Exploration works, how it can be automated, and how it is applied to the study of pattern-avoiding permutations to prove new results and reprove dozens of old ones. We will also discuss the new web database PermPAL that catalogs these results.
-
The Permutation Pattern Avoidance Library (PermPAL) is a database of algorithmically-derived theorems about permutation classes. These results are produced by the Combinatorial Exploration framework.
-
National Science Foundation research grant
This project aims to improve the proof-writing abilities of undergraduate students by training an innovative artificial intelligence model to provide feedback on student proofs to support both college students in proof-oriented mathematics courses and the post-secondary faculty who are teaching them.
-
Submitted.
A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is rational, and we give a procedure to construct a combinatorial finite state machine that allows one to compute this generating function. We then modify this procedure to compute generating functions for GSAWs under two probabilistic models. We perform Monte Carlo simulations to estimate the expected length and displacement for GSAWs on the quarter plane, half plane, full plane, and half-infinite strips of bounded height for which we cannot compute the generating function. Finally, we prove that the generating functions for Greek key tours (GSAWs on a finite grid that visit every vertex) on a half-infinite strip of fixed height are also rational, allowing us to resolve several conjectures.
-
Rutgers Experimental Mathematics Seminar (March 10, 2022)
Combinatorial structures are ubiquitous throughout mathematics. Graphs, permutations, words, and other such families of combinatorial objects often play a central role in work from many different fields. The study of enumerative combinatorics is concerned with the elucidation of structural properties of these families, including counting, classification, and limiting behavior.
Combinatorial Exploration is a framework that unifies the often ad-hoc methods used in enumerative combinatorics. In this talk we will explain how Combinatorial Exploration works, how it can be automated, and how it is applied to the study of pattern-avoiding permutations to prove new results and reprove dozens of old ones. We will also discuss the new web database PermPAL that catalogs these results.
