Combinatorial Exploration: An algorithmic framework to automate the proof of results in combinatorics

Are you interested in combinatorial mathematics? If so, I am happy to welcome you to the future! 

I am proud to share with you a new, 99-page preprint by three colleagues from my department (Christian Bean, Émile Nadeau and Henning Ulfarsson) and three of their collaborators (Michael Albert, Anders Claesson and Jay Pantone) that is the result of years of work that led to the birth of Combinatorial Exploration. Combinatorial Exploration is an algorithmic framework that can prove results that so far have required the ingenuity of human combinatorialists. More specifically, it can study, at the press of a button, the structure of combinatorial objects and derive their counting sequences and generating functions. The applicability and power of Combinatorial Exploration are witnessed by the applications to the domain of permutation patterns given in the paper. My colleagues use it to re-derive hundreds of results in the literature in a uniform manner and prove many new ones. See the Permutation Pattern Avoidance Library (PermPAL) and Section 2.4 of the article for a more comprehensive list of notable results. The paper also gives three additional proofs-of-concept, showing examples of how Combinatorial Exploration can prove results in the domains of alternating sign matrices, polyominoes, and set partitions. Last, but by no means least, the github repository at https://github.com/PermutaTriangle/comb_spec_searcher contains the open-source python framework for Combinatorial Exploration, and the one at https://github.com/PermutaTriangle/Tilings contains the code needed to apply it to the field of permutation patterns. 

 "Det er svært at spå, især om fremtiden", as Niels Bohr and Piet Hein famously said. However, let me stick my neck out and predict that this work will have substantial impact and will be the basis for exciting future work. 

 Congratulations to my colleagues! With my department chair hat on, I am very proud to see work of this quality stem from my department and humbled by what my colleagues have achieved already. As an interested observer, I am very excited to see what their algorithms will be able to prove in the future. For the moment, let's all enjoy what they have done already.