BiSC: An algorithm for discovering generalized permutation patterns

TL;DR

BiSC is a new algorithm that automatically generates conjectures about permutation sets in terms of avoided patterns, aiding in the discovery and proof of combinatorial theorems across various mathematical fields.

Contribution

The paper introduces BiSC, an algorithm that conjectures pattern avoidance descriptions for permutation classes, including known and new theorems, streamlining combinatorial research.

Findings

Successfully conjectured descriptions of known permutation classes

Discovered new theorems related to symmetry groups and tableaux

Automated pattern avoidance conjectures for complex permutation sets

Abstract

Theorems relating permutations with objects in other fields of mathematics are often stated in terms of avoided patterns. Examples include various classes of Schubert varieties from algebraic geometry (Billey and Abe 2013), commuting functions in analysis (Baxter 1964), beta-shifts in dynamical systems (Elizalde 2011) and homology of representations (Sundaram 1994). We present a new algorithm, BiSC, that, given any set of permutations, outputs a conjecture for describing the set in terms of avoided patterns. The algorithm automatically conjectures the statements of known theorems such as the descriptions of smooth (Lakshmibai and Sandhya 1990) and forest-like permutations (Bousquet-M{\'e}lou and Butler 2007), Baxter permutations (Chung et al. 1978), stack-sortable (Knuth 1975) and West-2-stack-sortable permutations (West 1990). The algorithm has also been used to discover new theorems and conjectures related to the dihedral and alternating subgroups of the symmetric group, Young tableaux, Wilf-equivalences, and sorting devices.