2×2 Monotone Grid Classes Are Finitely Based

In recent years, the emerging theory of grid classes has led to some of the major structural and enumerative developments in the study of permutation patterns. Particular highlights include the characterisation of all possible “small” growth rates (Huczynska and Vatter, 2006; Kaiser and Klazar, 2003; Vatter, 2011) and the subsequent result that all classes with these growth rates have rational generating functions (Albert et al., 2015). To support results such as these, the study of grid classes themselves has gained importance. Restricting one’s attention to monotone grid classes, it is known that the structure of the matrix defining a grid class determines both its growth rate (Bevan, 2015), and whether it is well-partially-ordered (Murphy and Vatter, 2003). One remaining open question about monotone grid classes concerns their bases, that is, the sets of minimal forbidden permutations of the classes. Backed up by some computational evidence, it is widely believed that all monotone grid classes are finitely based, but this is only known to be true for certain families, most notably those whose row-column graphs(i) are forests (Albert et al., 2013). To date, the only other instances of monotone grid classes that are known to have a finite basis are two 2 × 2 grid classes. The first concerns the class of skew-merged permutations, Av(2143, 3412), in (Stankova, 1994), while the second is in Waton’s PhD thesis (Waton, 2007). Inspired by Waton’s approach, we show that a certain family of (non-monotone) 2 × 2 grid classes are all finitely based, from which we can conclude the following result.