Recent Results in Pattern-avoiding Permutations

In the late 1960s Don Knuth asked which permutations could be sorted by various data structures. For a simple stack, he proved that permutations avoiding the pattern 312 can be so sorted, and further, that the number of such permutations of length n is given by Cn, the n th Catalan number. Knuth went on to pose the same question about other data structures that are widely used in sorting operations, namely deques, and two stacks, both in parallel and series. All these problems remain unsolved to this day, though there have been important developments. However while it is known that the number of permutations sortable by these structures grows exponentially, with a structure-dependent growth constant, the value of this constant is unknown in each case. We will discuss our efforts to estimate the constant in a number of cases, and also discuss the sub-leading asymptotic behaviour. Knuth’s work and subsequent developments saw the area of patternavoiding permutation develop into an interesting area of combinatorics in its own right. The fundamental question asked is “how many permutations of length n avoid a given pattern?” Here the pattern is a sub-permutation. The answer to this question is known for the 3! = 6 patterns of length three. In every case the answer is given by the n Catalan number. For the 4! = 24 permutations of length four, it is known that these fall into 3 distinct classes. Two of these have been completely solved, and it is known that the growth constants in the two solved cases are 8 and 9 exactly. For the third class, those avoiding the pattern 1324, very little is exactly known. By extensive computer enumeration, and careful numerical work, we show that the growth constant is 11.60 ± 0.01 and obtain a good estimate of the asymptotic behaviour, which is quite different to that for the two solved cases. Finally, we comment on permutations avoiding the vincular pattern abcd−e. Several cases of such patterns remain unsolved. From extensive enumerations we are able to make conjectures for the exact value of the growth constants for several of the unsolved cases. These turn out to be transcendental numbers. We then prove that the corresponding generating functions cannot be D-finite, contrary to earlier conjectures that all such patterns are D-finite. This is joint work with Andrew Conway.