Differential equations in the theory of permutation patterns

A permutation pattern is a permutation of a totally ordered set. We deal with permutations in one-line notation. An occurrence of a pattern p in a permutation P is a subsequence of elements of P whose relative order is the same as that of the elements in p. As an example, the permutation 461352 has three occurrences of the pattern 321, namely the subsequences 432, 632 and 652.
The study of permutation patterns originated in theoretical computer science by Donald Knuth in 1969. However, the first systematic study was done in 1985 by Simon and Schmidt, who completely classified the avoidance of patterns of length three. Since then about 2000 papers related to the field have been published.
In my talk I will explain, by means of examples, what differential equations (DEs) have to do with enumerative problems related to the permutation patterns. I will give a few examples of DEs appearing in the research, including those ones for which solutions are unknown.