Counting subwords in flattened involutions and Kummer functions

In this paper, we consider the problem of counting subword patterns in flattened involutions, which extends recent work on set partitions. We determine generating function formulas for the distribution of τ on the set I_n of involutions of size n in all cases in which τ is a subword of length three. In the cases of 123 and 132, the exponential generating function for the distribution may be expressed in terms of the Kummer functions. In the cases of 213 and 312, we consider, instead, the ordinary generating function for the distribution and show that it satisfies a functional equation in two parameters that can be solved explicitly in the avoidance case. Recurrences are also given for the general patterns 12...k, (k-2)k(k-1), and k12...(k-1), and in the first two cases, it is shown that the distribution on I_n for the number of occurrences of the patterns in the flattened sense is P-recursive for all k≥3.