  In this paper, we compute the distribution of the first letter statistic on
nine avoidance classes of permutations corresponding to two pairs of patterns
of length four. In particular, we show that the distribution is the same for
each class and is given by the entries of a new Schr\"oder number triangle.
This answers in the affirmative a recent conjecture of Lin and Kim. We employ a
variety of techniques to prove our results, including generating trees, direct
bijections and the kernel method. For the latter, we make use of in a creative
way what we are trying to show in three cases to aid in solving a system of
functional equations satisfied by the associated generating functions.
