  In this paper, we compute and demonstrate the equivalence of the joint
distribution of the first letter and descent statistics on six avoidance
classes of permutations corresponding to two patterns of length four. This
distribution is in turn shown to be equivalent to the distribution on a
restricted class of inversion sequences for the statistics that record the last
letter and number of distinct positive letters, affirming a recent conjecture
of Lin and Kim. Members of each avoidance class of permutations and also of the
class of inversion sequences are enumerated by the $n$-th large Schr\"oder
number and thus one obtains a new bivariate refinement of these numbers as a
consequence. We make use of auxiliary combinatorial statistics, special
generating functions (specific to each class) and the kernel method to
establish our results. In some cases, we utilize the conjecture itself in a
creative way to aid in solving the system of functional equations satisfied by
the associated generating functions.
