Equivalence of the descents statistic on some (4,4)-avoidance classes of permutations

In this article, we compute the generating function of the joint distribution of the first letter and descents statistics on six avoidance classes of permutations corresponding to two patterns of length four thereby demonstrating their equivalence. 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 nth large Schröder number, and thus, one obtains a new bivariate refinement of these numbers as a consequence. We make use of auxiliary combinatorial statistics to establish a system of recurrences for the distribution in question in each case and define special generating functions (specific to the class) based on the system. In some cases, we utilize the conjecture itself in a creative way to aid in solving the functional equations satisfied by these associated generating functions and in others use the kernel method.