Further enumeration results concerning a recent equivalence of restricted inversion sequences
Résumé
Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class In(≥,≠,>) is the same as that of n−1−asc on the class In(>,≠,≥), which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on In(≥,≠,>) of asc and desc along with two other parameters, and do the same for n−1−asc and desc on In(>,≠,≥). By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result |In(≥,≠,>)|=|In(>,≠,≥)|. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.
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