The study of Mahonian statistics dated back to 1915 when MacMahon showed that the major index and the inverse number have the same distribution on a set of permutations with length n. Since then, many Mahonian statistics have been discovered and much effort have been done to find the equidistribution between two Mahonian statistics on permutations avoiding length-3 classical patterns. In recent years, Amini and Do et al. have done extensive research with various methods to prove the equidistributions, ranging from using generating functions, Dyck paths, block decompositions, to bijections. In this thesis, we will solve the conjectured equidistribution between bast and foze on Av(312) using the bijection method, as well as refine two established results by Do et al. with a combinatorial approach.

翻译：对马霍尼型统计的研究可追溯至1915年，当时麦克马洪证明了在长度为n的排列集合中，主指标与逆序数具有相同的分布。此后，众多马霍尼型统计量被发现，学者们致力于寻找避免长度为3的经典模式的排列中两类马霍尼型统计量的等分布性。近年来，Amini与Do等人采用生成函数、Dyck路径、块分解及双射等多种方法，对等分布性进行了广泛研究。本文将通过双射方法解决Av(312)上bast与foze统计量等分布性的猜想，并利用组合方法对Do等人的两个已有结论进行改进。