ABSTRACT

We establish the connection between the two most famous Mahonian statistics, namely the number of inversions of a permutation, and the major index of a permutation. We extend their definitions to multisets. We introduce Gaussian coefficients, review a few of their occurrences, then we use them to prove that these two statistics are equidistributed for permutations of multisets as well. We also prove Euler's pentagonal number theorem on integer partitions and show how to apply it to find a formula for the number of permutations of a given length with a given number of inversions. As an application of inversions, we consider their roles in computing determinants.