Distributions of the k-major index in random words and permutations

The most noted permutation statistics, inversion number and major index, have been studied extensively in recent decades. Many results found regarding their joint distribution and normality have been found. We focus on the k-major index, a natural interpolation between the inversion number and major index, which exhibits elegant properties of equidistribution. We generalize normality and equidistribution to words with fixed alphabet size and large length. We then draw analogs from the fluctuation of the k-major index on a random, fixed permutation or word as k varies to standard Brownian motion. We prove a fundamental similarity of covariance independence, and find other connections. Finally, we observe limiting shapes of joint distributions of other pairs of potentially promising statistics.