We introduce and study new refinements of inversion statistics for permutations, such as $k$-step inversions, (the number of inversions with fixed position differences) and non-inversion sums (the sum of the differences of positions of the non-inversions of a permutation). We also provide a distribution function for non-inversion sums, a distribution function for $k$-step inversions that relates to the Eulerian polynomials, and special cases of distribution functions for other statistics we… Expand