  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 introduce, such as
(\leqk)-step inversions and (k1,k2)-step inversions (that fix the value
separation as well as the position). We connect our refinements to other work,
such as inversion tops that are 0 modulo a fixed integer d, left boundary sums
of paths, and marked meshed patterns. Finally, we use non-inversion sums to
show that for every number n > 34, there is a permutation such that the dot
product of that permutation and the identity permutation (of the same length)
is n.
