Refined Inversion Statistics on Permutations

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 $(\le\!\!k)$-step inversions and $(k_1,k_2)$-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 $n$.