Refined inversion statistics on permutations

TL;DR

This paper introduces new refined inversion statistics for permutations, analyzes their distributions, and connects them to existing combinatorial concepts, ultimately showing that all integers greater than 34 can be represented as dot products with permutations.

Contribution

It presents novel permutation statistics like k-step inversions and non-inversion sums, along with their distribution functions and connections to other combinatorial structures.

Findings

Distribution functions for non-inversion sums and k-step inversions

Connections to Eulerian polynomials and other combinatorial patterns

Existence of permutations with specific dot product values for all n > 34

Abstract

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.