Refined inversion statistics on permutations
Abstract
We introduce and study new refinements of inversion statistics for permutations, such as kstep inversions, (the number of inversions with fixed position differences) and noninversion sums (the sum of the differences of positions of the noninversions of a permutation). We also provide a distribution function for noninversion sums, a distribution function for kstep 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 noninversion 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2011
 DOI:
 10.48550/arXiv.1106.1995
 arXiv:
 arXiv:1106.1995
 Bibcode:
 2011arXiv1106.1995S
 Keywords:

 Mathematics  Combinatorics;
 05A05;
 05A15
 EPrint:
 27 pages, 3 figures, 6 tables