Refined inversion statistics on permutations
Electronic Journal of Combinatorics, Volume 19 (2012)
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 (\leq
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.
Updates
- Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker and Amanda Welch prove that the cosine map is a homomesy in their study of the FindStat database in the preprint Homomesies on permutations - an analysis of maps and statistics in the FindStat database
- Nathan Chapelier-Laget and Thomas Gerber find connections between the topics in our paper to entropy and atomic length in the preprint Atomic length in Weyl groups
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Presentations
- Counting special inversions in permutations, Mathematics Colloquium, University of Iceland, Reykjavik, Iceland, November 2010 (Joshua presented)
- Counting special inversions in permutations, Mathematics Colloquium, California State University, Long Beach, California, December 2010 (Henning presented)
- Refined inversion statistics on permutations, 2012 Joint Mathematics Meetings, Boston, MA, January 2012 (Joshua presented)