2024/04/25 by Cornwell, Christopher, McNew, Nathan
#05A05 #05A15 #57K10 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2404.16755
In 1977 Diaconis and Graham proved two inequalities relating different measures of disarray in permutations, and asked for a characterization of those permutations for which equality holds in one of these inequalities. Such a characterization was first given in 2013. Recently, another characterization was given by Woo, using a topological link in \mathbb R3 that can be associated to the cycle diagram of a permutation. We show that Woo's characterization extends much further: for any permutation, the discrepancy in Diaconis and Graham's inequality is directly related to the Euler characteristic of the associated link. This connection provides a new proof of the original result of Diaconis and Graham. We also characterize permutations with a fixed discrepancy in terms of their associated links and find that the stabilized-interval-free permutations are precisely those whose associated links are nonsplit.