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Mathematics > Combinatorics
Title: Links and the Diaconis-Graham Inequality
(Submitted on 25 Apr 2024)
Abstract: 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 $\mathbb R^3$ 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.
Submission history
From: Christopher Cornwell [view email][v1] Thu, 25 Apr 2024 17:11:52 GMT (23kb)
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