Author:
Cornwell Christopher,McNew Nathan
Abstract
AbstractIn 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 $${\mathbb {R}}^3$$
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.
Publisher
Springer Science and Business Media LLC
Reference12 articles.
1. Berman, Y., Tenner, B.E.: Pattern-functions, statistics, and shallow permutations. Electron. J. Combin. 29(4), Paper No. 4.43 (2022). https://doi.org/10.37236/10858
2. Callan, D.: Counting stabilized-interval-free permutations. J. Integer Seq. 7(1), Article 04.1.8 (2004)
3. Cornwell, C.R., McNew, N.: Unknotted cycles. Electron. J. Combin. 29(3), Paper No. 3.50, 26 (2022)
4. Diaconis, P., Graham, R.L.: Spearman’s footrule as a measure of disarray. J. Roy. Statist. Soc. Ser. B 39(2), 262–268 (1977)
5. Eliashberg, Y.: Legendrian and transversal knots in tight contact 3-manifolds. In: Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), pp. 171–193 (1993)