Abstract
For permutations \({\pi}\) and \({\tau}\) of lengths \({|\pi|\le|\tau|}\) , let \({t(\pi,\tau)}\) be the probability that the restriction of \({\tau}\) to a random \({|\pi|}\) -point set is (order) isomorphic to \({\pi}\) . We show that every sequence \({\{\tau_j\}}\) of permutations such that \({|\tau_j|\to\infty}\) and \({t(\pi,\tau_j)\to 1/4!}\) for every 4-point permutation \({\pi}\) is quasirandom (that is, \({t(\pi,\tau_j)\to 1/|\pi|!}\) for every \({\pi}\)). This answers a question posed by Graham.
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Daniel Král’: Part of the work leading to this invention has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259385. Oleg Pikhurko: This author was supported by the European Research Council (grant agreement no. 306493) and the National Science Foundation of the USA (grant DMS-1100215).
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Král’, D., Pikhurko, O. Quasirandom permutations are characterized by 4-point densities. Geom. Funct. Anal. 23, 570–579 (2013). https://doi.org/10.1007/s00039-013-0216-9
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DOI: https://doi.org/10.1007/s00039-013-0216-9