{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,11,15]],"date-time":"2024-11-15T14:40:21Z","timestamp":1731681621507,"version":"3.28.0"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T00:00:00Z","timestamp":1719446400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T00:00:00Z","timestamp":1719446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2024,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {R}}^3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that can be associated to the cycle diagram of a permutation. We show that Woo\u2019s characterization extends much further: for any permutation, the discrepancy in Diaconis and Graham\u2019s 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.<\/jats:p>","DOI":"10.1007\/s00493-024-00107-1","type":"journal-article","created":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T15:05:47Z","timestamp":1719500747000},"page":"1149-1167","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Links and the Diaconis\u2013Graham Inequality"],"prefix":"10.1007","volume":"44","author":[{"given":"Christopher","family":"Cornwell","sequence":"first","affiliation":[]},{"given":"Nathan","family":"McNew","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,6,27]]},"reference":[{"key":"107_CR1","doi-asserted-by":"publisher","unstructured":"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","DOI":"10.37236\/10858"},{"issue":"1","key":"107_CR2","volume":"7","author":"D Callan","year":"2004","unstructured":"Callan, D.: Counting stabilized-interval-free permutations. J. Integer Seq. 7(1), Article 04.1.8 (2004)","journal-title":"J. Integer Seq."},{"issue":"3","key":"107_CR3","doi-asserted-by":"publisher","DOI":"10.37236\/11016","volume":"29","author":"CR Cornwell","year":"2022","unstructured":"Cornwell, C.R., McNew, N.: Unknotted cycles. Electron. J. Combin. 29(3), Paper No. 3.50, 26 (2022)","journal-title":"Electron. J. Combin."},{"issue":"2","key":"107_CR4","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1111\/j.2517-6161.1977.tb01624.x","volume":"39","author":"P Diaconis","year":"1977","unstructured":"Diaconis, P., Graham, R.L.: Spearman\u2019s footrule as a measure of disarray. J. Roy. Statist. Soc. Ser. B 39(2), 262\u2013268 (1977)","journal-title":"J. Roy. Statist. Soc. 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