{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T22:54:29Z","timestamp":1693349669975},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2019,10,14]],"date-time":"2019-10-14T00:00:00Z","timestamp":1571011200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline1\" \/><jats:tex-math>$$\\tau \\in {S_3}$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline2\" \/><jats:tex-math>$$\\mu _n^\\tau $$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> denote the uniformly random probability measure on the set of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline3\" \/><jats:tex-math>$$\\tau $$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-avoiding permutations in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline4\" \/><jats:tex-math>$${S_n}$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline5\" \/><jats:tex-math>$${\\mathbb {N}^*} = {\\mathbb {N}} \\cup \\{ \\infty \\} $$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with an appropriate metric and denote by <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline6\" \/><jats:tex-math>$$S({\\mathbb{N}},{\\mathbb{N}^*})$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> the compact metric space consisting of functions <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline7\" \/><jats:tex-math>$$\\sigma {\\rm{  = }}\\{ {\\sigma _i}\\} _{i = 1}^\\infty {\\rm{  }}$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> from <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline8\" \/><jats:tex-math>$$\\mathbb {N}$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline9\" \/><jats:tex-math>$${\\mathbb {N}^ * }$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> which are injections when restricted to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline10\" \/><jats:tex-math>$${\\sigma ^{ - 1}}(\\mathbb {N})$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>; that is, if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline11\" \/><jats:tex-math>$${\\sigma _i}{\\rm{  = }}{\\sigma _j}$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline12\" \/><jats:tex-math>$$i \\ne j$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, then <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline13\" \/><jats:tex-math>$${\\sigma _i} = \\infty $$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Extending permutations <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline14\" \/><jats:tex-math>$$\\sigma \\in {S_n}$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> by defining <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline15\" \/><jats:tex-math>$${\\sigma _j} = j$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline16\" \/><jats:tex-math>$$j \\gt n$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we have <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline17\" \/><jats:tex-math>$${S_n} \\subset S({\\mathbb{N}},{{\\mathbb{N}}^*})$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. For each <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline18\" \/><jats:tex-math>$$\\tau \\in {S_3}$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we study the limiting behaviour of the measures <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline19\" \/><jats:tex-math>$$\\{ \\mu _n^\\tau \\} _{n = 1}^\\infty $$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline20\" \/><jats:tex-math>$$S({\\mathbb{N}},{\\mathbb{N}^*})$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We obtain partial results for the permutation <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline21\" \/><jats:tex-math>$$\\tau  = 321$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and complete results for the other five permutations <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548319000270_inline22\" \/><jats:tex-math>$$\\tau \\in {S_3}$$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0963548319000270","type":"journal-article","created":{"date-parts":[[2019,10,14]],"date-time":"2019-10-14T08:05:29Z","timestamp":1571040329000},"page":"137-152","source":"Crossref","is-referenced-by-count":1,"title":["The Infinite limit of random permutations avoiding patterns of length three"],"prefix":"10.1017","volume":"29","author":[{"given":"Ross G.","family":"Pinsky","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,10,14]]},"reference":[{"key":"S0963548319000270_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2013.12.004"},{"key":"S0963548319000270_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-07965-3"},{"key":"S0963548319000270_ref2","doi-asserted-by":"publisher","DOI":"10.1201\/9780203494370"},{"key":"S0963548319000270_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000171"},{"key":"S0963548319000270_ref3","volume-title":"Limit Distributions for Sums of Independent Random Variables","author":"Gnedenko","year":"1968"},{"key":"S0963548319000270_ref1","doi-asserted-by":"publisher","DOI":"10.1214\/17-AOP1223"},{"key":"S0963548319000270_ref4","doi-asserted-by":"publisher","DOI":"10.1214\/10-AOP536"},{"key":"S0963548319000270_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2012.01.001"},{"key":"S0963548319000270_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20677"},{"key":"S0963548319000270_ref10","doi-asserted-by":"publisher","DOI":"10.1214\/18-AOP1286"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548319000270","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,14]],"date-time":"2020-01-14T05:56:53Z","timestamp":1578981413000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548319000270\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,14]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["S0963548319000270"],"URL":"https:\/\/doi.org\/10.1017\/s0963548319000270","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,14]]}}}