{"id":339350,"date":"2016-12-20T10:49:45","date_gmt":"2016-12-20T18:49:45","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=339350"},"modified":"2018-10-16T21:11:51","modified_gmt":"2018-10-17T04:11:51","slug":"representing-permutations-moves","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/representing-permutations-moves\/","title":{"rendered":"Representing Permutations with Few Moves"},"content":{"rendered":"<p>Consider a finite sequence of permutations of the elements 1, . . . , <em>n<\/em>, with the property that each element changes its position by at most 1 from any permutation to the next. We call such a sequence a <em>tangle<\/em>, and we define a <em>move<\/em> of element <em>i<\/em> to be a maximal subsequence of at least two consecutive permutations during which its positions form an arithmetic progression of common difference +1 or \u22121. We prove that for any initial and final permutations, there is a tangle connecting them in which each element makes at most 5 moves, and another in which the total number of moves is at most 4<em>n<\/em>. On the other hand, there exist permutations that require at least 3 moves for some element, and at least 2<em>n<\/em> \u2212 2 moves in total. If we further require that every pair of elements exchange positions at most once, then any two permutations can be connected by a tangle with at most O(log <em>n<\/em>) moves per element, but we do not know whether this can be reduced to O(1) per element, or to O(<em>n<\/em>) in total. A key tool is the introduction of certain restricted classes of tangle that perform pattern-avoiding permutations.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider a finite sequence of permutations of the elements 1, . . . , n, with the property that each element changes its position by at most 1 from any permutation to the next. We call such a sequence a tangle, and we define a move of element i to be a maximal subsequence of [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-339350","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Cornell University 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