[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83774-en":3,"doc-seo-83774-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83774,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","When Arrow Patterns Meet Classical Patterns","The work bridges permutation cycle notation and one-line notation through the arrow pattern introduced by Berman and Tenner. It builds on Archer and Laudone’s systematic study of arrow-pattern avoidance, which left three conjectures unresolved. This paper completes the enumeration of permutations avoiding both a classical pattern of length 3 and a fixed arrow pattern of length 3, confirming the first two conjectures. It settles the remaining conjecture via two independent bijective proofs using restricted Foata-type constructions and Dyck-path correspondences.","arXiv :2607 .04094v1 [math .CO] 5 Jul 2026  \nWHEN ARROW PATTERNS MEET CLASSICAL PATTERNS  \nSHISHUO FU˚ AND ZHENGHE YANG  \nAbstract. Seeking to bridge the structural divide between a permutation’s cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures.  \nIn this paper, we resolve all three conjectures. First, we enumerate all six subclasses of permutations that simultaneously avoid a classical pattern of length 3 and a fixed arrow pattern of length 3, thereby confirming the first two conjectures. Second, we settle the third conjecture (which involves a different arrow pattern) by providing two independent proofs. These proofs rely on a restriction of Biane’s bijection to non-nesting involutionsand Krattenthaler’s bijection from 321-avoiding permutations to Dyck paths, respectively.  \n1. Introduction  \nLet Sn denote the set of permutations on rns :“ t1 , 2 , . . . , nu. Given two permutations σ P Sn and π P Sm , we say that σ contains π as a (classical) pattern, if there exist 1 ď i 1 ă i2 ă ¨ ¨ ¨ ă im ď n such that entries σi1 , σi2 , . . . , σim form a sequence that is order-isomorphic to π . Otherwise σ is said to avoid π . We use Snpπq to denote the set of n-permutations that avoid the pattern π . The problem of enumerating various classes of pattern-avoiding permutations has spawned a stunning amount of work in enumerative combinatorics; see Kitaev’s book exposition [13] for further information on this fast-developing field.  \nIn a recent study on the so-called “shallow” permutations, Berman and Tenner [4] introduced a new notion of permutation pattern called the arrow pattern, whose definition we will recall in next section. Interest in shallow permutations stems from their role in understanding the Diaconis-Graham inequality [7 , 8 , 15 , 22], which involves three fundamental permutation statistics: length, reflection length, and depth (or total displacement) . As revealed by Berman and Tenner [4], the arrow pattern serves as a natural framework to simultaneously capture the structural information required by all three statistics.  \nArcher and Laudone initiated in [1] the enumeration of arrow pattern avoiding permutations. Towards the end of their paper, they paired arrow pattern avoidance with classical pattern avoidance and made the following three intriguing conjectures. Let π be a classical pattern and α be an arrow pattern, then for every n P N, we denote by an pπ, αq :“ |Snpπ, αq| the number of n-permutations that avoid simultaneously two patterns π and α . Further notations and some preliminary results will be given in Section 2.  \nDate: July 7, 2026 .  \nKey words and phrases . permutation pattern, arrow pattern, Fibonacci numbers, Catalan numbers, Motzkin paths.  \n2020 Mathematics Subject Classification. 05A05, 05A15, 05A19 .  \n˚ Corresponding author: Shishuo Fu.  \n2 S. FU AND Z. YANG  \nConjecture 1.1 ( [1, Conjecture 7.1]) . For n ě 2, we have  \n(1) an p123 , p12; 1 Ñ 3qq “ 2n ´ n,  \n(2) an p321 , p12; 1 Ñ 3qq “ F2n´1 ,  \n(3) an p321 , p12; 1 Ñ 2qq “ Mn ,  \nwhere Fn is the n-th Fibonacci number [19, A000045] and Mn is the n-th Motzkin number [19, A001006] .  \nMotivated by this conjecture, in the current paper we carry out a complete enumeration of Snpπ, αq, where π ranges over all six classical patterns of length 3 and α “ p12; 1 Ñ 3q. The results are summarized in Table 1 and the proofs are given in Section 3. Recall that Cn :“ n1`1 `2nn˘ is the n-th Catalan number [19, A000108] . In particular, this confirms items (1) and (2) from Conjecture 1.1. The remaining part (3) is proved and refined/generalized in Section 4, in two ways both of which are bijective in nature; see Theorems 4.4 and 4.6. We conclude the paper with some remarks that hopefully could stimulate future research.  \n\n| π | an pπ, p12; 1 Ñ ","cbCairVGHW90cXzZ","https://ap.wps.com/l/cbCairVGHW90cXzZ","pdf",583718,2,1,17,"English","en",105,"# Introduction\n# Conjectures and Enumeration Results\n# Preliminaries","[{\"question\":\"What problem does the paper address regarding permutation patterns?\",\"answer\":\"The paper studies permutations that avoid simultaneously a classical pattern and an arrow pattern, aiming to resolve conjectures about their counts.\"},{\"question\":\"Which conjectures are resolved in this paper?\",\"answer\":\"All three conjectures from Archer and Laudone’s study are resolved: two are confirmed through a complete enumeration, and the third is proved in two independent ways.\"},{\"question\":\"What mathematical tools are used to prove the results?\",\"answer\":\"The proofs use bijective techniques, including a restriction of Biane’s bijection to non-nesting involutions and a correspondence between 321-avoiding permutations and Dyck paths via Krattenthaler’s bijection.\"}]",1784190332,43,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"when-arrow-patterns-meet-classical-patterns","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/when-arrow-patterns-meet-classical-patterns/83774/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address regarding permutation patterns?","Question",{"text":75,"@type":76},"The paper studies permutations that avoid simultaneously a classical pattern and an arrow pattern, aiming to resolve conjectures about their counts.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which conjectures are resolved in this paper?",{"text":80,"@type":76},"All three conjectures from Archer and Laudone’s study are resolved: two are confirmed through a complete enumeration, and the third is proved in two independent ways.",{"name":82,"@type":73,"acceptedAnswer":83},"What mathematical tools are used to prove the results?",{"text":84,"@type":76},"The proofs use bijective techniques, including a restriction of Biane’s bijection to non-nesting involutions and a correspondence between 321-avoiding permutations and Dyck paths via Krattenthaler’s 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