[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84249-en":3,"doc-seo-84249-105":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},84249,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","On Possible Values of the Group Complexity Function of Infinite Words","A group complexity framework generalizes factor complexity and abelian complexity for infinite words. For each length n, factors are grouped into equivalence classes under permutations from selected subgroups Gn of the symmetric group Sn, producing values between abelian and factor complexities. The work defines a universal group complexity property requiring every intermediate complexity integer to be achievable for all n. Sturmian words satisfy this property, and additional results address minimal aperiodic ternary words and eventually periodic words.","arXiv :2607 .07620v 1 [ cs .DM] 8 Jul 2026  \nOn possible values of the group complexity function  \nof infinite words  \nMaksim Launer, Svetlana Puzynina, Ekaterina Voloshinova Saint Petersburg State University, Russia  \nmlauner_official@bk.ru, s.puzynina@gmail.com, evoloshinova@gmail.com  \nAbstract  \nA classical notion of a factor complexity of an infinite word is defined as a function p(n) counting, for each n, the number of distinct factors (or blocks of consecutive letters) of the word of length n. The notion has various generalizations and variants. For example, the abelian complexity pab (n) counts the number of distinct factors of each length n up to abelian equivalence, i.e., only the numbers of occurrences of letters are taken into account, and not their order. The notion of a group complexity generalizes both notions of a factor and an abelian complexities. Namely, given a sequence ω = (Gn) of subgroups of the symmetric group Sn , the group complexity pω (n) of a word counts the number of classes of factors of each length n of the word, where words obtained from one another by permutations from Gn are put in the same class. Taking Gn = Sn , we obtain the abelian complexity, and taking Gn = Id, we recover the factor complexity. Clearly, the group complexity value is between the abelian and the factor complexities. In this paper, we are interested in the following property of words. We say that an infinite word has universal group complexity if for each length n and for each k satisfying pasb (n) ⩽ k ⩽ ps (n), there exists a group G ∈ Sn such that pGs(n) = k. In other words, all “intermediate” values of complexity can be obtained. We show that Sturmian words satisfy the universal group complexity property, while they are not the only ones. We also study the universal group complexity property for aperiodic ternary words of minimal complexity and for eventually periodic words.  \n1 Introduction  \nFor each infinite word w, its factor complexity function counts, for each n, the number of distinct factors of w of length n. This notion was introduced in 1938 in a seminal paper by Morse and Hedlund [15] on symbolic dynamics. Among other results, Morse and Hedlund gave a relation between factor complexity and periodicity in infinite words; namely, they proved that each aperiodic infinite word w has factor complexity at least n + 1 for each length n. They further showed that an infinite word w has complexity n + 1 for each length n if and only if w is binary, aperiodic and balanced, i.e. , w is a Sturmian word (see also [8, 16]) . Thus Sturmian words are those aperiodic words of the lowest factor complexity. They arise naturally in many different areas  \nof mathematics including combinatorics, algebra, number theory, ergodic theory and dynamical systems. Sturmian words also have applications in theoretical physics as 1-dimensional models of quasi-crystals, and in theoretical computer science where they are used in computer graphics as digital approximation of straight lines. For more on Sturmian words, we refer to Chapter 2 in [14] .  \nProblems in the study of factor complexity of infinite words include characterizing complexities of important families of words, such as morphic [17] and Toeplitz [5] words, and the study of words of linear complexity [4, 13] . An important longstanding open problem in combinatorics on words is an inverse problem of characterizing possible complexity functions of infinite words; see, e.g., [2] and a recent characterization in an asymptotic form [10] . For more on factor complexity we refer to a book chapter [6] .  \nThere exist multiple generalizations and extensions of the notion of a complexity function including the abelian complexity [19] . Two finite words are said to be abelian equivalent if they are permutations of each other. In other words, in abelian combinatorics on words we consider commutative images of words, so that the order of letter is not taken into account. The abelian complexity","cbCain1y93moXT5K","https://ap.wps.com/l/cbCain1y93moXT5K","pdf",474267,3,1,28,"English","en",105,"# Abstract\n# Introduction\n## Factor, abelian, and group complexity\n## Universal group complexity property\n# Main results and word classes","[{\"question\":\"What is the group complexity function for an infinite word?\",\"answer\":\"Given a sequence ω=(Gn) of subgroups of Sn, the group complexity pω(n) counts, for each length n, the number of equivalence classes of word factors under permutations from Gn.\"},{\"question\":\"How does group complexity relate to factor complexity and abelian complexity?\",\"answer\":\"Factor complexity is obtained by taking Gn as the identity subgroup, while abelian complexity is obtained by taking Gn=Sn; for any Gn, the group complexity lies between the abelian and factor complexities.\"},{\"question\":\"What does it mean for a word to have universal group complexity?\",\"answer\":\"For every length n and each integer k between the abelian complexity and the factor complexity, there exists a subgroup G∈Sn such that the resulting group complexity equals k.\"}]",1784194364,71,{"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},"on-possible-values-of-the-group-complexity-function-of-infinite-words","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-possible-values-of-the-group-complexity-function-of-infinite-words/84249/",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","2026-07-24","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 is the group complexity function for an infinite word?","Question",{"text":75,"@type":76},"Given a sequence ω=(Gn) of subgroups of Sn, the group complexity pω(n) counts, for each length n, the number of equivalence classes of word factors under permutations from Gn.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does group complexity relate to factor complexity and abelian complexity?",{"text":80,"@type":76},"Factor complexity is obtained by taking Gn as the identity subgroup, while abelian complexity is obtained by taking Gn=Sn; for any Gn, the group complexity lies between the abelian and factor complexities.",{"name":82,"@type":73,"acceptedAnswer":83},"What does it mean for a word to have universal group complexity?",{"text":84,"@type":76},"For every length n and each integer k between the abelian complexity and the factor complexity, there exists a subgroup G∈Sn such that the resulting group complexity equals k.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]