A new characteristic property of rich words

A NEW CHARA CTERISTIC PR OPER TY OF R ICH WORDS MICHELANGELO BUCC I , AL ESSANDR O DE LUCA, AMY GLEN, AND LUCA Q. ZAMBONI A B S T R AC T . Originally i ntrodu ced and studied by the third and fourth author s together with J. Justin and S. W idmer (2008), ric h wor ds constitute a n ew class of finite and infinite words ch aracterized by contain ing the maximal number of distinct palindrom es. Several characterizations of r ich words ha ve already been established. A p articularly nice ch aracteristic prop erty is that all ‘com- plete retur ns’ to palindr omes are palindro mes. In this note, we prove that rich words are also characterized by the pro perty that each factor is uniquely deter- mined by its longest palindro mic prefix and its lon gest palindro mic suffix. 1. I N T RO D U C T I O N In [ 2 ], X. Droubay , J . Justin, and G. Pirillo proved th at any finite word w of length | w | contains at mos t | w | + 1 distinct palindromes (in cluding the empt y word). Inspired by this result, the third and fourth authors together with J. Justin and S. W idm er recently initiated a uni fied study o f fini te and infinite words that are characterized by containing the maxi mal number of dis tinct palind romes (see [4]). Such words are called r ich wor ds in view o f t heir ‘palindromic rich- ness’. More precisely , a fi n ite word w is rich if and only if it has e xactly | w | + 1 distinct palindrom ic factors. For example, abac is rich, whereas abca is not. An infinite word is rich if all of its factors are rich. Rich words ha ve appeared in many di f ferent contexts; they include epistur- mian words, complementation-symmetri c sequences, symbolic codings of t ra- jectories of sy mmetric interval exchange t ransformations, and a certain class of words asso ciated with β -expansions w here β is a sim ple Parry number . An- other special class of rich words consists of S. Fischler’ s sequences with “abun- dant palindrom ic prefixes”, w hich were introduced and studied in [3] i n rela- tion to Diophantin e approx imation. Some ot her sim ple examples of rich words Date : May 28, 2008 . 2000 Mathematics Subject Classification. 68R15. K ey wor ds and phrases. co mbinato rics on words; palindro mes; rich w o rds; return words. Corresponding author: Am y Glen. 1 2 MICHELANGELO B UCC I, ALESSANDRO DE LUCA, AMY GLEN, AND LUCA Q. ZAMBONI include: non-recurrent infinite words like abbb b · · · and ab aabaaaba aaab · · · ; the periodic infinite words: ( aab k aabab )( aab k aabab ) · · · , with k ≥ 0 ; the non- ultimately periodic recurrent infinit e word ψ ( f ) where f = abaababaab a · · · is the F i bonacci wor d and ψ is the morphism : a 7→ aab k aabab , b 7→ bab ; and the recurrent, b ut not uniformly recurrent, infinite word generated by the morphism: a 7→ aba , b 7→ bb . See [4] for further examples and refere n ces. Let u be a non-empt y factor of a finite or infinit e word w . W e s ay t hat u is unioccurr ent in w if u has e xactly one occurrence in w . Otherwise, if u h as more than o ne occurrence in w , then there exists a factor r of w ha vi ng exactly two di stinct occurrences of u , one as a prefix and one as a su ffi x. Such a factor r is called a comp lete r eturn to u in w . For example, aabcbaa i s a comp lete return t o aa in the rich word: aabcbaaba . In [4], it was shown that rich words are characterized by the property th at all complete returns to palin dromes are palindromes. The following proposition collects together all of the cha racteristic properties of rich words that were pre vious ly established in [2] and [4]. Pr oposition 1. F or any finite or infin ite wor d w , the following conditions ar e equivalent: i) w is rich; ii) every factor u of w contains exactly | u | + 1 distinct palindr omes; iii) f or each factor u of w , every pr efix (r esp. suffix) of u has a unio ccurr ent palindr o mic suf fix (r esp. pr efix); iv) eve ry pr efix of w has a uni occurr ent palindromic s uffix; v) for each pa lindr omi c factor p of w , every complete r eturn to p in w is a palindr o me. Remark 2 . The equiv alences: i) ⇔ ii), i) ⇔ i ii), and i) ⇔ iv) we re pro ved in [2]. Explicit charac t erizations of periodic rich infinite w ords and rec urrent bal- anced rich i nfinite words ha ve also been establis hed i n [4]. More recently , we proved the following connection between palindromic richness and complexity . Pr oposition 3. [1] F or any infinit e wor d w whose set of factors is closed under r eversal, the following c o nditions ar e equivalent: • all complete r eturns to palindromes a r e palindromes; • P ( n ) + P ( n + 1) = C ( n + 1 ) − C ( n ) + 2 f or all n ∈ N , wher e P (r esp. C ) denot es the palindromi c complexity (r esp. factor compl exity ) function of w , which counts the number of di stinct palindr om ic factors (r esp. f ac- tors) of each length in w . A NEW CHARA CT ERISTIC P R OP ER TY OF RICH WORDS 3 From the perspecti ve of richness, t he above p roposition can be vi e wed as a characterization of r ecurr ent rich infinite words since any rich infinite word is recurrent if and only if its set of factors i s closed under rev ersal (see [4]). Inter- estingly , the proof of Proposition 3 relied upon another characterization of rich words, stated belo w . Pr oposition 4. [1] A finite o r infini te wor d w is ri ch if a nd only if, for each factor v of w , every factor of w be ginning with v and ending with ˜ v and contai ning no other occurr ences of v or ˜ v is a palindr ome. In this note, we establish yet another interesting characteristic property of rich words. Our main results are the following two theorems. Theor em 5. F or any finite or infinite wor d w , the foll owing conditions are equiv- alent: (A) w is rich; (B) each non-pal indr omic f actor u of w is uniquely determined by a pair ( p, q ) o f distinct pa lindr omes such that p and q ar e not factors of each other and p (r esp. q ) is the longest palindr omic pr efix (r esp. suffix) of u . Theor em 6. A finit e o r infi nite wor d w is rich if and only if each factor of w is uniquely determined by its longest palindr omi c pr efix and its longest palindromic suffix. 2. T E R M I N O L O G Y A N D N OTA T I O N Giv en a finite word w = x 1 x 2 · · · x m (where each x i is a letter), the length of w , denoted b y | w | , is equal t o m . W e denote by ˜ w the re versal of w , g iv en by ˜ w = x m · · · x 2 x 1 (the “mirror image” of w ). If w = ˜ w , then w i s called a palindr o me . By conv enti on, the empty wor d ε i s assumed to be a palindrome. A finite word z is a factor of a finite or infinite word w if w = uz v for some words u , v . In the special case u = ε (resp. v = ε ), we call z a pr efix (resp. suffix ) of w . If u 6 = ε and v 6 = ε , t hen we say that z i s an interior factor of w = u z v . A p r oper factor (r esp. pr oper pr efix , pr oper suffix ) of a word w is a factor (resp. prefix, suf fix) of w th at is shorter than w . 3. P R O O F O F T H E O R E M 5 The following two lemmas establish that (A) implies (B). 4 MICHELANGELO B UCC I, ALESSANDRO DE LUCA, AMY GLEN, AND LUCA Q. ZAMBONI Lemma 7. S uppose w is a finite or infinite ric h wor d and let u be any non- palindr o mic facto r of w with longest palindr omic p r efix p and longest p alin- dr omic suffix q . Then p 6 = q , and p a nd q ar e not factors of eac h ot her . Pr oof. By Proposit ion 1 , p and q are unioccurrent factors of u . Thus, since u i s not a palindrome (and hence | u | > max { | p | , | q |} ), i t follows immediately that p 6 = q , and p and q are not factors of each other .  Lemma 8. Suppose w is a finite or infinite rich wor d. If u a nd v ar e f actors of w with the same longest pa lindr omi c pr efix p an d the sa me l ongest palin dr omic suffix q , then u = v . Pr oof. W e first observe th at if u or v i s a palindrome, then u = p = q = v . So let us now assume that neither u nor v is a palindrom e. Suppose to the cont rary that u 6 = v . Then u and v are clearly not factors of each other s ince neither u nor v is equal to p o r q , and p and q are unioccurrent in each of u and v (by Propos ition 1). Let z be a factor of w of minimal length containing both u and v . A s u and v are not factors of ea ch other , we may assume withou t los s of generality that z begins with u and ends wit h v . Then z contains at least two distinct occurrences of p (as a prefix of each of u and v ). In particular , z begins with a complete return r 1 to p wit h | r 1 | > | u | because p is unioccurrent in u by Proposition 1. Mo reove r , r 1 is a palindrome by the richness of w , and hence r 1 ends with ˜ u since u i s a proper prefix of r 1 . Similarly , z ends with a com plete return r 2 to q with | r 2 | > | v | since q is unioccurrent in v by Proposition 1. Hence, since r 2 is a palindrome (by the richness of w ) and v i s a proper p refix of r 2 , it follows that r 2 begins with ˜ v . So w e have shown that ˜ u and ˜ v are (distinct) interior factors of z . Let u s first suppo se that an occurrence of ˜ v is followed by an occurrence of ˜ u i n z (i.e., z has an interior factor beginning with ˜ v and ending wi th ˜ u ). Then, since q is a unioccurrent prefix of each of the (distin ct) fa cto rs ˜ v and ˜ u , we deduce that z contains (as an interior f actor) a complete return r 3 to q beginning with ˜ v . In particular , as r 3 is a palindrome (by richness), r 3 ends wit h v . Thus, z has a proper prefix beginning with u and ending with v , contradicting the minimali ty of z . On th e ot her hand, if z has an i nterior factor beginning w ith ˜ u and ending with ˜ v , then using the same reasoning as abov e, we d educe that z has a p roper s uffi x beginning wit h u and ending with v . But again, this contradicts the minim ality of z ; whence u = v .  A NEW CHARA CT ERISTIC P R OP ER TY OF RICH WORDS 5 The proof of “(A) ⇒ (B)” is now complete. The next lemm a p roves that (B) implies (A). Lemma 9. Suppose w is a fin ite or infinite wor d with the pr operty t hat e ach non-palindromic factor u of w is uniquely determined by a pair ( p, q ) of dist inct palindr o mes such that p and q ar e not factors of each ot her and p (re s p. q ) is the longest palindr omic pr efix (r esp. suffix) of u . Then w is rich. Pr oof. T o prove that w is rich, it s uffi ces to show that each prefix of w has a unioccurrent palindromi c suf fix (see Proposition 1). Let u be any prefix of w and l et q be the longest palin dromic s uffi x of u . W e first ob serve that if u is a palindrome t hen u = q , and hence q is unioccurrent in u . No w let us sup pose that u is n ot a palindrom e and let p be th e longest palindromic prefix of u . If q is not unioccurrent in u , then, as p and q are not factors of e ach other (by the giv en property of w ), we deduce that u has a proper factor v beginning wit h p and ending with q and not containing p or q as an interior factor . Moreover , we observe that p is th e l ongest palindrom ic prefix of v ; otherwise p would o ccur in the interio r o f v (as a suffix o f a longer p alindromic prefix of v ). Similarly , we d educe t hat q i s the longest palind romic s uffi x o f v . So v has the same longest palind romic prefix and th e same lon gest pali ndromic suffi x as u , a contradict ion. Wh ence q is u nioccurrent in u . This comp letes the proof of the lemma.  Note. Likewise, in the case when w is finite, o ne can easily show that each suffix of w has a unioccurrent palin dromic prefix; whence w is rich by Proposition 1. 4. P R O O F O F T H E O R E M 6 Lemma 8 proves that each factor of a rich word is uni quely determined by its longest palindromic prefix and its longest palindromic suffix. Con versely , su ppose w i s a finite or infinit e word with th e property that each factor of w is uni quely d etermined by i ts longest palindrom ic prefix and i ts longest palindromic suffix. T o pro ve t hat w is rich, we could u se v ery simi- lar reasoning as in the proof of Lemm a 9. But for the sake o f interest, we g iv e a s lightly di ff erent proof. Specifically , we show that all compl ete returns to any palindromic factor of w are palindromes; whence w is rich by Propositio n 1. Let p be any pali ndromic factor of w and let u s suppo se to the contrary that w contains a non-pali ndromic complete retu rn r to p . Then r 6 = pp and the two occurrences of p in r cannot overlap. Otherwise r = pz − 1 p for some word z such 6 MICHELANGELO B UCC I, ALESSANDRO DE LUCA, AMY GLEN, AND LUCA Q. ZAMBONI that p = z f = g z = ˜ z ˜ g = ˜ p ; whence z = ˜ z and r = g ˜ z ˜ g = g z ˜ g , a pal indrome. So r = pv p for some non-palind romic word v . W e easily see th at p is b oth the longest palindromic prefix and the longest pal indromic suffix of r ; otherwise p would occur in the interior of r as a s uffi x of a longer palindrom ic prefix of r , or as a prefix o f a lon ger palindromic s uffi x o f r . As r 6 = p , we have reached a contradiction t o the fact that p is the only factor o f w having i tself as both its longest palindromic prefix and its l ongest palindromic suffix. Th us, all c o mplete returns to p in w are palind romes. This compl etes the proof of Theorem 6.  R E F E R E N C E S [1] M. Bucci, A. D e Luc a, A. Glen, L.Q. Zambon i, A co nnection between palind romic and factor complexity using retu rn words, Adv . in Appl. Math. , to appear, arXiv:0802.133 2 . [2] X. Dro ubay , J. Justin, G. Pirillo, Episturmian words and some constructio ns of de Luca and Rauzy , Theor et. Comput. Sci. 255 (200 1) 539–553. [3] S. Fisch ler , Palindro mic prefixes and episturmian words, J. Combin . Theory Ser . A 113 (2006 ) 1 281–1 304. [4] A. Glen, J. Justin, S. W idm er , L.Q. Zam boni, Palindro mic richn ess, Eur o pean J. Combin. (in press), doi:10. 1016 /j.ejc.2008.0 4.006 D I PA RT I M E N T O D I M AT E M AT I C A E A P P L I C A Z I O N I “ R . C AC C I O P P O L I ” , U N I V E R S I T ` A D E G L I S T U D I D I N A P O L I F E D E R I C O I I , V I A C I N T I A , M O N T E S . A N G E L O , I - 8 0 1 2 6 N A P O L I , I T A L Y E-mail address : micbucc i@unina.it D I PA RT I M E N T O D I M AT E M AT I C A E A P P L I C A Z I O N I “ R . C AC C I O P P O L I ” , U N I V E R S I T ` A D E G L I S T U D I D I N A P O L I F E D E R I C O I I , V I A C I N T I A , M O N T E S . A N G E L O , I - 8 0 1 2 6 N A P O L I , I T A L Y E-mail address : alessan dro.deluca@ unina.it L A C I M , U N I V E R S I T ´ E D U Q U ´ E B E C ` A M O N T R ´ E A L , C . P . 8 8 8 8 , S U C C U R S A L E C E N T R E - V I L L E , M O N T R ´ E A L , Q U ´ E B E C , H 3 C 3 P 8 , C A N A DA E-mail address : amy.gle n@gmail.com U N I V E R S I T ´ E D E L YO N , U N I V E R S I T ´ E L Y O N 1 , C N R S U M R 5 2 0 8 I N S T I T U T C A M I L L E J O R D A N , B ˆ A T I M E N T D U D O Y E N J E A N B R A C O N N I E R , 4 3 , B L V D D U 1 1 N O V E M B R E 1 9 1 8 , F - 6 9 6 2 2 V I L L E U R B A N N E C E D E X , F R A N C E R E Y K JA V I K U N I V E R S I T Y , S C H O O L O F C O M P U T E R S C I E N C E , K R I N G L A N 1 , 1 0 3 R E Y K - JA V I K , I C E L A N D E-mail address : luca.za mboni@wanad oo.fr