In previous work, Currie and Rampersad showed that the growth of the number
of binary words avoiding the pattern xxx^R was intermediate between polynomial
and exponential. We now show that the same holds for the growth of the number
of binary words avoiding the pattern xx^Rx. Curiously, the analysis for xx^Rx
is much simpler than that for xxx^R. We derive our results by giving a
bijection between the set of binary words avoiding xx^Rx and a class of
sequences closely related to the class of "strongly unimodal sequences."Comment: 4 page

Currie, James D.

Rampersad, Narad

English

arXiv

In previous work, Currie and Rampersad showed that the growth of the number
of binary words avoiding the pattern xxxR was intermediate between polynomial and
exponential. We now show that the same result holds for the growth of the number
of binary words avoiding the pattern xxRx. Curiously, the analysis for xxRx is much
simpler than that for xxxR. We derive our results by giving a bijection between the
set of binary words avoiding xxRx and a class of sequences closely related to the class
of “strongly unimodal sequences”.NSERCcs.uwaterloo.ca/journals/JIS/VOL18/Currie/currie12.pd

WinnSpace Repository

23 11Article 15.10.3Journal of Integer Sequences, Vol. 18 (2015),236147Binary Words Avoiding xxRx andStrongly Unimodal SequencesJames Currie and Narad RampersadDepartment of Mathematics and StatisticsUniversity of Winnipeg515 Portage AvenueWinnipeg, Manitoba R3B 2E9Canadaj.currie@uwinnipeg.can.rampersad@uwinnipeg.caAbstractIn previous work, Currie and Rampersad showed that the growth of the numberof binary words avoiding the pattern xxxR was intermediate between polynomial andexponential. We now show that the same result holds for the growth of the numberof binary words avoiding the pattern xxRx. Curiously, the analysis for xxRx is muchsimpler than that for xxxR. We derive our results by giving a bijection between theset of binary words avoiding xxRx and a class of sequences closely related to the classof “strongly unimodal sequences”.1 IntroductionIn this paper we give an exact characterization of the binary words avoiding the patternxxRx. Here the notation xR denotes the reversal or mirror image of x. For example, theword 0101 1010 0101 is an instance of xxRx, with x = 0101. A natural language example ofan instance of xxRx is the French word “resserres”1 (meaning “sheds” or “storerooms”). The1This example is due to the anonymous referee, who also provided the following literary reference: “Lesresserres sont exclusivement re´serve´es aux pigeons.” (E´mile Zola, Le Ventre de Paris, G. Charpentier, Paris,1873)1set of binary words avoiding the related pattern xxxR has been the subject of recent study.This study began with the work of Du, Mousavi, Schaeffer, and Shallit [3], who observedthat there exist infinite periodic binary words that avoid xxxR and provided an example ofan aperiodic infinite binary word that avoids xxxR. Answering a question of Du et al., thepresent authors derived upper and lower bounds for the number of binary words of lengthn that avoid xxxR and showed that the growth of this quantity was neither polynomial norexponential [2]. This result was the first time such an intermediate growth rate had beenshown in the context of pattern avoidance.In the present work we analyze the binary words avoiding xxRx. A preliminary investi-gation, focusing on infinite words, was initiated by Du and Shallit [4]. Here we look into theenumeration of finite binary words avoiding xxRx. Once more, we are able to show a growthrate for the number of such words that is neither polynomial nor exponential. Surprisingly,the analysis is much simpler than what was required to show the analogous result for thepattern xxxR. Indeed we are able to obtain a complete characterization of the binary wordsavoiding xxRx by describing a correspondence between such words and sequences of inte-gers that are very closely related to strongly unimodal sequences of integers. These latterhave recently been studied by Rhoades [6]. This correspondence provides a rather pleasingconnection between avoidability in words and the classical theory of partitions. Finally, weconclude this work with some remarks concerning the non-context-freeness of the languagesof binary words that avoid the patterns xxxR and xxRx respectively.2 Enumerating binary words avoiding xxRxLetL = {w ∈ {0, 1}∗ : w avoids xxRx},and let L = L0∪L1, where L0 (resp., L1) consists of the words in L that begin with 0 (resp.,1), along with the empty word.Note that the words in L avoid both 000 and 111. Observe that any w ∈ {0, 1}∗ thatavoids 000 and 111 has a unique representation of the formw = A0a1a1A1a2a2A2 · · ·Ak−1akakAk, (1)where each Ai is a prefix (possibly empty) of either 010101 · · · or 101010 · · · and each ai ∈{0, 1}. Given such a factorization, we define an associated sequence f(w) = (n0, . . . , nk),where• n0 = |A0a1|,• ni = |aiAiai+1|, for 0 < i < k, and• nk = |akAk|.2For example, if w = 010100101101010011, then f(w) = (5, 4, 6, 2, 1).Let Uˆ denote the set of all sequences of the form (d1, d2, . . . , dm) for which there existsj ∈ {1, 2, . . . ,m} such that either(Type 1) 0 < d1 < · · · < dj−1 < dj > dj+1 > · · · > dm > 0, or(Type 2) 0 < d1 < · · · < dj−1 = dj > dj+1 > · · · > dm > 0.The weight of any such sequence is the sum d1+· · ·+dm. We also include the empty sequencein Uˆ . Type 1 sequences are called strongly unimodal sequences.2 Note that the set Uˆ canalso be equivalently defined as follows: Uˆ consists of all sequences (d1, d2, . . . , dm) for whichthere is no j such that both dj−1 ≥ dj and dj ≤ dj+1.We show the following.Theorem 1. The map f defines a one-to-one correspondence between the words in L0 oflength n and the sequences in Uˆ of weight n.Proof. Let w ∈ {0, 1}∗ be a word starting with 0. Let f(w) = (n0, . . . , nk) and letw = A0a1a1A1a2a2A2 · · ·Ak−1akakAk,be the factorization given in (1).We first show that if f(w) /∈ Uˆ then w /∈ L0. Since f(w) /∈ Uˆ there is some j such thatboth nj−1 ≥ nj and nj ≤ nj+1. Define B1, B2, and B3 as follows:• if j = 1 then B1 is the suffix of A0a1 of length n1; otherwise, B1 is the suffix ofaj−1Aj−1aj of length nj;• B2 = ajAjaj+1;• if j = k − 1 then B3 is the prefix of akAk of length nk−1; otherwise B3 is the prefix ofaj+1Aj+1aj+2 of length nj.The conditions on nj−1, nj , and nj+1 ensure that B1, B2, and B3 can be defined. However,we now see that B1 = BR2 = B3, where B1 is either (01)nj/2 or (10)nj/2. The word w thuscontains an instance of xxRx, and hence is not in L0.Next we show that if w /∈ L0, then f(w) /∈ Uˆ . First note that if w has a factor w′ suchthat f(w′) /∈ Uˆ , then f(w) /∈ Uˆ . We may therefore suppose that w = vvRv and contains nosmaller instance of the pattern xxRx. Then there are indices i < j such that• v = A0a1 · · ·Ai−1ai,• vR = aiAi · · ·Aj−1aj, and2The “U” in Uˆ therefore refers to “unimodal”, and the “hat” to the fact that this set additionally includesthe Type 2 sequences.3• v = ajAj+1 · · · akAk.If k = 2 we necessarily have f(w) = (n0, n0, n0) /∈ Uˆ , so suppose k > 2. Since w contains nosmaller instance of xxRx, we must have n0 < n1. However, we also have nj−1 = nj = n0 andnj+1 = n1. Thus, we have nj−1 = nj < nj+1 and so f(w) /∈ Uˆ .We now turn to the enumeration of the sequences in Uˆ . Let uˆ(n) denote the number ofsequences in Uˆ of weight n. Recall that a partition of n is a sequence (t1, t2, . . . , tk) suchthat1 ≤ t1 ≤ t2 ≤ · · · ≤ tk and t1 + t2 + · · ·+ tk = n.A partition of n into distinct parts is a partition of n where the ti are all distinct. A sequence(d1, d2, . . . , dm) in Uˆ can be represented by a pair (λ, µ) of partitions into distinct parts, wherethe partition λ gives the increasing part of the sequence and the partition µ, read in thereverse order, gives the decreasing part of the sequence. Let pd(n) denote the number ofpartitions of n into distinct parts. Then the generating function for partitions into distinctparts is given by ∑n≥0pd(n)qn =∞∏j=1(1 + qj).The numbers uˆ(n) are thus almost given by the coefficients of the square of this function,i.e., the function ∑n≥0u˜(n)qn =∞∏j=1(1 + qj)2.Unfortunately, the quantity u˜(n) double-counts every sequence of Type 1: once for the casewhere the maximal element of the sequence comes from λ, and once for the case where itcomes from µ. It follows then that for n ≥ 1, we haveu˜(n)/2 ≤ uˆ(n) ≤ u˜(n). (2)Of course u˜(n)/2 is an underestimate for uˆ(n), since, although it corrects for the double-counting of Type 1 sequences, it halves the number of Type 2 sequences. However, intuitivelyone sees that the number of Type 2 sequences is relatively small, and consequently uˆ(n) israther close to u˜(n)/2.Let c(n) be the number of words in L of length n. From (2), Theorem 1, and the factthat c(n) = 2uˆ(n), we have the following.Corollary 2. For n ≥ 1, the number c(n) of binary words of length n that avoid xxRxsatisfiesu˜(n) ≤ c(n) ≤ 2u˜(n).4Rhoades [6] has recently determined the asymptotics of u˜(n); viz.,u˜(n) =√3(24n− 1)3/4 exp(π6√24n− 1)(1 +(π2 − 9)4π(24n− 1)1/2 +O(1n)).This shows, as claimed, that the growth of c(n) is intermediate between polynomial andexponential. The following table gives the values of c(n) and u˜(n) for small values of n.n 0 1 2 3 4 5 6 7 8 9 10 11 12c(n) 1 2 4 6 10 16 24 34 50 72 100 138 188u˜(n) 1 2 3 6 9 14 22 32 46 66 93 128 176The sequences c(n) and u˜(n) are sequences A261204 and A022567 respectively of TheOnline Encyclopedia of Integer Sequences, available online at http://oeis.org . We wouldalso like to note that the number c(n) grows significantly faster than the number of binarywords of length n that avoid xxxR, whose order of growth was previously estimated to be,roughly speaking, on the order of elog2 n [2].3 Non-context-freeness of the language LRecall that in previous work we showed that the language S of binary words avoiding xxxRhas intermediate growth [2]. Adamczewski [1] observed that this implies that S is not acontext-free language, since it is well known that context-free languages have either poly-nomial or exponential growth [7]. He asked if there is a “direct proof” of the non-context-freeness of S. This seems to be difficult; we have not been able to come up with such aproof. Indeed, it even seems to be rather difficult to give a direct proof that S is not aregular language.Adamczewski’s observation applies just as well to the language L of binary words avoidingxxRx: the intermediate growth shown above implies that L is not context-free. However,unlike for S, it is relatively easy to give a direct proof that L is not context-free. First, weobserve that since the class of context-free languages is closed under intersection with regularlanguages and under finite transduction, if the language L is context-free then the languageL ∩ (01)+(10)+(01)+(10)+ = {(01)i(10)j(01)k(10)ℓ : (i < j or k < j) and (j < k or ℓ < k)}is context-free, and in turn, the languageL′ = {aibjckdℓ : (i < j or k < j) and (j < k or ℓ < k)}is context-free. However, one can show using Ogden’s lemma [5] that the latter is notcontext-free. A sketch of the proof is as follows. Let n be the constant of Ogden’s lemmaand consider the word anbn+1cn+2dn+3 ∈ L′, where the block of a’s is “marked”. The first ofthe two blocks to be pumped must therefore be contained in the block of a’s. If the second5block to be pumped does not contain a b, then pumping up produces a word w′ for whichthe number of b’s is both less than the number of a’s and less than the number of c’s. Ifthe second block to be pumped does contain a b, then pumping up produces a word w′ forwhich the number of c’s is both less than the number of b’s and less than the number of d’s.In both cases the word w′ is not in L′. Therefore L′ is not context-free, and consequently Lis not context-free.4 AcknowledgmentsBoth authors are supported by NSERC Discovery Grants. We would like to thank theanonymous referee for his/her helpful feedback.References[1] B. Adamczewski, Personal communication.[2] J. Currie and N. Rampersad, Growth rate of binary words avoiding xxxR. Preprintavailable at http://arxiv.org/abs/1502.07014 .[3] C. F. Du, H. Mousavi, L. Schaeffer, and J. Shallit, Decision algorithms for Fibonacci-automatic words, with applications to pattern avoidance. Preprint available athttp://arxiv.org/abs/1406.0670 .[4] C. F. Du and J. Shallit, Avoiding patterns of morphic and antimorphic involutions.Unpublished manuscript, dated October 16 2013.[5] W. Ogden, A helpful result for proving inherent ambiguity, Math. Systems Theory 2(1968), 191–194.[6] R. Rhoades, Asymptotics for the number of strongly unimodal sequences, Int. Math. Res.Not. IMRN (2014), no. 3, 700–719.[7] V. I. Trofimov, Growth functions of some classes of languages (Russian), Cybernetics(1981), no. 6, i, 9–12, 149.2010 Mathematics Subject Classification: Primary 68R15.Keywords: pattern with reversal, avoidability in words, strongly unimodal sequence.(Concerned with sequences A022567 and A261204.)6Received August 12 2015; revised version received August 24 2015. Published in Journal ofInteger Sequences, September 14 2015.Return to Journal of Integer Sequences home page.7

Binary Words Avoiding xxRx and Strongly Unimodal Sequences

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Binary words avoiding xx^Rx and strongly unimodal sequences

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