On the distribution of characteristic parameters of words II

Theoret. Informatics Appl.36(2002) 97–127 DOI: 10.1051/ita:2002005

ON THE DISTRIBUTION OF CHARACTERISTIC PARAMETERS OF WORDS II

Arturo Carpi

and Aldo de Luca

Abstract. The characteristic parameters Kw and Rw of a word w over a finite alphabet are defined as follows: Kwis the minimal natural number such that w has no repeated suffix of length Kw and Rw is the minimal natural number such that w has no right special factor of lengthRw. In a previous paper, published on this journal, we have studied the distributions of these parameters, as well as the distribution of the maximal length of a repetition, among the words of each length on a given alphabet. In this paper we give the exact values of these distributions in a special case. However, these values give upper bounds to the distributions in the general case. Moreover, we study the most frequent and the average values of the characteristic parameters and of the maximal length of a repetition over the set of all words of lengthn. Mathematics Subject Classification. 68R15, 68R05.

Introduction

In a recent paper [4], which hereafter will be also referred to as CP, we have studied some properties of the distributions of two basic parameters which can be associated with any finite wordwon a given alphabetA. These parameters, called characteristic parameters and denoted byK_wandR_w, are defined as follows: K_w is the length of the shortest unrepeated suffix ofwandR_wis the minimal natural

Keywords and phrases: Special factor, characteristic parameter, repeated factor.

∗ The work for this paper has been supported by the Italian Ministry of Education under Project COFIN 2001 – Linguaggi Formali e Automi: teoria ed applicazioni.

1 Dipartimento di Matematica e Informatica, Universit`a di Perugia, via Vanvitelli 1, 06123 Perugia, Italy; e-mail: [email protected]

2Dipartimento di Matematica dell’Universit`a di Roma “La Sapienza”, piazzale Aldo Moro 2, 00185 Roma, Italy and Centro Interdisciplinare “B. Segre”, Accademia dei Lincei, via della Lungara 10, 00100 Roma, Italy; e-mail:[email protected]

c EDP Sciences 2002

number such that w has no right special factor of length R_w. We recall that a factor u of a wordw is (right) special if there exist two distinct letters a and b such that uaandubare both factors ofw.

As shown in a series of papers [1–5] characteristic parameters give a great amount of information about the structure of a word. For instance, the maximal lengthG_w of a repeated factor of a non-empty wordwis given by

G_w= max{Rw, K_w} −1. (1)

In CP we studied how the values of the characteristic parameters, as well as of some other related quantities, are distributed among the words of each length. More precisely, ifAis a fixedd-letter alphabet, for any pair of natural numbersiandn, we denote byD_R(i, n),D_K(i, n), andD_G(i, n) the number of wordswof lengthn on the alphabet Asuch that, respectively,R_w, K_w, andG_w is equal to i. In the case of a binary alphabet, the values of D_R(i, n)/2,D_K(i, n)/2, and D_G(i, n)/2 for small values ofi andn are reported in Tables 1–3 of CP. The following basic relation between D_R andD_G holds: for alli, n >0 one has

D_R(i, n+ 1) =D_R(i, n) + (d−1)D_G(i−1, n). (2) Moreover, fori, n >0 one has

D_G(i−1, n)≤D_R(i, n) +D_K(i, n),

where equality holds if and only ifi > n/2. We also showed that wheniis fixed and ngrows,D_R(i, n) andD_K(i, n) are non-decreasing. This is not true forD_G(i, n), because one has D_G(i, n)6= 0 if and only ifi < n≤i+dⁱ⁺¹.

In CP we studied the “diagonal behaviour” of D_R, D_K, and D_G, i.e., the behaviour ofD_R(i, n),D_K(i, n), andD_G(i, n) when variablesiandnare simulta- neously increased by 1. We showed that, for anyi, n≥0,

D_K(i, n)≤D_K(i+ 1, n+ 1), (3) where equality holds if and only ifi > n/2. In other terms, for any fixedm≥0, the values ofD_K on the points of a diagonal line (t, m+t)_t≥0 are initially increasing and ultimately constant. A similar property holds for functionsD^∗_GandD_K^∗ where for alli, n≥0,D_G^∗(i, n) andD^∗_K(i, n) denote respectively the number of the words of lengthnsuch thatG_w≥i andK_w≥i. Moreover, for t, m≥0, one has

D_R(t, m+t)≤D_R(m,2m), (4)

where the “=” sign holds if and only ift≥m. Similarly, form >0 andt≥0 one has

D_G(t, m+t)≤D_G(m,2m), (5)

where the “=” sign holds if and only ift≥m−1. Wheni≥n/2 some noteworthy relations hold. In particular, one has fori≥n/2>0

D_R(i, n) = (d−1)dⁿ⁻ⁱ

n−iX

t=1

d^−tD_K(t,2t−1) (6) and, fori≥ bn/2c ≥1

D_R(i, n) = (d−1)D_G^∗(i−1, n−1). (7) In this paper we continue the analysis of the distributions of characteristic param- eters of words. In Section 2 we obtain explicit arithmetic expressions, involving the M¨obius function, for D_R(i, n),D_K(i, n),D_G(i, n),D^∗_K(i, n), andD^∗_G(i, n), at least when i > n/2. In view of the “diagonal behaviour”, these expressions give upper bounds to the values of the preceding maps, in the general case.

Another result concerns the counting of repetitions. Byrepetition of lengthm in a wordwwe mean any unordered pair of distinct occurrences of the same factor of lengthminw. We show that the total number of repetitions of lengthmin all the words of length non ad-letter alphabet is given by

d^n−m

n−m+ 1 2

.

This and other related results are of interest for applications, since repetitions play an essential role in algorithms for text compression and sequence assembly [5,8,10].

In the last two sections we study the behaviour of D_R(i, n), D_K(i, n), and D_G(i, n) when the length n is fixed and i varies. In Section 3 we are mainly interested in the average values hRin, hKin, hGin of R_w, K_w, and G_w on the words w of length n on a d-letter alphabet, with d ≥ 2. We study the most frequent values of the characteristic parameters and of the maximal length of a repetition in the set of words of lengthnand use these results for evaluating the average values. We show that hGin andhKin are upperbounded, respectively, by d2 log_dne −1/2 and dlog_dne+ 2 while hRin is lowerbounded by blog_d(n−1)c.

Moreover,

n→∞lim hKin

log_dn = lim

n→∞(hRin− hGin) = 1 and

n→∞lim(hGin− hGi_n−1) = lim

n→∞(hRin− hRi_n−1) = lim

n→∞(hKin− hKi_n−1) = 0.

We also obtain upper bounds to the number of symmetric words (cf. Sect. 3 of CP) of length smaller than nand to the number of semiperiodic words [2] of lengthn. Moreover, we prove that the fraction of the words of lengthnwhich are periodic-like [3] is exactly given byhKin− hKin−1.

In Section 4 we show that the points of maximum ofD_G(i, n), viewed as a func- tion ofiwithnfixed but sufficiently large, lie betweenblog_dnc −1 andd2 log_dn+ log_dlog_dne −1. Similarly, the points of maximum ofD_R(i, n) and D_K(i, n) lie, respectively, between blog_dn−log_dlog_dnc −4 and d2 log_dn+ log_dlog_dne and between 0 and dlog_dn+ log_dlog_dne+ 2.

1. Preliminaries

Let A be a non-empty set, or alphabet, of cardinality d > 0. We denote by A^∗ the set of all finite sequences of elements ofA, including the empty sequence, denoted by. The elements of Aare usually calledletters and those ofA^∗ words.

The word is calledempty word. We setA⁺ =A^∗\ {}. A wordw∈A⁺ can be written uniquely as a sequence of letters as

w=a1a2· · ·a_n,

with a_i ∈ A, 1 ≤ i ≤ n, n > 0. The integer n is called the length of w and denoted by |w|. By definition, the length of is equal to 0. For anyn≥0 we set Aⁿ={w∈A^∗| |w|=n}.

A wordwis calledprimitive if it cannot be written asw=u^r withu6=and r >1. Two words u, v∈A^∗ areconjugate if there exist wordsr, s∈A^∗ such that u=rsandv=sr. As is well known (see [9]), conjugacy is an equivalence relation in A^∗. Moreover, any conjugate of a primitive word is primitive. A conjugacy class of a primitive word will be called primitive.

Letw∈A^∗. The wordu∈A^∗ is afactor (orsubword) ofwif there exist words λ, µ such that w=λuµ. A factoruof w is called proper ifu 6=w. Ifw =uµ, for some wordµ(resp.w=λu, for some word λ), then uis called a prefix (resp.

suffix) ofw. For any word w, we denote respectively by Fact(w), Pref(w), and Suff(w) the sets of its factors, prefixes, and suffixes.

Let u∈ Fact(w). Any pair (λ, µ)∈ A^∗×A^∗ such that w =λuµ is called an occurrence of uin w. If λ = (resp. µ = ), then the occurrence ofu is called initial (resp.terminal). An occurrence is calledinternal if it is neither initial nor terminal. A factor uofwisrepeated if it has at least two distinct occurrences in w, otherwise it is calledunrepeated.

A wordsis called aright (resp.left)special factor ofwif there exist two letters x, y ∈A,x6=y, such that sx, sy∈Fact(w) (resp.xs, ys∈Fact(w)).

With each word w one can associate the word k_w (resp. h_w) defined as the shortest suffix (resp. prefix) ofwwhich is an unrepeated factor ofw.

In the following, for any non-empty wordw, we shall denote byk⁰_w (resp.h⁰_w) the longest repeated suffix (resp. prefix) ofw. One has, trivially, k_w =xk_w⁰ and h_w=h⁰_wy withx, y∈A.

For any word w, we shall consider the parametersK_w=|kw| and H_w =|hw|.

Moreover, we shall denote by R_w the minimal natural number such that there is no right special factor ofwof lengthR_w and byL_w the minimal natural number such that there is no left special factor ofwof lengthL_w.

For any w∈A⁺ we setB_w={a∈A|k⁰_wa∈Fact(w)}. Thus,B_w is the set of letters ofAextending on the rightk_w⁰ inw. Moreover, we setB=A. In Section 2 of CP we proved that for allw∈A^∗and any x∈B_w one has

K_wx=K_w+ 1 and R_wx=R_w. (8) Letw=a1· · ·a_n,a_r∈A, 1≤r≤n. Arepetition of length q >0 inwis any pair (i, j), 1≤i < j ≤n−q+ 1, such that

a_ia_i+1· · ·a_i+q−1=a_ja_j+1· · ·a_j+q−1. (9) Moreover, a repetition of length 0 is any pair (i, j) with 1≤i < j≤n+ 1.

The maximal length of a repetition in w, that is the maximal length of a re- peated factor ofw, is denoted byG_w. As proved in Section 3 of CP, for allw∈A⁺ one has

G_w+ 1≤ |w| ≤G_w+d^Rw. (10) In particular, ifd >1 one derives

G_w≥ blog_d|w|c −1. (11) The following lemmas will be useful in the sequel:

Lemma 1.1. Letd >1. For anyw∈A⁺ one has G_w+R_w≥2blog_d|w|c −1.

Proof. Letw∈Aⁿ. By equation (10),G_w+R_w≥G_w+ log_d(n−G_w). Since the second derivative of the functionx+ log_d(n−x) is negative, the minimal value of this function in the interval [blog_dnc −1, n−1] is equal to

min{blog_dnc −1 + log_d(n− blog_dnc+ 1), n−1}·

By equations (10) and (11),blog_dnc −1≤G_w≤n−1 so that one derives G_w+R_w≥min{blog_dnc −1 + log_d(n− blog_dnc+ 1), n−1} · (12) Since ford≥2 one hasn≥dblog_dnc, one gets

n−1≥2blog_dnc −1 (13)

and

n− blog_dnc+ 1≥n−n

d+ 1>n d,

so that

blog_dnc −1 + log_d(n− blog_dnc+ 1)>blog_dnc −1 + log_dn

d ≥2blog_dnc −2.

(14) By equations (12, 13), and (14) one obtainsG_w+R_w>2blog_dnc −2, from which the conclusion follows.

We observe that the lower bound in the preceding lemma is effectively reached.

In fact, if wis a de Bruijn word of orderm (cf. Sect. 3 of CP) one hasR_w =m, G_w=m−1, andm=blog_d|w|c.

Lemma 1.2. Let d > 1. For any w ∈ A⁺ such that R_w < blog_d|w|c one has R_w+K_w≥2blog_d|w|c.

Proof. By Lemma 1.1 one hasG_w≥2blog_d|w|c −1−R_w≥ blog_d|w|c> R_w. This impliesK_w=G_w+ 1 so thatR_w+K_w≥2blog_d|w|c.

Let us denote by P_w(q) the number of all repetitions of length q in w. For instance, in the case of the wordw=aabaababbab, as one easily verifies, one has P_w(1) = 25,P_w(2) = 10,P_w(3) = 3,P_w(4) = 1, andP_w(5) = 0.

Lemma 1.3. For anyw∈A⁺ andq≥0, one has P_w(q)≥G_w−q+ 1.

Proof. The result is trivially true ifq= 0 orq > G_w. Thus suppose 0< q≤G_w. Let us write w as w = a1· · ·a_n with a_r ∈ A, 1 ≤ r ≤ n. Since G_w is the maximal length of a repeated factor of w there exist integers i and j such that 1≤i < j≤n−G_w+ 1 and

a_ia_i+1· · ·a_i+Gw−1=a_ja_j+1· · ·a_j+Gw−1. Thus, theG_w−q+ 1 pairs

(i, j),(i+ 1, j+ 1), . . . ,(i+G_w−q, j+G_w−q) are repetitions of lengthq ofw. Hence,P_w(q)≥G_w−q+ 1.

2. Exact computations

In the sequel we shall assume that the alphabetA contains at least two letter, i.e.,d >1.

In this section we give explicit arithmetic expressions for D_R(i, n), D_K(i, n), D_G(i, n), D^∗_K(i, n), and D^∗_G(i, n), involving the M¨obius function, at least when i > n/2. In view of the diagonal behaviour of these functions, these expressions give upper bounds to the values of the preceding maps, when i≤n/2.

Letw =a1a2· · ·a_n be a word, a_i∈A, i= 1, . . . , n. We recall (cf. [9]) that a positive integer p≤nis called aperiod of wif for alli, j ∈[1, n] such thati≡j (mod p), one hasa_i =a_j. For any word w, we denote by π_w itsminimal period.

A word wis calledperiodic if|w| ≥2π_w.

The notion of period is also related to the notion of border of a word. A word uis called aborder ofwif it is both a proper prefix and a proper suffix ofw. The longest border of the word w will be called themaximal border of w. It is well known (cf. [9]) that the maximal border of a wordwhas length|w| −π_w.

Let ψ : N+ → N+ be the function counting, for any positive integer n the number of primitive words of length n on the alphabetA. As is well known [9], for anyn, ψ(n) is given by

ψ(n) =X

m|n

µ(m)d^mn,

where µis the M¨obius function (see, for instance [7]).

Lemma 2.1. Letn andpbe positive integers such that n≥2p−2. The number of words of length n having minimal period pis given byψ(p).

Proof. Letube a primitive word of lengthpand prolonguon the right in a word w of lengthn having period p. Let us show thatp is the minimal period of w.

Indeed, suppose thatwhas a minimal periodq < p. Sincen≥2p−2≥p+q−1, by the theorem of Fine and Wilf [6],w has also the period gcd(p, q) which has to be equal to q, sinceqis the minimal period ofw. Thus,p=rqwithr >1. Since u has the period q, it follows that u is not primitive, which is a contradiction.

Conversely, let w be a word of length n having the minimal periodp. Then the prefix uof lengthpofwhas to be primitive as, otherwise, the minimal period of w would be less thanp.

In conclusion, if n≥2p−2, the number of words of length nhaving minimal periodpcoincides with the number of primitive words of lengthp,i.e., ψ(p).

Lemma 2.2. Letm be a positive integer andwa word. One has K_w≥m if and only if there exists u∈Suff(w)such that|u|=π_u+m−1.

Proof. Let us suppose that K_w ≥ m. Thus w has a repeated suffix v of length m−1. Let ube the shortest suffix ofw with two occurrences ofv. This implies that v is a border of u with no internal occurrence in u. Moreover, v is the longest border of u, otherwisev would have an internal occurrence in u. Hence, the minimal period of uis given byπ_u=|u| − |v|=|u| −m+ 1.

Conversely, suppose that there exists u∈Suff(w) such that|u|=π_u+m−1.

Then, uhas a maximal borderv of length|v|=|u| −π_u =m−1. The wordv is a repeated suffix ofwso thatK_w≥ |v|+ 1 =m.

Lemma 2.3. Letwbe a word andm >|w|/2. Then there is at most one suffixu of w such that|u|=π_u+m−1.

Proof. Suppose that u, u⁰ ∈Suff (w) are such that |u|=π_u+m−1,|u⁰|=π_u⁰+ m−1, and|u⁰| ≤ |u|. Since|u| ≤ |w| ≤2m−1, it follows thatπ_u≤mso that

|u⁰|=π_u0+m−1≥π_u0 +π_u−1.

Since u⁰ is a suffix ofu, u⁰ has also the period π_u. By the theorem of Fine and Wilf [6], it follows that u⁰ has the period gcd(π_u, π_u⁰). Thus, since π_u⁰ is the minimal period of u⁰ one hasπ_u⁰ = gcd(π_u, π_u⁰), so that π_u is a multiple ofπ_u⁰. Since π_u ≤ |u⁰| and π_u is a multiple of π_u⁰ it follows that π_u⁰ is a period of u.

Consequently,π_u⁰ =π_u and|u|=|u⁰|which implies u=u⁰. We recall that the mapD^∗_K is defined for alli, n≥0 by

D_K^∗(i, n) = Card({w∈Aⁿ |K_w≥i}) =X

m≥i

D_K(m, n).

By equation (3) one easily derives (cf.Sect. 5 of CP) that for alli, n≥0,

D_K^∗(i, n)≤D^∗_K(i+ 1, n+ 1), (15) where equality holds if and only if i > n/2.

Proposition 2.4. Letm andnbe integers with0≤m≤n. One has

D^∗_K(m, n)≤^n−m X+1 i=1

d^n−m−i+1ψ(i),

where equality holds if and only ifm > n/2. In particular, for0≤m≤none has D^∗_K(m, n)≤(n−m+ 1)d^n−m+1.

Proof. First, we suppose that m > n/2. Then, for anyi= 1, . . . , n−m+ 1, one has i+m−1 ≥ 2i−1 so that, by Lemma 2.1 there are exactlyψ(i) words of minimal period i and length i+m−1. These words can be prolonged on the left into d^n−m−i+1ψ(i) words of length nsatisfying the condition in Lemma 2.2.

Since m > n/2, by using Lemma 2.3, one derives that, starting from distinct values of the period i, distinct words are obtained. We conclude that the total number of words w of length n such that K_w ≥ m, i.e., D^∗_K(m, n) is given by P_n−m+1

i=1 d^n−m−i+1ψ(i).

If, on the contrary, m≤ n/2, then by equation (15) and the first part of the proof one has

D^∗_K(m, n)< D_K^∗(m+n+ 1,2n+ 1) =

n−mX+1 i=1

d^n−m−i+1ψ(i).

To conclude the proof, we observe that for any i≥1, triviallyψ(i)≤dⁱ, so that

n−m+1X

i=1

d^n−m−i+1ψ(i)≤^n−m+1X

i=1

d^n−m+1= (n−m+ 1)d^n−m+1.

In the sequel, we follow the convention that a sumP_s

i=ta_i holds 0 if t > s.

Proposition 2.5. Letm andnbe integers with0≤m≤n. One has D_K(m, n)≤ψ(n−m+ 1) +d^n−m(d−1)

n−mX

i=1

d⁻ⁱψ(i),

where equality holds if and only if m > n/2.

Proof. One can write D_K(m, n) =X

i≥m

D_K(i, n)− X

i≥m+1

D_K(i, n) =D^∗_K(m, n)−D_K^∗(m+ 1, n).

Ifm > n/2, by Proposition 2.4 one has D_K(m, n) =

n−mX+1 i=1

d^n−m−i+1ψ(i)−^n−mX

i=1

d^n−m−iψ(i)

=ψ(n−m+ 1) +d^n−m(d−1)

n−mX

i=1

d⁻ⁱψ(i).

If, on the contrary,m≤n/2, then, by an iterated application of equation (3), one derives

D_K(m, n)< D_K(m+n+ 1,2n+ 1).

Sincem+n+ 1>(2n+ 1)/2, by the previous argument it follows

D_K(m, n)< ψ(n−m+ 1) +d^n−m(d−1)

n−mX

i=1

d⁻ⁱψ(i),

which concludes the proof.

Let us introduce now the functionω defined for anyp≥0 by ω(p) =

Xp t=1

d^−t−1ψ(t)(d+ (d−1)(p−t)). (16)

Proposition 2.6. Letm, nbe integers with 0≤m≤n. One has D_R(m, n)≤(d−1)d^n−mω(n−m), where equality holds if and only if m≥n/2.

Proof. The statement is trivial for n= 0. Let us then supposen >0.

First we consider the casem≥n/2. By equation (6) one has D_R(m, n) = (d−1)d^n−m

n−mX

t=1

d^−tD_K(t,2t−1).

For 1≤t≤n−mone has (2t−1)/2< t≤2t−1 so that, by Proposition 2.5 D_K(t,2t−1) =ψ(t) +d^t−1(d−1)

t−1

X

i=1

d⁻ⁱψ(i).

Thus,

D_R(m, n) = (d−1)d^n−m

n−mX

t=1

d^−tψ(t) +d−1 d

n−mX

t=1 t−1

X

i=1

d⁻ⁱψ(i)

!

. (17) Since

n−mX

t=1 t−1

X

i=1

d⁻ⁱψ(i) =

n−mX

i=1

(n−m−i)ψ(i)d⁻ⁱ, equation (17) becomes

D_R(m, n) = (d−1)d^n−m

n−mX

t=1

d^−tψ(t)

1 +d−1

d (n−m−t)

= (d−1)d^n−mω(n−m).

In the general case, by equation (4) one derivesD_R(m, n)≤D_R(n−m,2(n−m)) and, by the previous argument,D_R(m, n)≤(d−1)d^n−mω(n−m), where equality holds if and only if m≥n/2. This proves our assertion.

We recall that the mapD^∗_G is defined fori≥0 andn >0 by D^∗_G(i, n) = Card({w∈Aⁿ|G_w≥i}) =X

m≥i

D_G(m, n).

As proved in Section 5 of CP, for anyi≥0 andn >0 one has

D^∗_G(i, n)≤D^∗_G(i+ 1, n+ 1), (18) where equality holds if and only if i≥ bn/2c.

Proposition 2.7. Letm, nbe integers with 0≤m < n. One has D_G^∗(m, n)≤d^n−mω(n−m),

where equality holds if and only if m≥ bn/2c.

Proof. Let us first suppose that bn/2c ≤ m < n. Since m+ 1 ≥ (n+ 1)/2, by Proposition 2.6 one has D_R(m+ 1, n+ 1) = (d−1)d^n−mω(n−m). Moreover, by equation (7), D_R(m+ 1, n+ 1) = (d−1)D_G^∗(m, n). This impliesD_G^∗(m, n) = d^n−mω(n−m).

In the case m < bn/2c, by equation (18), one derives D^∗_G(m, n) < D^∗_G(m+ n,2n). Sincem+n≥n, one hasD^∗_G(m+n,2n) =d^n−mω(n−m), and this proves the assertion.

The previous proposition gives an upper bound to the number of words having at least one repeated factor of length m. Observe that, sinceψ(t)≤d^t,t≥1, for any p >0, one has

ω(p)<

Xp t=1

d^−tψ(t)(1 +p−t)<

Xp t=1

(1 +p−t) = p+ 1

2

.

Thus, a less sharp but simpler upper bound toD^∗_G(m, n) is given by the following corollary. A similar upper bound was proved recently in [10].

Corollary 2.8. For0≤m < nthe following holds:

D^∗_G(m, n)< d^n−m

n−m+ 1 2

.

An interpretation of this upper bound can be given in terms of the total number P(m, n) of repetitions of lengthmin all the words of lengthn,i.e.,

P(m, n) = X

w∈Aⁿ

P_w(m).

Proposition 2.9. Let n andm be integers such that 0 ≤m < n. The following holds:

P(m, n) =d^n−m

n−m+ 1 2

.

Proof. Ifm= 0, then the result is trivial. Thus, we supposem >0. For any pair of integersiandj such that 1≤i < j ≤n−m+ 1, we count the wordsw=a1· · ·a_n, a_r∈A, 1≤r≤n, satisfying equation (9) withq=m. Let us prove that a wordw satisfying equation (9) is uniquely determined by the worda1· · ·a_j−1a_j+m· · ·a_n. Indeed, by equation (9) one hasa_j+p=a_i+p, 0≤p≤m−1. One derivesa_j=a_i. If we suppose of knowing all the letters up to a_j+p−1, 0 < p ≤ m−1, then a_j+p =a_i+p and a_i+p is already known sincei+p < j+p. This proves that the

number of the wordswsatisfying equation (9) is given byd^n−m. Since there are (n−m+ 1)(n−m)/2 pairs (i, j) satisfying the condition 1≤i < j ≤n−m+ 1 the result follows.

From Proposition 2.9, one obtains a different proof of Corollary 2.8 since, triv- ially,D^∗_G(m, n)≤P(m, n).

Example 2.10. Let us consider a binary alphabet and let m = 5 and n = 18.

In this case by means of a computer one obtains D^∗_G(5,18) = P18

i=5D_G(i,18) = 223 250 (cf. Tab. 3 of CP). By Proposition 2.7 one has D_G^∗(5,18)≤2¹³ω(13) = D^∗_G(12,25) = 363 874< P(5,18) = 745 472.

Proposition 2.11. Let mandn be integers with0≤m≤n. One has D_G(m, n)≤ψ(n−m)

+ (d−1)d^n−m−2

n−m−X1 t=1

d^−tψ(t)(2 + (d−1)(n−m−t+ 1)),

where equality holds if and only if m≥ bn/2c.

Proof. Ifm≥ bn/2c, by Proposition 2.7 one has

D_G(m, n) =D^∗_G(m, n)−D^∗_G(m+ 1, n) =d^n−mω(n−m)−d^n−m−1ω(n−m+ 1).

In view of equation (16), by simple algebraic manipulations, one derives D_G(m, n) =ψ(n−m)

+ (d−1)d^n−m−2

n−m−X1 t=1

d^−tψ(t)(2 + (d−1)(n−m−t+ 1)).

In the general case, by equation (5) one derivesD_G(m, n)≤D_G(n−m−1,2(n− m)−1), where equality holds if and only ifm≥ bn/2c. By the previous argument the result follows.

In conclusion of this section, we compute the exact value of D_K(2, n), n≥0, in the case of a binary alphabet. We denote by Fib_n the sequence of Fibonacci numbers, defined by

Fib0= 0, Fib1= 1, Fib_n+1= Fib_n+ Fib_n−1, n≥1.

Proposition 2.12. Let d= 2. For all n >1one has 1

2D_K(2, n) = Fib_n−1+n−2.

Proof. For n = 2, the result is trivial since D_K(2,2) = 2 and Fib1 = 1. Let us suppose n >2.

We have to count the number of words w ∈ {a, b}ⁿ having K_w = 2. For symmetry reasons, we shall count only the words ending by the lettera.

First, we consider the casek_w=ba. Thus the lettera, but notba, has to appear in the prefix ofwof lengthn−2. Any occurrence of ain this prefix either is the prefix ofwof length 1 or is preceded by anothera. Hence, the only possible words of this kind area^jb^mawithj, m >0 andj+m+ 1 =n. Their number isn−2.

Now, we consider the case k_w =aa. We observe that one has k_w =aa if and only if w=buor w=abuwith k_u =aa. Indeed, since|w| >2, wcannot begin byaa, so that eitherw=buor w=abu. Moreover,aais an unrepeated suffix of wif and only if it is an unrepeated suffix ofu. Let us denote by g(n) the number of wordsw∈ {a, b}ⁿ such thatk_w =aa. The previous argument shows that, for n >2,g(n) =g(n−1) +g(n−2). Sinceg(1) = 0 = Fib0 andg(2) = 1 = Fib1, it follows that g(n) = Fib_n−1.

Therefore, the total number of words w of length n ending by a and having K_w= 2 is given by Fib_n−1+n−2.

From the previous proposition, in the case d = 2 one easily derives that for n≥4 one hasD_K(2, n) =D_K(2, n−1) +D_K(2, n−2)−2(n−5). An interesting problem is to determine in the casei >2 similar recursive relations forD_K(i, n).

3. Average values

In this section we shall be mainly interested in the average values hRin,hKin, hGin of R_w, K_w, and G_w on the words w of lengthn on the alphabet A. First we evaluate the most frequent values of the characteristic parameters and of the maximal length of a repetition in the set of words of length n. From this, we show that hGin and hKin are upperbounded, respectively, by d2 log_dne −1/2 and dlog_dne+ 2 while hRin is lowerbounded by blog_d(n−1)c. Moreover, one has lim_n→∞hKin/log_dn = 1, lim_n→∞(hRin− hGin) = 1, and lim_n→∞(hGin− hGin−1) = lim_n→∞(hRin − hRin−1) = lim_n→∞(hKin − hKin−1) = 0. We also obtain upper bounds to the number of symmetric words of length smaller thann and to the number of semiperiodic words [2] of lengthn. Moreover, we show that the number of periodic-like words of lengthnis equal todⁿ(hKin− hKi_n−1).

We start with the following proposition showing that the words of length n having a repeated factor of length significantly larger than 2 log_dn are a small fraction of all words of lengthn. This result was first proved in [10].

Proposition 3.1. Letn >1 andr≥0. The following holds:

Card({w∈Aⁿ |G_w≥ d2 log_dne+r})< 1 2d^n−r.

Proof. Let us setm=d2 log_dne+r. Ifm≥n, the result is trivially true, because the set {w∈Aⁿ |G_w ≥m} is empty. Let us then supposem < n. Sincem >0,

we can write

n−m+ 1 2

d^n−m<n²

2 d^n−m<1 2d^n−r. From Corollary 2.8 the result follows:

By the previous proposition one has that the number of words w of lengthn such that G_w < d2 log_dne+r is greater than dⁿ(1−d^−r/2). In particular, if one takes r=blog_dlog_dnc, then by the preceding formula and equation (11) one derives that, for a sufficiently large n, the maximal length of repetitions in the overwhelming majority of the words ofAⁿ will lie in the interval

[blog_dn−1c, 2 log_dn+ log_dlog_dn]. Proposition 3.2. Letn >0 andr≥0. The following holds:

Card({w∈Aⁿ|K_w≥ dlog_dne+r})≤d^n−r+1.

Proof. If r = 0, the result is trivially true. Let us then supposer > 0 and set m=dlog_dne+r. Ifm > n, then the set{w∈Aⁿ|K_w≥m}is empty so that the statement follows. Let us then supposem≤n. By Proposition 2.4, one has

Card({w∈Aⁿ |K_w≥m})≤(n−m+ 1)d^n−m+1≤nd^n−m+1.

Since m=dlog_dne+rone has nd^n−m+1 ≤d^n−r+1, which proves the statement.

Now, we introduce the sequenceφ_n of real numbers defined for anyn >1 by φ_n= log_dn−2 log_dlog_dn= log_d n

log²_dn· (19)

Proposition 3.3. One has

n→lim+∞

1

dⁿCard({w∈Aⁿ |K_w≤φ_n}) = 0.

Proof. First, we verify that for 1≤i≤none has 1

dⁿ Card({w∈Aⁿ|K_w≤i})≤

1− 1 dⁱ

_bn/ic−1

. (20)

Indeed, setq=bn/icandr=n−iq. Any wordw∈Aⁿ such thatK_w≤ican be factorized as

w=s⁰s1s2· · ·s_q,

with s_q ∈Aⁱ,s1, . . . , s_q−1∈Aⁱ\ {sq},s⁰ ∈A^r, since s_q is unrepeated inw. The number of words of this kind isd^rdⁱ(dⁱ−1)^q−1, so that

Card({w∈Aⁿ|K_w≤i})≤d^rdⁱ(dⁱ−1)^q−1.

Dividing bydⁿ, one obtains equation (20). Using the inequality 1 +x≤e^xholding for all real number x, one derives

1− 1

dⁱ

_bn/ic−1

≤e^{−d−ibn/i−1c}. (21) Since bothφ_n andd^−φnn/φ_n are diverging sequences, if one replacesibybφncin the right hand side of equation (21), one obtains a sequence which vanishes when ndiverges. Thus, the conclusion follows by takingi=bφncin equation (20).

By taking r =blog_dlog_dnc in Proposition 3.2 and using Proposition 3.3, one derives that

n→lim+∞

1

dⁿCard({w∈Aⁿ|log_dn−2 log_dlog_dn < K_w<log_dn+ log_dlog_dn}) = 1.

Thus, for a sufficiently large n, the minimal length of an unrepeated suffix in the overwhelming majority of the words ofAⁿ will be in the interval

[log_dn−2 log_dlog_dn, log_dn+ log_dlog_dn].

Now consider the map D^∗_R defined for alli, n≥0 (see Sect. 5 of CP) by D^∗_R(i, n) = Card({w∈Aⁿ|R_w≥i}) = X

m≥i

D_R(m, n).

Since for anyw∈Aⁿ one hasR_w≤G_w+ 1, one derives that for 0< m≤n D^∗_R(m, n)≤D^∗_G(m−1, n). (22) Consequently, for a sufficiently large n the maximal length of right special fac- tors in the overwhelming majority of the words ofAⁿ will not exceedd2 log_dn+ log_dlog_dne.

Proposition 3.4. Letn >0 andr >0. The following holds:

Card({w∈Aⁿ|R_w≤ blog_dnc −r})≤d^n−r+2. Proof. By Lemma 1.2, ifR_w≤ blog_dnc −r, then

K_w≥2blog_dnc −R_w≥ blog_dnc+r≥ dlog_dne+r−1.

Hence, by Proposition 3.2, the result follows.

By takingr=blog_dlog_dncin Propositions 3.4 and 3.1 and using equation (22), one derives that

n→lim+∞

1

dⁿCard({w∈Aⁿ|log_dn−log_dlog_dn < R_w<2 log_dn+ log_dlog_dn}) = 1.

In other terms, for a sufficiently largenthe maximal length of right special factors in the overwhelming majority of the words of Aⁿ lies in the interval

[log_dn−log_dlog_dn, 2 log_dn+ log_dlog_dn].

Now, for anyn >0, let us denote byhGin,hRin, andhKin, the average values of the parametersG_w,R_w, andK_w on the words of lengthn,i.e.

hGi_n= 1 dⁿ

X

w∈Aⁿ

G_w, hRi_n= 1 dⁿ

X

w∈Aⁿ

R_w, hKi_n= 1 dⁿ

X

w∈Aⁿ

K_w.

Note that, for anyn >0, one has hGin = 1

dⁿ Xn

i=0

iD_G(i, n) = 1 dⁿ

Xn i=1

D^∗_G(i, n). (23)

In a similar way, one has hRin = 1

dⁿ Xn

i=0

iD_R(i, n) = 1 dⁿ

Xn

i=1

D^∗_R(i, n) (24)

and

hKin= 1 dⁿ

Xn

i=0

iD_K(i, n) = 1 dⁿ

Xn

i=1

D^∗_K(i, n). (25) In the cased= 2, the values ofhRin,hKin, andhGin for 1≤n≤26 are given in Table 1.

We recall that, as proved in Sections 2 and 3 of CP, for allw∈A⁺the following relations hold:

K_w≤K_xw ≤1 +K_w, R_w≤R_xw≤1 +R_w, G_w≤G_xw≤1 +G_w. (26) Proposition 3.5. For anyn >0 one has:

hKin<hKin+1<1 +hKin, hRin<hRin+1<1 +hRin, hGin<hGin+1 <1 +hGin.

Proof. By equation (26), for anyw ∈A⁺ and any x∈ A one hasK_w ≤K_xw ≤ 1 +K_w. Moreover, for anyn >0 there exist certainly a wordw∈Aⁿ and letters x, y ∈ A such that K_w < K_xw and K_yw <1 +K_w. For instance, one can take w=aⁿ, x=a, andy=b6=a. Since

dⁿ⁺¹hKin+1= X

u∈Aⁿ⁺¹

K_u= X

w∈Aⁿ

X

x∈A

K_xw,

one derives

d X

w∈Aⁿ

K_w< dⁿ⁺¹hKin+1< d X

w∈Aⁿ

(1 +K_w).

Dividing bydⁿ⁺¹ one deriveshKin<hKin+1 <1 +hKin.

Table 1. hRin, hKin, andhGin, 1≤n≤26.

n hRin hKin hGin

1 0.0000 1.0000 0.0000 2 0.5000 1.5000 0.5000 3 1.0000 2.0000 1.2500 4 1.6250 2.3750 1.6250 5 2.1250 2.7500 2.1875 6 2.6563 3.0625 2.5938 7 3.1250 3.3438 2.9688 8 3.5469 3.5859 3.2969 9 3.9219 3.8125 3.6523 10 4.2871 4.0117 3.9648 11 4.6260 4.1895 4.2422 12 4.9341 4.3516 4.4980 13 5.2161 4.5005 4.7446 14 5.4803 4.6372 4.9758 15 5.7281 4.7633 5.1934 16 5.9607 4.8800 5.3961 17 6.1784 4.9886 5.5889 18 6.3836 5.0903 5.7730 19 6.5783 5.1856 5.9480 20 6.7632 5.2754 6.1146 21 6.9389 5.3602 6.2736 22 7.1062 5.4405 6.4255 23 7.2658 5.5167 6.5705 24 7.4182 5.5893 6.7094 25 7.5638 5.6586 6.8427 26 7.7033 5.7249 6.9709