Characteristic Parameters and Special Trapezoidal Words

TL;DR

This paper investigates the properties of trapezoidal words, focusing on their characteristic parameters and classifications, including their prefixes, symmetry, and special types, extending previous work on Sturmian words.

Contribution

It provides new characterizations of right special and strictly bispecial trapezoidal words, expanding understanding of their structure and parameters.

Findings

Characterization of right special trapezoidal words

Identification of strictly bispecial trapezoidal words

Analysis of prefixes' symmetry and periodicity

Abstract

Following earlier work by Aldo de Luca and others, we study trapezoidal words and their prefixes, with respect to their characteristic parameters $K$ and $R$ (length of shortest unrepeated suffix, and shortest length without right special factors, respectively), as well as their symmetric versions $H$ and $L$. We consider the distinction between closed (i.e., periodic-like) and open prefixes, and between Sturmian and non-Sturmian ones. Our main results characterize right special and strictly bispecial trapezoidal words, as done by de Luca and Mignosi for Sturmian words.

Equations

$$w=pxux\cdot yuyq$$

$$O{C}_w(n)=\{\begin{array}{l}1\text{ if w[n] is closed,}\\ 0\text{ if w[n] is open.}\end{array}$$

$${\mathrm{OC}}_w=\prod _{n=0}^{\infty }{1}^{k_n}{0}^{k_n^{\prime }}$$

$$D_w:=H_w-L_w=|{\mathrm{OC}}_w{|}_1-|{\mathrm{OC}}_w{|}_0$$

Full text

1 Introduction

Sturmian words are certainly among the most studied objects in combinatorics on words, thanks to their natural definition, interesting characterizations, and numerous applications in several fields; see [1, 11] for surveys. An infinite word is Sturmian if it has exactly $n+1$ distinct factors (blocks of consecutive letters) for each length $n\geq 0$.

Trapezoidal words, introduced in [4, 8], are a natural finite analogue of Sturmian words. They have at most $n+1$ factors of each length $n$, so that the graph of their factor complexity function is in the shape of an isosceles trapezoid (or triangle), whence their name.

The original definition of trapezoidal words, however, uses characteristic parameters $K$ and $R$. For a finite word $w$, $K_{w}$ denotes the length of the shortest unrepeated suffix of $w$, whereas $R_{w}$ denotes the smallest integer $n\geq 0$ such that $w$ has no right special factor of length $n$. A word $w$ is then trapezoidal if and only if its length $|w|$ verifies $|w|=K_{w}+R_{w}$ (with $|w|\geq K_{w}+R_{w}$ true in general, see [8]). Finite Sturmian words, i.e., factors of Sturmian words, are trapezoidal, but there exist non-Sturmian trapezoidal words such as $aabb$.

In [2], the property of being closed (aka periodic-like) or open is considered for trapezoidal words. In particular, the case of (prefixes of) the Fibonacci word was completely characterized, and this was later (cf. [5]) extended to all characteristic Sturmian words. Our first aim, explored in Section 3, is to extend some of those arguments to the general trapezoidal case, with respect to the values of characteristic parameters for closed and open prefixes.

In [7], a characterization was given for right (resp. left) special Sturmian words, i.e., finite words $w$ over $A=\{a,b\}$ such that both extensions $wa,wb$ (resp. $aw,bw$) are Sturmian. Previously, in [9], strictly bispecial ones (that is, words $w$ such that $awa,awb,bwa,bwb$ are all Sturmian) had been characterized; in particular, these turn out to be the noteworthy family of central words. In [6], bispecial (i.e., simultaneously right and left special) Sturmian words were characterized. Our main objective in Section 4 is to give similar characterizations for special trapezoidal words. Special words in a language are often useful for dealing with enumerative and structural questions.

2 Notation and Preliminaries

Let $A=\{a,b\}$ be an alphabet. The free monoid of all words over $A$ under concatenation is denoted by $A^{*}$; its neutral element is the empty word $\varepsilon$. For $w\in A^{*}$ and $x\in A$, $|w|_{x}$ denotes the number of occurrences of $x$ in $w$.

If $w=pvs\in A^{*}$, we say that $v$ is a factor of $w$, $p$ is a prefix, and $s$ is a suffix. A border of $w$ is a word that is simultaneously a proper prefix and a suffix of $w$. The definitions of factor and prefix also apply to (right-)infinite words over $A$. A word $u$ is a right (resp. left) special factor of a finite or infinite word $w$ over $A$ if $ua$ and $ub$ (resp. $au$, $bu$) are both factors of $w$.

As anticipated above, an infinite word is Sturmian if it has exactly $n+1$ distinct factors of each length $n\geq 0$. Therefore Sturmian words are the simplest aperiodic words in terms of factor complexity, and this is one of the reasons of interest in their study (see [1, 11]). Equivalently, a binary infinite word is Sturmian if and only if it has exactly one right (resp. left) special factor of each length. In particular, a Sturmian word is called standard or characteristic if all its left special factors occur as prefixes.

Among the many known characterizations of factors of Sturmian words, or finite Stumian words, perhaps the most famous and widely used one deals with balance; $w\in A^{*}$ is Sturmian if and only if there is no word $u$ such that $aua$ and $bub$ are both factors of $w$. Such a pair of factors for a non-Sturmian word is called a pathological pair (cf. [2, 4]).

Central words are palindromic prefixes of characteristic Sturmian words. They enjoy many equivalent definitions and interesting properties (cf. [7, 1]). In particular, a word is central if and only if it can be written as $a^{n}$, $b^{n}$, or $uabv=vbau$ for some integer $n\geq 0$ and $u,v\in A^{*}$; in the latter case, $u$ and $v$ are central words themselves.

The parameters $K_{w}$ and $R_{w}$ defined in the previous section were introduced in [8], along with their “left” counterparts $H_{w}$ (length of the shortest unrepeated prefix) and $L_{w}$ (smallest $n\geq 0$ such that $w$ has no left special factor of length $n$). As already stated, a finite word is trapezoidal if $|w|=K_{w}+R_{w}$, or equivalently if $|w|=H_{w}+L_{w}$ (cf. [8, 4]).

Another noteworthy parameter is the (minimal) period $\pi_{w}$ of a word $w$, which can be defined by $\pi_{w}=|w|-|v|$ where $v$ is the longest border of $w$. Central words can also be characterized in terms of periods; $w\in A^{*}$ is central if and only if $|w|=\pi_{wa}+\pi_{wb}-2$ (cf. [7]). It is known (see for example [10]) that periodic extensions of a finite Sturmian word $w$, i.e., words $w^{\prime}$ such that $w$ is a factor of $w^{\prime}$ and $\pi_{w^{\prime}}=\pi_{w}$, are still Sturmian; however, this property does not extend to trapezoidal words.

Example 1

Let $w=aaababa$. Then $H_{w}=R_{w}=3$ and $L_{w}=K_{w}=4$, so that $w$ is trapezoidal. Its period is $\pi_{w}=6$, but the periodic extension $w^{\prime}=abaaababa=abw$ is not trapezoidal, as $|w^{\prime}|=9>4+4=K_{w^{\prime}}+R_{w^{\prime}}$.

The following theorem is essentially a restatement of [4, Theorem 5]. It characterizes non-Sturmian trapezoidal words as products of two periodic extensions of the elements in a pathological pair.

Theorem 2.1

A word $w\in A^{*}$ is trapezoidal non-Sturmian if and only if it can be written as

[TABLE]

where $u$ is a central word, $A=\{x,y\}$, $\pi_{pxux}=\pi_{ux}$, and $\pi_{yuyq}=\pi_{yu}$. Furthermore, $R_{w}=|pxux|$ and $K_{w}=|yuyq|$.

Note that for such a word, $\{xux,yuy\}$ is actually the shortest pathological pair (cf. [2]).

Example 2

The trapezoidal word $w=aaababa$ considered in Example 1 can be written as $aaa\cdot baba$, where $aaa$ and $baba$ are periodic extensions (to the left and to the right, respectively) of the elements of the pathological pair $\{aaa,bab\}$.

The non-trapezoidal word $w^{\prime}=abaaababa$ does not verify the condition in Theorem 2.1. Indeed, its only pathological pair is $(aaa,bab)$, and writing $w^{\prime}=abaaa\cdot baba$ we obtain $\pi_{abaaa}=4\neq 1=\pi_{aa}$.

3 Closed and Open Trapezoidal Words

A finite, nonempty word $w$ is said to be closed (or periodic-like in earlier works) if it has a border $u$ with no internal occurrences, that is, a factor occurring exclusively as a prefix and as a suffix; another common terminology for describing this situation is that $w$ is a complete (first) return to $u$. In particular, single letters are closed, their border being the empty word.

A non-closed word is said to be open. Equivalently, $w$ is open if and only if its longest repeated prefix (resp. suffix) is a right (resp. left) special factor (cf. [8]).

The following result was proved in [3, Proposition 3.6].

Proposition 1

All closed trapezoidal words are Sturmian.

The following result, showing a basic connection between the behavior of $H$ and the property of being closed or open, is essentially known (see [5, Lemma 6 and Remark 8]). We report a proof for the sake of completeness.

Lemma 1

Let $w\in A^{*}$ and $x\in A$. Then $H_{wx}=H_{w}+1$ if $wx$ is closed, and $H_{wx}=H_{w}$ if $wx$ is open.

Proof

Trivial if $w=\varepsilon$. Let then $w$ be nonempty, and $hy$ ($y\in A$) be its shortest unrepeated prefix, so that $H_{w}=|hy|$. If $wx$ is closed, then its longest border has to be longer than $h$ since $h$ has internal occurrences in $wx$, and not longer than $hy$ since otherwise $hy$ would reoccur in $w$. Hence $x=y$ and $H_{wx}=|hy|+1=H_{w}+1$ as desired. If $wx$ is open, then $hy$ cannot have internal occurrences in $wx$, since it is unrepeated in $w$, and it cannot be a suffix either, otherwise $wx$ would be closed. Hence $hy$ is unrepeated in $wx$, i.e., $H_{wx}=H_{w}$. ∎

Lemma 2

Let $wx$ be a trapezoidal word, $x\in A$. Then $L_{wx}=L_{w}$ if $wx$ is closed, and $L_{wx}=L_{w}+1$ if $wx$ is open.

Proof

Follows from Lemma 1 as $|w|=H_{w}+L_{w}$ and $|wx|=H_{wx}+L_{wx}$. ∎

Clearly, the following symmetric statement holds for left extensions $xw$.

Lemma 3

Let $w\in A^{*}$ and $x\in A$. Then $K_{xw}=K_{w}+1$ if $xw$ is closed, and $K_{xw}=K_{w}$ if $xw$ is open. Moreover, if $xw$ is trapezoidal, then $R_{xw}=R_{w}$ if $xw$ is closed, and $R_{xw}=R_{w}+1$ otherwise.

For a trapezoidal word $w$, the equality $\{H_{w},L_{w}\}=\{K_{w},R_{w}\}$ holds (cf. [8]). The following theorem, proved in [2, Proposition 4.4], is more precise.

Theorem 3.1

Let $w$ be a trapezoidal word. Then $H_{w}=K_{w}$ and $L_{w}=R_{w}$ if $w$ is closed, whereas $H_{w}=R_{w}$ and $L_{w}=K_{w}$ if $w$ is open.

Corollary 1

Let $wx$ be a trapezoidal word and $x\in A$. Then $K_{wx}=K_{w}+1$ and $R_{wx}=R_{w}$, unless

• •

$w$* is closed and $wx$ is open, or*

• •

$w$* is open and $wx$ is closed,*

in which cases we have $K_{wx}=R_{w}+1$ and $R_{wx}=K_{w}$ instead.

Proof

Consequence of Lemmas 1, 2, and Theorem 3.1.

Proposition 2

Let $w\in A^{*}$, $y\in A$. If $wy$ is trapezoidal but not Sturmian, then $w$ is open.

Proof

If $w$ is not Sturmian, by Proposition 1 we are done. Let then $w$ be Sturmian and assume it is closed, by contradiction. By Proposition 1, $wy$ is open. Writing $wy=pxux\cdot yuyq$ as in Theorem 2.1, we have $q=\varepsilon$ as $w$ is Sturmian, and $H_{w}=H_{wy}=R_{wy}=|pxux|$ by Lemma 1 and Theorems 3.1 and 2.1. It follows that $pxu$ is the longest border of $w$. As $x\neq y$ and $w$ ends in $yu$, this is clearly absurd. ∎

Let $w_{[n]}$ denote the prefix of $w$ of length $n$. The oc-sequence of a word $w$ is the characteristic sequence of its closed prefixes. In other terms, it is the binary word $OC_{w}$ such that

[TABLE]

The oc-sequence is a useful tool in studying the structure of finite and infinite words. For example, in [5], the following was proved:

Theorem 3.2

Let $w$ be an infinite word, and let

[TABLE]

for suitable positive integers $k_{n},k_{n}^{\prime}$, with $n\geq 0$. Then $k_{n}\leq k_{n}^{\prime}$ for all $n\geq 0$, with equality holding for all $n$ if and only if $w$ is a characteristic Sturmian word.

In terms of oc-sequences, an immediate consequence of Lemmas 1 and 2 is the following (see also [5, Remark 8]):

Proposition 3

For any word $w$, $H_{w}$ is the number of closed nonempty prefixes of $w$, i.e., $H_{w}=|\mathrm{OC}_{w}|_{1}$. If $w$ is trapezoidal, then $L_{w}=|\mathrm{OC}_{w}|_{0}$.

The following two results show the behavior of characteristic parameters $H$ and $L$ at the end of runs of 1 and 0 in the oc-sequence.

Proposition 4

Let $w\in A^{*}$ and $x\in A$ be such that $wx$ is an open trapezoidal word, while $w$ is closed. Then $L_{w}<H_{w}$.

Proof

Since $L_{wx}=L_{w}+1$ by Lemma 2, the longest left special factor $\ell x$ of $wx$ occurs as a suffix, and $\ell$ is the longest left special factor of $w$. Clearly, the suffix $\ell$ has internal occurrences in $w$, so that it is strictly shorter than the longest border $v$. This proves $L_{w}-1=|\ell|<|v|=H_{w}-1$, whence the assertion. ∎

Proposition 5

Let $w\in A^{*}$ and $x\in A$ be such that $wx$ is a closed trapezoidal word, while $w$ is open. Then $H_{w}\leq L_{w}$.

Proof

Since $H_{wx}=H_{w}+1$ by Lemma 1, the longest repeated prefix $v$ of $w$ occurs as a suffix, so that $H_{w}\leq K_{w}$. The assertion follows by Theorem 3.1.∎

While Theorem 3.2 gives local constraints for an oc-sequence (namely, each run of 1s is followed by a longer or equal run of 0s), our last three results can be viewed as more global constraints in the case of trapezoidal words. Considering the integer parameter

[TABLE]

gives an interesting way to picture this situation. Indeed by Proposition 3, if $w$ is trapezoidal then $D_{w}=|\mathrm{OC}_{w}|_{1}-|\mathrm{OC}_{w}|_{0}$, so that $D$ increases or decreases by 1 at each subsequent prefix, depending on whether it is closed or open; moreover by Propositions 4–5, $D$ is necessarily positive (resp. non-positive) when encountering the last closed (resp. open) prefix in a run.

Example 3

Let $w=baabaababab$. Then $w_{[n]}$ is closed for $n=1$ and $4\leq n\leq 8$, and open otherwise; that is, $\mathrm{OC}_{w}=10011111000$. As predicted by Propositions 4–5, $D$ reaches its (positive) local maxima, respectively 1 and 4, at $n=1$ and $n=8$, and its (non-positive) local minimum of $-1$ for $n=3$. Since $w$ is not Sturmian, by Proposition 1 any subsequent trapezoidal right extension will be open, leading to an indefinite decrease of $D$.

4 Special Trapezoidal Words

In analogy with the case of finite Sturmian words (cf. [9, 7]), we say that a trapezoidal word $w\in A^{*}$ is right (resp. left) special if $wa$ and $wb$ (resp. $aw,bw$) are both trapezoidal, and that $w$ is strictly bispecial if $awa,awb,bwa$, and $bwb$ are all trapezoidal.

Proposition 6

A right special trapezoidal word is Sturmian.

Proof

Let $w$ be a non-Sturmian trapezoidal word, then open by Proposition 1. If $z\in A$ and $wz$ is trapezoidal, then it is also not Sturmian (like $w$) and hence open. By Corollary 1, $R_{wz}=R_{w}$ and $K_{wz}=K_{w}+1$. By Theorem 2.1, it follows $wz=pxux\cdot yuyqz$ with $\pi_{yuyqz}=\pi_{yu}=\pi_{yuyq}$. This shows that $z$ is uniquely determined, so that $w$ cannot be right special. ∎

Symmetrically, one can prove that

Proposition 7

A left special trapezoidal word is Sturmian.

Theorem 4.1

A trapezoidal word $w$ is right special if and only if either of the following holds:

$w$* is a suffix of a central word, or* 2. 2.

$w=pxuxyu$* for a central word $u$, distinct letters $x,y$, and a word $p$ such that $\pi_{pxux}=\pi_{ux}$.*

Symmetrically, $w$ is a left special trapezoidal word if and only if it is either a prefix of a central word, or written as $w=uxyuyq$ for $x,y\in A$, $x\neq y$, and $\pi_{yuyq}=\pi_{yu}$.

Proof

As is well known (cf. [7]), both extensions $wa,wb$ of a word $w$ are Sturmian if and only if $w$ is a suffix of a central word. Let now $w$ be right special and such that one extension is not Sturmian. By Proposition 6, $w$ is Sturmian. As a consequence of Theorem 2.1, we must have $w=pxux\cdot yu$ where $A=\{x,y\}$, $u$ is some central word, and $p$ is such that $\pi_{pxux}=\pi_{ux}$.

Conversely, if $w=pxuxyu$ with $A=\{x,y\}$, $u$ central and $\pi_{pxux}=\pi_{ux}$, then $w$ is Sturmian as $\{xux,yuy\}$ is the only pathological pair in the trapezoidal non-Sturmian word $wy$; therefore, $wx$ must be Sturmian (and then trapezoidal) too.

The left special case is similar. ∎

The following theorem is a restatement of results in [6, 7]; it characterizes Sturmian words that are bispecial (as Sturmian words).

Theorem 4.2

Let $w\in A^{*}$. Then $wa,wb,aw,bw$ are all Sturmian if and only if $w=(uxy)^{n}u$ for some central word $u$, $\{x,y\}=A$ and a nonnegative integer $n$. Furthermore, $awa,awb,bwa,bwb$ are all Sturmian if and only if $w$ is central, i.e., $n=0$, whereas for $n>0$ exactly one such bilateral extension is not Sturmian, namely $xwy$.

Semicentral words were defined in [2] by the property of having their longest repeated prefix, longest repeated suffix, longest left special factor, and longest right special factor coincide. In the same paper, they were characterized as words $w$ such that $w=uxyu$ for some central word $u$ over $A=\{x,y\}$. Hence, they correspond to the case $n=1$ in the previous theorem.

Our final result is a characterization of strictly bispecial trapezoidal words.

Theorem 4.3

A trapezoidal word is strictly bispecial if and only if it is central or semicentral.

Proof

By Theorem 4.2, central words are strictly bispecial. Moreover, by the same theorem all bilateral extensions of a semicentral word $uxyu$ are Sturmian, except for $xuxyuy$ which is trapezoidal non-Sturmian by Theorem 2.1.

Conversely, if $w$ is a strictly bispecial trapezoidal word, then either all bilateral extensions are Sturmian, in which case $w$ is central by Theorem 4.2 and we are done, or at least one is not.

Assume, for instance, that $cwa$ is trapezoidal non-Sturmian, the other cases being similar. By Proposition 6, $cw$ is Sturmian, so that $cwb$ must be too. Symmetrically, as a consequence of Proposition 7, $dwa$ must be Sturmian as well (where $\{c,d\}=A$). In all cases, $wa,wb,aw,bw$ are all Sturmian. By Theorem 4.2, it follows $w=(uxy)^{n}u$ for some $n>0$; as a consequence of Theorem 2.1, we must have $n=1$. ∎

5 Concluding Remarks

A few related problems remain open. In particular, in [5] the oc-sequence for (prefixes of) characteristic Sturmian words was characterized, see Theorem 3.2. The general trapezoidal case, and even the non-standard Sturmian one, is still open. We believe our results may shed some light on the matter, as illustrated at the end of Section 3.

Regarding the preceding section, a simple and elegant characterization of (not necessarily strictly) bispecial trapezoidal words, such as Theorem 4.2 is for the Sturmian case, is still missing. Theorem 4.1 might be an ingredient for such a result.

Acknowledgments

We thank the anonymous referees for their many helpful comments. This paper is dedicated to the memory of our dear colleague Aldo de Luca (1941–2018).