Palindromes in two-dimensional Words

TL;DR

This paper explores the properties of two-dimensional palindromes, focusing on HV-palindromes, and proves a conjecture about their maximum number of distinct sub-arrays, also analyzing their minimal occurrence in infinite words.

Contribution

It introduces structural properties of HV-palindromes, proves a conjecture on their maximum count in finite words, and determines the minimal number in infinite words over finite alphabets.

Findings

Proved the maximum number of HV-palindromic sub-arrays in finite 2D words.

Established the least number of HV-palindromes in infinite 2D words.

Compared properties of 2D palindromes and HV-palindromes.

Abstract

A two-dimensional ($2$D) word is a $2$D palindrome if it is equal to its reverse and it is an HV-palindrome if all its columns and rows are $1$D palindromes. We study some combinatorial and structural properties of HV-palindromes and its comparison with $2$D palindromes. We investigate the maximum number number of distinct non-empty HV-palindromic sub-arrays in any finite $2$D word, thus, proving the conjecture given by Anisiua et al. We also find the least number of HV-palindromes in an infinite $2$D word over a finite alphabet size $q$.

Tables

Table 1. Table 1. Max HV-palindromes in a binary word of size ( m , n ) 𝑚 𝑛 (m,n)

$m\times n$	Max $\left(HV\right)$	Our Bound	$m\times n$	Max $\left(HV\right)$	Our Bound
3 $\times$ 2	6	6	3 $\times$ 6	20	22
3 $\times$ 3	10	10	4 $\times$ 2	8	8
3 $\times$ 4	13	14	4 $\times$ 3	13	14
3 $\times$ 5	17	18	4 $\times$ 4	19	20

Equations

$$u\text{obar}v=\begin{array}{cccccc}{u}_{1,1}& \cdots & u_{1,n_1}& v_{1,1}& \cdots & v_{1,n_2}\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ u_{m,1}& \cdots & u_{m,n_1}& v_{m,1}& \cdots & v_{m,n_2}\end{array}$$

$$u\ominus v=\begin{array}{ccc}{u}_{1,1}& \cdots & u_{1,n}\\ ⋮& \ddots & ⋮\\ u_{m_1,1}& \cdots & u_{m_1,n}\\ v_{1,1}& \cdots & v_{1,n}\\ ⋮& \ddots & ⋮\\ v_{m_2,1}& \cdots & v_{m_2,n}\end{array}$$

$$w=\begin{array}{cccc}a& b& c& a\\ b& c& c& a\\ a& c& c& b\\ a& c& b& a\end{array}$$

$$u=\begin{array}{ccc}a& b& a\\ b& c& b\\ a& b& a\end{array}$$

$$u^R=[(x\text{obar}y{)}^{i\text{obar}}\text{obar}x{]}^R=x^R\text{obar}(y^R\text{obar}{x}^R{)}^{i\text{obar}}=(x^R\text{obar}{y}^R{)}^{i\text{obar}}\text{obar}{x}^R=(x\text{obar}y{)}^{i\text{obar}}\text{obar}x=u$$

$$\begin{array}{ccc}x& u& y\\ v& p& v^R\\ y& u^R& x\end{array}$$

$$\begin{array}{ccc}{x}_1& u_1& x_1\\ v_1& p_1& v_1^R\\ x_1& u_1^R& x_1\end{array}$$

$$\begin{array}{ccc}x& u& y\\ v& p& v^R\\ y& u^R& x\end{array}$$

$$v=\begin{array}{cccc}{w}_{1,1}& w_{1,2}& \cdots & w_{1,j}\\ w_{2,1}& w_{2,2}& \cdots & w_{2,j}\\ ⋮& ⋮& \ddots & ⋮\\ w_{i,1}& w_{i,2}& \cdots & w_{i,j}\end{array}=\begin{array}{cccc}{w}_{m-i+1,n-j+1}& w_{m-i+1,n-j+2}& \cdots & w_{m-i+1,n}\\ w_{m-i+2,n-j+1}& w_{m-i+2,n-j+2}& \cdots & w_{m-i+2,n}\\ ⋮& ⋮& \ddots & ⋮\\ w_{m,n-j+1}& w_{m,n-j+2}& \cdots & w_{m,n}\end{array}$$

$$w_{1+k,1}{w}_{1+k,2}\cdots w_{1+k,j}=(w_{m-k,n-j+1}{w}_{m-k,n-j+2}\cdots w_{m-k,n}{)}^R$$

$$\left\{\begin{array}{c}aa\\ aa\end{array},\begin{array}{c}ab\\ ba\end{array},\begin{array}{c}ba\\ ab\end{array},\begin{array}{c}bb\\ bb\end{array}\right\}$$

$${\circlearrowleft }_k^{Col}w=v_{n-k+1}\text{obar}\cdots \text{obar}{v}_n\text{obar}{v}_1\text{obar}{v}_2\text{obar}\cdots \text{obar}{v}_{n-k-1}\text{obar}{v}_{n-k}$$

$${\circlearrowleft }_k^{Row}w=u_{m-k+1}\ominus \cdots \ominus u_m\ominus u_1\ominus u_2\ominus \cdots \ominus u_{m-k-1}\ominus u_{m-k}$$

$$Conj(w)=\{{\circlearrowleft }_i^{Col}{\circlearrowleft }_j^{Row}w,1\le i\le n,1\le j\le m\}$$

$$Conj(w)=\left\{\begin{array}{c}abc\\ cbb\\ bbc\\ cba\end{array},\begin{array}{c}cbb\\ bbc\\ cba\\ abc\end{array},\begin{array}{c}bbc\\ cba\\ abc\\ cbb\end{array},\begin{array}{c}cba\\ abc\\ cbb\\ bbc\end{array},\begin{array}{c}bca\\ bbc\\ bcb\\ bac\end{array},\begin{array}{c}bbc\\ bcb\\ bac\\ bca\end{array},\begin{array}{c}bcb\\ bac\\ bca\\ bbc\end{array},\begin{array}{c}bac\\ bca\\ bbc\\ bcb\end{array},\begin{array}{c}cab\\ bcb\\ cbb\\ acb\end{array},\begin{array}{c}bcb\\ cbb\\ acb\\ cab\end{array},\begin{array}{c}cbb\\ acb\\ cab\\ bcb\end{array},\begin{array}{c}acb\\ cab\\ bcb\\ cbb\end{array}\right\}$$

$$PALConj(w)=\left\{\begin{array}{c}abc\\ cbb\\ bbc\\ cba\end{array},\begin{array}{c}bbc\\ cba\\ abc\\ cbb\end{array}\right\},HVPALConj(w)=\varnothing$$

$$PALConj(v)=HVPALConj(v)=\left\{\begin{array}{c}abba\\ aaaa\\ aaaa\\ abba\end{array},\begin{array}{c}baab\\ aaaa\\ aaaa\\ baab\end{array},\begin{array}{c}aaaa\\ abba\\ abba\\ aaaa\end{array},\begin{array}{c}aaaa\\ baab\\ baab\\ aaaa\end{array}\right\}$$

$$0\le |PALConj(w)|\le \{\begin{array}{ll}4,& \text{if m,n are even},\\ 1,& \text{if m,n are odd},\\ 2,& \text{otherwise}.\end{array}$$

$$PALConj(w)=\{{\circlearrowleft }_{i_1}^{Col}{\circlearrowleft }_{j_1}^{Row}w,{\circlearrowleft }_{i_1}^{Col}{\circlearrowleft }_{j_2}^{Row}w,{\circlearrowleft }_{i_2}^{Col}{\circlearrowleft }_{j_1}^{Row}w,{\circlearrowleft }_{i_2}^{Col}{\circlearrowleft }_{j_2}^{Row}w\}$$

$$PALConj(w)=\{{\circlearrowleft }_{i_1}^{Col}{\circlearrowleft }_{j_1}^{Row}w,{\circlearrowleft }_{i_2}^{Col}{\circlearrowleft }_{j_1}^{Row}w\}$$

$$PALConj(w)=\{{\circlearrowleft }_{i_1}^{Col}{\circlearrowleft }_{j_1}^{Row}w,{\circlearrowleft }_{i_1}^{Col}{\circlearrowleft }_{j_2}^{Row}w\}$$

$$PALConj(w)=\{{\circlearrowleft }_{i_1}^{Col}{\circlearrowleft }_{j_1}^{Row}w\}$$

$$0\le |HVPALConj(w)|\le \{\begin{array}{ll}4,& \text{if m,n are even},\\ 1,& \text{if m,n are odd},\\ 2,& \text{otherwise}.\end{array}$$

$$a_{n}=\left\{\begin{array}[]{llll}\frac{m}{2}((\frac{m}{2}+1)n-m+2),&&&if\;m\;is\;even\\ (n-2)(\frac{m+1}{2})^{2}+2m,&&&if\;m\;is\;odd.\end{array}\right.$$

$$a_k=a_{k-1}+\sum _{i=1}^m\lceil \frac{i}{2}\rceil ⟹a_k=a_2+(k-2)\sum _{i=1}^m\lceil \frac{l}{2}\rceil$$

$$a_{k}=\left\{\begin{array}[]{llll}\frac{m}{2}((\frac{m}{2}+1)k-m+2),&&&if\;m\;is\;even\\ (k-2)(\frac{m+1}{2})^{2}+2m,&&&if\;m\;is\;odd.\end{array}\right.$$

$$a_k=a_{k-2}+\frac{m(m+1)}{2}+\sum _{i=1}^m\lceil \frac{i}{2}\rceil$$

$$a_k=a_{k-2}+\sum _{l=1}^m\lceil \frac{l}{2}\rceil +\sum _{i=1}^m\lceil \frac{i}{2}\rceil =a_{k-2}+2\sum _{l=1}^m\lceil \frac{l}{2}\rceil$$

$$\left\{\begin{array}[]{llll}\infty,&&&if\;|Alph(w)|=1\\ 20,&&&if\;|Alph(w)|=2\\ q,&&&if\;|Alph(w)|=q,\;where\;q\geq 3.\end{array}\right.$$

Full text

1. Introduction

Palindromes are extensively studied in $1$-dimensional words by several authors (See [1, 4, 13, 14]). There is an increasing interest in the combinatorial properties of palindromes in mathematics, theoretical computer science, and biology. The notion of $\theta$-palindrome was defined in [5] and studied in [19] and [7] independently. Some properties that link the palindromes to classical notions such as that of primitive words are present in [20]. Due to their symmetrical properties, this concept was generalized to two-dimension. Such a construction has significance in detecting bilateral symmetry of an image and face recognition technologies ([8, 16]).

Identifying palindromes in arrays dates back to $1994$, when authors in [18], described an array to be a palindrome if all its rows and columns are $1$D palindromes. These structures are referred to as HV-palindromes in [3], where H and V stand for horizontal and vertical respectively. It was much later in $2001$ when Berthé et al., [6] formally defined a $2$D palindrome to be an array which is equal to its reverse. It can be easily observed that an array whose all rows and columns are $1$D palindromes is equal to its reverse. Hence, HV-palindromes is a sub-class of $2$D palindromes. Recently, there has been a rise in research that deals with the concept of $2$D palindromes. A relation between $2$D palindromes and $2$D primitive words was studied in [21]. An algorithm for finding maximal $2$D palindromes was given in [11]. The maximum and the least number of $2$D palindromic sub-arrays in a given array was studied in [3, 23] and [24] respectively.

The main idea of this paper is to study the structure of a special type of $2$D word called an HV-palindrome. The motivation of studying this structure came from the conjecture mentioned in [3] which speculates about the maximum number of HV-palindromic sub-arrays in a $2$D word of size $(2,n)$ for a given $n$. We settle this conjecture in affirmative and generalize the result to $2$D words of larger sizes.

This paper is organized as follows. Section $3$ deals with the characterization of a word to be an HV-palindrome. Further, the $2$D words whose all $2$D palindromes are HV-palindromes are also characterized. In Section $4$, we count the number of possible $2$D palindromes and HV-palindromes for a given array size and investigate the number of $2$D palindromic and HV-palindromic conjugates of a $2$D word. In Section $5$, we find the upper and lower bounds on the number of HV-palindromic sub-arrays in a $2$D word. We end the paper with some concluding remarks.

2. Basic definitions and notations

An alphabet $\Sigma$ is a finite non-empty set of symbols. A $1$D word is defined to be a sequence of symbols from $\Sigma$. $\Sigma^{*}$ denotes the set of all words over $\Sigma$ including the empty word $\lambda$. $\Sigma^{+}=\Sigma\setminus\lambda$. The length of a word $w\in\Sigma^{*}$ is the number of symbols in a word and is denoted by $|w|$. The reversal of $w=a_{1}a_{2}\cdots a_{n}$ is defined to be a string $w^{R}=a_{n}\cdots a_{2}a_{1}$ where $a_{i}\in\Sigma$. $Alph(w)$ is the set of all sub-words of $w$ of length $1$. A word $w$ is said to be a palindrome if $w=w^{R}$. The concepts of prefix, suffix, primitivity, and conjugates are as usual. For all other concepts in formal language theory and combinatorics on words, the reader is referred to [17, 22].

2.1. Two-dimensional arrays

A two-dimensional word $w=[w_{i,j}]_{1\leq i\leq m,1\leq j\leq n}$ over $\Sigma$ of size $(m,n)$ is defined to be a two-dimensional rectangular array of letters. If both $m$ and $n$ are infinite, then $w$ is an infinite $2$D word. A factor of $w$ is a sub-array/sub-word of $w$. In the case of $2$D words, an empty word is a word of size $(0,0)$, and we use the notation $\lambda$ to denote such a word. The set of all $2$D words including the empty word $\lambda$ over $\Sigma$ is denoted by $\Sigma^{**}$ whereas, $\Sigma^{++}$ is the set of all non-empty $2$D words over $\Sigma$. Note that, the words of size $(m,0)$ and $(0,m)$ for $m>0$ are not defined.

** Definition 2.1****.**

Let $u=[u_{i,j}]$ and $v=[v_{i,j}]$ be two $2D$ words over $\Sigma$ of size $(m_{1},n_{1})$ and $(m_{2},n_{2})$, respectively.

1. (1)

The column concatenation of $u$ and $v$ $($denoted by $\obar)$ is a partial operation, defined if $m_{1}=m_{2}=m$, and it is given by

[TABLE]

The column closure of $u$ $($denoted by $u^{*\obar})$ is defined as $u^{*\obar}=\bigcup_{i\geq 0}u^{i\obar}$ where $u^{0\obar}=\lambda,u^{1\obar}=u,u^{n\obar}=u\obar u^{(n-1)\obar}$. As $n\rightarrow\infty$, we get $u^{\infty\obar}$. 2. (2)

The row concatenation of $u$ and $v$ $($denoted by $\ominus)$ is a partial operation defined if $n_{1}=n_{2}=n$, and it is given by

[TABLE]

The row closure of $u$ $($denoted by $u^{*\ominus})$ is defined as $u^{*\ominus}=\bigcup_{i\geq 0}u^{i\ominus}$ where $u^{0\ominus}=\lambda,u^{1\ominus}=u,u^{n\ominus}=u\ominus u^{(n-1)\ominus}$. As $n\rightarrow\infty$, we get $u^{\infty\ominus}$.

In [2], a prefix of a $2$D word $w$ is defined to be a rectangular sub-block that contains one corner of $w$, whereas suffix of $w$ is defined to be a rectangular sub-array that contains the diagonally opposite corner of $w$. However, we consider prefix/suffix of a $2$D word $w$ to be a rectangular sub-array that contains the top-left/bottom-right corner of $w$. This was formally defined in [21] as follows.

** Definition 2.2****.**

Given $u\in\Sigma^{**}$, $v\in\Sigma^{**}$ is said to be a prefix of $u$ $($respectively, suffix of $u)$, denoted by $v\leq_{p}u$ $($respectively $v\leq_{s}u)$ if $u=(v\ominus x)\obar y$ or $u=(v\obar x)\ominus y$ $($respectively, $u=y\obar(x\ominus v)$ or $u=y\ominus(x\obar v))$ for $x,y\in\Sigma^{**}$.

A border of a $2$D word is a sub-array that occurs as both the prefix and suffix.

** Definition 2.3****.**

Let $w=[w_{i,j}]_{1\leq i\leq m,1\leq j\leq n}$ be a $2$D word. The reverse and transpose of $w$, denoted by $w^{R}$ and $w^{T}$ respectively are defined as

$w^{R}=\begin{matrix}w_{m,n}&w_{m,n-1}&\cdots&w_{m,1}\\ w_{m-1,n}&w_{m-1,n-1}&\cdots&w_{m-1,1}\\ \vdots&\vdots&\ddots&\vdots\\ w_{1,n}&w_{1,n-1}&\cdots&w_{1,1}\end{matrix}\;\;\;\;\;\;w^{T}=\begin{matrix}w_{1,1}&w_{2,1}&\cdots&w_{m,1}\\ w_{1,2}&w_{2,2}&\cdots&w_{m,2}\\ \vdots&\vdots&\ddots&\vdots\\ w_{1,n}&w_{2,n}&\cdots&w_{m,n}\end{matrix}$

If $w=w^{R}$, then $w$ is said to be a two-dimensional $(2$D$)$ palindrome ([6, 11]).

** Example 2.4****.**

Let $\Sigma=\{a,b,c\}$ and

[TABLE]

As $w=w^{R}$, so $w$ is a $2$D palindrome.

Note that the rows and columns of a $2$D palindrome are not always $1$D palindromes. Also, if all columns and rows of a finite $2$D word are palindromes, then the word is a $2$D palindrome. Such palindromes are referred to as HV-palindromes in [3].

** Example 2.5****.**

Let $\Sigma=\{a,b\}$ and

[TABLE]

As every row and column of $u$ is a $1$D palindrome, so $u$ is an HV-palindrome.

We recall the notion of horizontal and vertical palindromes from [23].

** Definition 2.6****.**

The horizontal palindromes of a $2$D word $w$ are the palindromic factors of $w$ of size $(1,j)$ where $j\geq 1$ and $vertical\;palindromes$ of $w$ are the palindromic factors of $w$ of size $(i,1)$ where $i\geq 2$.

The palindromes of size $(1,1)$ are trivial and rest are non-trivial. Note that, all horizontal and vertical palindromes are HV-palindromes. We now recall the notion of the center of a $2D$ word as defined in [11].

** Definition 2.7****.**

Center is the position that results in an equal number of columns to the left and right, as well as an equal number of rows above and below, i.e. in a word of size $(m,n)$, if $m$ and $n$ are odd, then the center is at location $(\lceil\frac{m}{2}\rceil,\lceil\frac{n}{2}\rceil)$. If $m$ or/and $n$ is even, the center is in between rows or/and columns respectively.

** Example 2.8****.**

The center of $w$ in Example 2.4 is in between the rows and columns of the sub-array $cc\ominus cc$. However, for the word $u$ from Example 2.5, the center of $u$ is the sub-word c.

Throughout the paper, by ‘the number of palindromes’, we mean the number of non-empty distinct palindromic factors. For more information pertaining to two-dimensional word concepts, we refer the reader to [6, 10, 11, 12].

3. Structure of an HV-palindrome

By definition, if a $2$D word $w$ of size $(m,n)$ is a palindrome, then $w=w^{R}$ and $w$ admits two symmetries namely identity and $180^{\circ}$ rotation. Hence, given a $2$D palindrome $w=[w_{i,j}]$ of size $(m,n)$, let $p_{1}$ be the prefix of size $(\lfloor\frac{m}{2}\rfloor,n)$ and $p_{2}=w_{\lceil\frac{m}{2}\rceil,1}w_{\lceil\frac{m}{2}\rceil,2}\cdots w_{\lceil\frac{m}{2}\rceil,n}$. Then, $w$ is of the form

1. (1)

$p_{1}\ominus p_{1}^{R}$, if $m$ is even. 2. (2)

$p_{1}\ominus p_{2}\ominus p_{1}^{R}$, if $m$ is odd.

In addition to the symmetries in a $2$D palindrome, an HV-palindrome is preserved under reflections about horizontal and vertical axis containing the center of the word. Due to such symmetrical properties, we give the exact structure of an HV-palindrome.

** Theorem 3.1****.**

$($Structure theorem of HV-palindromes$)$

Given an HV-palindrome $w=[w_{i,j}]$ of size $(m,n)$, let $u=u_{1}\ominus u_{2}\ominus\cdots\ominus u_{\lfloor\frac{m}{2}\rfloor}$ be the prefix of $w$ of size $(\lfloor\frac{m}{2}\rfloor,\lfloor\frac{n}{2}\rfloor)$,  $v={u_{1}}^{R}\ominus{u_{2}}^{R}\ominus\cdots\ominus{u_{\lfloor\frac{m}{2}\rfloor}}^{R}$,  $p_{1}=w_{1,\lceil\frac{n}{2}\rceil}w_{2,\lceil\frac{n}{2}\rceil}\cdots w_{\lfloor\frac{m}{2}\rfloor,\lceil\frac{n}{2}\rceil}$ and $p_{2}=w_{\lceil\frac{m}{2}\rceil,1}w_{\lceil\frac{m}{2}\rceil,2}\cdots w_{\lceil\frac{m}{2}\rceil,\lfloor\frac{n}{2}\rfloor}$. Then, $w$ is of one of the following forms.

[TABLE]

where $x=w_{\lceil\frac{m}{2}\rceil,\lceil\frac{n}{2}\rceil}.$

Proof.

We give the proof of the case when $m$ and $n$ are both even and the rest of the cases follow similarly. Let $w=[w_{i,j}]$ be an HV-palindrome of size $(m,n)$ and $u=u_{1}\ominus u_{2}\ominus\cdots\ominus u_{\frac{m}{2}}$ be the prefix of $w$ of size $(\frac{m}{2},\frac{n}{2})$. Now, as every row of $w$ is a palindrome, $u_{1}u_{1}^{R}\ominus u_{2}u_{2}^{R}\ominus\cdots\ominus u_{\frac{m}{2}}u_{\frac{m}{2}}^{R}=u\obar v$ is the prefix of $w$ of size $(\frac{m}{2},n)$. Also, as every column of $w$ is a palindrome, then $w=(u\obar v)\ominus(v^{R}\obar u^{R})$. ∎

An HV-palindrome of size $(m,n)$ can be constructed from an HV-palindrome of size $(m+1,n+1)$ by removing its $\lceil\frac{m}{2}\rceil^{th}$ row and $\lceil\frac{n}{2}\rceil^{th}$ column. Such a construction in the case of a $2$D palindrome is given in [21]. We also observe the following result.

** Lemma 3.2****.**

If $w$ is an HV-palindrome of size $(m,n)$, then the word obtained by the removal of first and last $k$ rows of $w$, for $1\leq k\leq\lfloor\frac{m}{2}\rfloor$ and first and last $r$ columns of $w$, for $1\leq r\leq\lfloor\frac{n}{2}\rfloor$ is an HV-palindrome of size $(m-2k,n-2r)$. This result is also true in the case of $2$D palindromes.

3.1. Characterization of an HV-palindrome

In this section, we first give two necessary and sufficient conditions for a $2$D word to be an HV-palindrome and then give a characterization of $2$D words such that all of their palindromic sub-words are HV-palindromes i.e., words with no non-HV-palindromic sub-arrays.

** Proposition 3.3****.**

Let $w=w_{1}\ominus w_{2}\cdots\ominus w_{m}=u_{1}\obar u_{2}\cdots\obar u_{n}$ be a $2$D word of size $(m,n)$, where $w_{i}$ and $u_{j}$ is the $i^{th}$ row and $j^{th}$ column of $w$ respectively. Then, $w$ is an HV-palindrome if and only if $w_{i}=w_{m-i+1}$ for $1\leq i\leq\lfloor\frac{m}{2}\rfloor$ and $u_{j}=u_{n-j+1}$ for $1\leq j\leq\lfloor\frac{n}{2}\rfloor$.

Proof.

Let $w=w_{1}\ominus w_{2}\cdots\ominus w_{m}=u_{1}\obar u_{2}\cdots\obar u_{n}$ be an HV-palindrome. This implies each $w_{i}$ and $u_{j}$ is a palindrome. Every row of $w$ is a palindrome if and only if $u_{j}=u_{n-j+1}$ for $1\leq j\leq\lfloor\frac{n}{2}\rfloor$. Every column is a palindrome if and only if $w_{i}=w_{m-i+1}$ for $1\leq i\leq\lfloor\frac{m}{2}\rfloor$. ∎

One can easily observe that the above result does not hold true in the case of $2$D palindromes that are not HV-palindromes. For example, the word $w$ given in Example 2.4 does not satisfy the conditions given in Proposition 3.3.

It is well known that a 1D word $u$ is a $1$D-palindrome if and only if $u=(xy)^{i}x$ for some 1D palindromes $x,y\in\Sigma^{*}$ and $i\geq 1$. In general, it was proved in [20], that if $\theta$ is an antimorphic involution, then $w\in P_{\theta}$ iff $w=\alpha(\beta\alpha)^{i}$ for some $a,b\in P_{\theta}$ and $i\geq 0$ where $P_{\theta}$ denote the set of all $\theta$-palindromes. For $2$D palindromic words, the condition was shown to be only sufficient in [21]. We now show that the condition is both necessary and sufficient for an HV-palindrome.

** Proposition 3.4****.**

Let $x,y$ be HV- palindromes. Then, $u=(x\obar y)^{i\obar}\obar x$ or $u=(x\ominus y)^{i\ominus}\ominus x$, $i\geq 1$ if and only if $u$ is an HV- palindrome.

Proof.

Let $u=(x\obar y)^{i\obar}\obar x$, then

[TABLE]

Hence, $u$ is a $2$D palindrome. A similar proof follows for $u=(x\ominus y)^{i\ominus}\ominus x$. Now as $x$ and $y$ are HV-palindromes, then $x\obar y\obar x$ is an HV-palindrome. Hence, $u$ is an HV-palindrome for $i\geq 1$.

Conversely, let $u=u_{1}\ominus u_{2}\ominus\cdots\ominus u_{r}$ be an HV-palindrome of size $(r,s)$, then by Proposition 3.3, $u_{i}=u_{r-i+1}$ for $1\leq i\leq\lfloor\frac{r}{2}\rfloor$ and every row of $u$ is a palindrome. Let $x=u_{1}$ and $y=u_{2}\ominus u_{3}\ominus\cdots\ominus u_{r-1}$. By Lemma 3.2, $y$ is an HV-palindrome. Thus, $u=(x\ominus y)\ominus x$. ∎

All letters from the alphabet $\Sigma$ are trivially palindromes and hence every $2$D word has palindromic sub-words. Also, all trivial palindromes are HV-palindromes and all HV-palindromes are $2$D palindromes. However, a $2$D palindrome may or may not be an HV-palindrome. A word may or may not contain non-HV palindromes. In the following, we give a characterization of $2$D words such that all of its palindromic sub-words are HV-palindromes. The word $u$ in Example 2.5 is one such word.

** Theorem 3.5****.**

All $2$D palindromic sub-words of a $2$D word $w$ are HV-palindromes if and only if $w$ has no sub-word of the form

[TABLE]

where $x,y\in\Sigma$ such that $x\neq y$, $u$ and $v^{T}$ are $1$D words and $p$ is a $2$D palindrome.

Proof.

Let $w$ be a $2$D word with no sub-word of the form

[TABLE]

where $x\neq y$ such that $x,y\in\Sigma$, $u$ and $v^{T}$ are $1$D words and $p$ is a $2$D palindrome. We show that all of its $2$D palindromic sub-words are HV-palindromes. Let $p^{\prime}$ be a $2$D palindromic sub-word of $w$ of size $(r,s)$, with $r,s\geq 2$, then it must be of the form

[TABLE]

where $x_{1}\in\Sigma$, $u_{1}$ and $v_{1}$ are $1$D words and $p_{1}$ is a $2$D palindrome. We show that $p^{\prime}$ is an HV-palindrome. Consider the first and the last row of $p^{\prime}$. We show that they are the same. If not, consider the first position where they are different, say it is the $i^{th}$ position. Let the $i^{th}$ position of the first and last row of $p^{\prime}$ be $a$ and $b$ respectively, where $a\neq b\in\Sigma$. As $p^{\prime}$ is a $2$D palindrome, then the $(s-i+1)^{th}$ position of the first and last row of $p^{\prime}$ are $b$ and $a$ respectively. Then $p^{\prime}$ has a sub-word $\begin{matrix}p^{\prime}_{1,i}&\cdots&p^{\prime}_{1,s-i+1}\\ \vdots&\ddots&\vdots\\ p^{\prime}_{r,i}&\cdots&p^{\prime}_{r,s-i+1}\end{matrix}$ of the form $\begin{matrix}a&u_{2}&b\\ v_{2}&p_{2}&v_{2}^{R}\\ b&u_{2}^{R}&a\end{matrix}$ where $a\neq b$, $u_{2}$ and $v_{2}^{T}$ are $1$D words and $p_{2}$ is a $2$D palindrome which is a contradiction. Hence, the first row and the last row of $p^{\prime}$ are same. Similarly, we can show that the $j^{th}$ and $(r-j+1)^{th}$ row of $p^{\prime}$ are same for $1\leq j\leq\lfloor\frac{r}{2}\rfloor$. Now, consider the word $(p^{\prime})^{T}$ which is a palindrome of size $(s,r)$. Apply the same procedure on $(p^{\prime})^{T}$ to show that the $j^{th}$ and ${(s-j+1)}^{th}$ row of $(p^{\prime})^{T}$ are same for $1\leq j\leq\lfloor\frac{s}{2}\rfloor$. This implies $j^{th}$ and ${(s-j+1)}^{th}$ column of $p^{\prime}$ is same for $1\leq j\leq\lfloor\frac{s}{2}\rfloor$. Thus, by Proposition 3.3, $p^{\prime}$ is an HV-palindrome.

Conversely, if all palindromic sub-words of $w$ are HV-palindromes, then any sub-word of the form

[TABLE]

where $x,y\in\Sigma$ are distinct, $u$ and $v^{T}$ are $1$D words and $p$ is a $2$D palindrome is itself a non-HV-palindrome which is a contradiction. ∎

3.2. Borders in a 2D palindrome

In [15], the behaviour of borders of $1$D words was investigated and was shown that the asymptotic probability that a random word has a given maximal border length $k$ is a constant, depending only on k and the alphabet size. The concept of $\theta$-bordered and $\theta$-unbordered words for an arbitrary morphic or antimorphic involution $\theta$ was reviewed in [19]. Here, we analyze the structure of borders in a $2$D palindrome.

** Proposition 3.6****.**

Every border of a $2$D palindrome is a $2$D palindrome.

Proof.

Let $w=[w_{i,j}]$ be a $2$D palindrome of size $(m,n)$. Let $v$ be a border of size $(i,j)$, then

[TABLE]

Now, as $w$ is a palindrome, then for $0\leq k\leq i-1$

[TABLE]

Hence, by Equations (1) and (2), $v$ is a $2$D palindrome. ∎

** Corollary 3.7****.**

Every border of an HV-palindrome is a $2$D palindrome.

Note that the border of an HV-palindrome need not be an HV palindrome. For example, in the word $aba\ominus bab\ominus aba$, the sub-array $ab\ominus ba$ is a border but is not an HV-palindrome.

It is clear that the maximum number of borders in a word of size $(m,n)$ is $mn$ and is achieved when $|Alph(w)|=1$. Let $BOR(w)$ be the set of all borders of $w$, then we have the following result.

** Lemma 3.8****.**

Let $w$ be a word of size $(m,n),\;m,n\geq 2$.

1. (1)

*If $w$ is a *$2$D palindrome, then $2\leq|BOR(w)|\leq mn$. 2. (2)

If $w$ is an HV-palindrome, then $4\leq|BOR(w)|\leq mn$.

Proof.

Let $w$ be a word of size $(m,n)$, $m,n\geq 2$.

1. (1)

If $w$ is a $2$D palindrome, then the prefix of size $(1,1)$ and $(m,n)$ is a border of $w$. The word $abb\ominus bbb\ominus bba$ achieves the lower bound. 2. (2)

If $w$ is an HV-palindrome, then the prefixes of size $(1,1)$, $(1,n)$, $(m,1)$ and $(m,n)$ are borders of $w$. The word $abba\ominus bbbb\ominus abba$ has only four borders.

∎

4. Counting Palindromes

It can be easily observed that given an alphabet of size $q$, there are $q^{\lceil\frac{n}{2}\rceil}$ distinct $1$D palindromes of length $n$. In this section, we first count the number of distinct $2$D palindromes (HV-palindromes) that can be obtained over a given alphabet $\Sigma$. We then determine the number of $2$D palindromes (HV-palindromes) that can appear in the conjugacy class of a given $2$D word.

** Theorem 4.1****.**

Let $\Sigma$ be a finite alphabet such that $|\Sigma|=q$. Then, there are

1. (1)

$q^{i}$*, where $i=\lceil\frac{mn}{2}\rceil$ distinct *$2$D palindromes of size $(m,n)$. 2. (2)

$q^{j}$, where $j=\lceil\frac{m}{2}\rceil\lceil\frac{n}{2}\rceil$ distinct HV-palindromes of size $(m,n)$.

Proof.

Let $w$ be a $2$D palindrome of size $(m,n)$ such that $w=w_{1}\ominus w_{2}\ominus\cdots\ominus w_{m}$, where $w_{i}$ is the $i^{th}$ row of $w$. Since $w=w^{R}$, the $1$D word $u=w_{1}w_{2}\cdots w_{m}$ is a $1$D palindrome. Let $v$ be the prefix of $u$ of length $\lceil\frac{mn}{2}\rceil$. Then, $v^{R}$ is a suffix of $u$. Thus, we have $\lceil\frac{mn}{2}\rceil$ distinct choices of letters from $\Sigma$ and therefore, there are $q^{\lceil\frac{mn}{2}\rceil}$ distinct $2$D palindromes of size $(m,n)$ over $\Sigma$. If $w$ is an HV-palindrome of size $(m,n)$, then by Proposition 3.3, we have $\lceil\frac{m}{2}\rceil\lceil\frac{n}{2}\rceil$ distinct choices of letters in the prefix of $w$ of size $(\lceil\frac{m}{2}\rceil,\lceil\frac{n}{2}\rceil)$. Hence, there are $q^{\lceil\frac{m}{2}\rceil\lceil\frac{n}{2}\rceil}$ distinct HV-palindromes of size $(m,n)$ over $\Sigma$. ∎

We illustrate with the following example.

** Example 4.2****.**

Consider the binary alphabet $\Sigma=\{a,b\}$. The set of all $2$D palindromes of size $(2,2)$ is

[TABLE]

This set has four $2$D palindromes and two HV-palindromes.

4.1. Palindromes in the conjugacy class of a word

It was proved in [14] that the conjugacy class of a $1$D word contains at most two $1$D palindromes. The concept of $\theta$ conjugacy on words was defined in [19].

We consider the concept of conjugacy class in two-dimensional words and count the number of $2$D palindromes and HV-palindromes. We recall the definition of conjugates of an array from [25].

** Definition 4.3****.**

Let $u_{1},u_{2},\cdots u_{m}$ and $v_{1},v_{2},\cdots v_{n}$ be respectively the $m$ rows and $n$ columns of a word $w$ of size $(m,n)$. The cyclic rotation of $k$ columns, for $1\leq k\leq n$, denoted by $\circlearrowleft^{Col}_{k}$ is defined as the word

[TABLE]

Similarly, the cyclic rotation of $k$ rows, for $1\leq k\leq m$, denoted by $\circlearrowleft^{Row}_{k}$ is defined as the word

[TABLE]

Then, the conjugacy class of $w$, denoted by $Conj(w)$ is defined as

[TABLE]

Note that, given any $2$D word of size $(m,n)$, the number of elements in its conjugacy class can be at most $mn$. We illustrate with the following example.

** Example 4.4****.**

Consider the 2D word $w=abc\ominus cbb\ominus bbc\ominus cba$ of size $(4,3)$. Then,

[TABLE]

** Remark 4.5****.**

For a 2D word $w$ of size $(m,n)$, if $|Alph(w)|=1$, $|Conj(w)|=1$ and if $|Alph(w)|=mn$, $|Conj(w)|=mn$. However, the converse need not be true as illustrated in Example 4.4. If $w$ is a $2$D palindrome with $|Alph(w)|=\lceil\frac{mn}{2}\rceil$ or an HV-palindrome with $|Alph(w)|=\lceil\frac{m}{2}\rceil\lceil\frac{n}{2}\rceil$, then $|Conj(w)|=mn$.

We count the maximum number of $2$D palindromes (HV-palindromes) in the conjugacy class of a $2$D word. We call such conjugates as palindromic (HV-palindromic) conjugates and denote them by $PALConj(w)\;(HVPALConj(w))$. Note that, for the $w$ given in Example 4.4 we have,

[TABLE]

We give another example of a $2$D word $v=abba\ominus aaaa\ominus aaaa\ominus abba$ of size $(4,4)$ where $|HVPALConj(v)|=|PALConj(v)|=4$.

[TABLE]

We first recall the following result for $1$D words from [14].

** Theorem 4.6****.**

A conjugacy class of a $1$D word contains at most two palindromes and it has exactly two if and only if it contains a word of the form $(uu^{R})^{i}$, where $uu^{R}$ is a primitive word and $i\geq 1$.

Consider a $2$D word $w=v_{1}\obar v_{2}\obar\cdots\obar v_{n}\;=\;u_{1}\ominus u_{2}\ominus\cdots\ominus u_{m}$, where $v_{i}$ is the $i^{th}$ column and $u_{j}$ is the $j^{th}$ row of $w$. The word $w$ can be considered as a $1$D word over the alphabet of columns i.e., the set $A=\{v_{1},v_{2},\ldots,v_{n}\}$ and over the alphabet of rows i.e., the set $B=\{u_{1},u_{2},\ldots,u_{m}\}$.

** Remark 4.7****.**

Note that a $2$D word is a $2$D palindrome if and only if it is a $1$D palindrome over its alphabet of columns and over its alphabet of rows.

We now have the following result.

** Theorem 4.8****.**

Let $w$ be a $2$D word of size $(m,n)$, then

[TABLE]

Proof.

It is clear that there exist words $($for example, $w=aa\ominus ab)$ with no palindromic conjugates. Also, note that for a $2$D word $w$ of size $(m,n)$ such that $|Alph(w)|=1$, $|PALConj(w)|=1$ and for $w$, such that $|Alph(w)|=mn$, $|PALConj(w)|=0$.

We now find the number of palindromic conjugates in a word $w$ of size $(m,n)$. If $w$ has no palindromic conjugates, then we are done. Otherwise, assume that $v\in PALConj(w)$. Note that, $PALConj(w)=PALConj(v)$. Now, $v=v_{1}\obar v_{2}\obar\cdots\obar v_{n}\;=\;u_{1}\ominus u_{2}\ominus\cdots\ominus u_{m}$, where $v_{i}$ is the $i^{th}$ column of $v$ and $u_{j}$ is the $j^{th}$ row of $v$. As $v$ is a $2$D palindrome, by Remark 4.7, it is a $1$D palindrome over the alphabet of columns and over the alphabet of rows. We have the following cases.

• Case 1:

If $m$ and $n$ are both even, then by Theorem 4.6, as $n$ is even, there can be at most two $1$D palindromic conjugates of $v$ say $v$ and $v^{\prime}$ over its alphabet of columns. Each of them can be expressed as a $1$D word over its alphabet of rows say the set $B$. Again, as $m$ is even, by Theorem 4.6, there are at most two palindromic conjugates of $v$ and $v^{\prime}$ each as a $1$D word over $B$. In total, there are at most four palindromic conjugates in the conjugacy class of $w$. Hence, for some $i_{1},\;i_{2},\;j_{1}$ and $j_{2}$,

[TABLE]

• Case 2:

If $m$ is odd and $n$ is even, then $v$ can be expressed as a $1$D word over its alphabet of rows say the set $B$. As $m$ is odd, by Theorem 4.6, there is no palindromic conjugate of $v$, other than $v$ over $B$. Again, $v$ can be expressed as a $1$D word over the alphabet of columns say the set $A$ to obtain at most two palindromic conjugates over $A$. In total, there are at most two palindromic conjugates in the conjugacy class of $w$. Hence, for some $i_{1},\;i_{2}$ and $j_{1}$,

[TABLE]

Similar is the case when $m$ is even and $n$ is odd. Here, for some $i_{1},\;j_{1}$ and $j_{2}$,

[TABLE]

• Case 3:

If $m,\;n$ are both odd, by Theorem 4.6, there is no palindromic conjugate of $v$, other than $v$. Hence, for some $i_{1}$ and $j_{1}$,

[TABLE]

∎

Theorem 4.8 also holds in the case of HV-palindromes. We have the following result.

** Corollary 4.9****.**

Let $w$ be a $2$D word of size $(m,n)$, then

[TABLE]

** Remark 4.10****.**

We now find the exact structure of the words that achieve the above upper bound in the case of HV-palindromes. Let $v\in HVPALConj(w)$, then by the structure of $v$, $i_{1}=0,j_{1}=0,i_{2}=\frac{n}{2}$ and $j_{2}=\frac{m}{2}$.

1. Case 1:

If $m$ and $n$ are even, then to have four distinct palindromic conjugates, we have the following:

1. (a)

$v\;\neq\;\circlearrowleft^{Col}_{\frac{n}{2}}v$ and $\circlearrowleft^{Row}_{\frac{m}{2}}v\;\neq\;\circlearrowleft^{Col}_{\frac{n}{2}}\circlearrowleft^{Row}_{\frac{m}{2}}v\;\implies$ the prefix of size $(m,\frac{n}{2})$ is not an HV-palindrome. 2. (b)

$v\;\neq\;\circlearrowleft^{Row}_{\frac{m}{2}}v$ and $\circlearrowleft^{Col}_{\frac{n}{2}}v\;\neq\;\circlearrowleft^{Col}_{\frac{n}{2}}\circlearrowleft^{Row}_{\frac{m}{2}}v\;\implies$ the prefix of size $(\frac{m}{2},n)$ is not an HV-palindrome. 3. (c)

$w\;\neq\;\circlearrowleft^{Col}_{\frac{n}{2}}\circlearrowleft^{Row}_{\frac{m}{2}}v\;\implies$ the prefix of size $(\frac{m}{2},\frac{n}{2})$ is not a $2$D palindrome. 4. (d)

$\circlearrowleft^{Col}_{\frac{n}{2}}v\;\neq\;\circlearrowleft^{Row}_{\frac{m}{2}}v\;\implies$ the prefix of size $(\frac{m}{2},\frac{n}{2})$ is not a $2$D palindrome.

Hence, in this case $|PALConj(v)|=4$ iff the prefix of size $(\frac{m}{2},\frac{n}{2})$ of $v$ is not a $2$D palindrome. 2. Case 2:

For $m$ even and $n$ odd, $|PALConj(v)|=2$ iff the prefix of size $(\frac{m}{2},n)$ is not an HV-palindrome. 3. Case 3:

For $m$ odd and $n$ even, $|PALConj(v)|=2$ iff the prefix of size $(m,\frac{n}{2})$ is not an HV-palindrome.

5. Bounds on the number of palindromes

In this section, we find the maximum and the least number of HV-palindromes in any $2$D finite and infinite word respectively. We also compare these results with the existing results on $2$D palindromes.

5.1. On the maximum number of HV-palindromes

In this section, we find the maximum number of non-empty distinct HV-palindromes in a $2$D word $w$ of size $(m,n)$. It was proved in [1] and [23] that there are at most $n$ palindromes in a $1$D word of length $n$ and at most $2n+\lfloor\frac{n}{2}\rfloor-1$ palindromes in a two-row array of size $(2,n)$ respectively. Further, it was conjectured in [3] that the number of HV-palindromes in any $2$D word of size $(2,n)$ is less than or equal to $2n$. We give a proof of this conjecture.

** Theorem 5.1****.**

The maximum number of HV-palindromes in any $2$D word of size $(2,n)$ is $2n,\;n\geq 1$.

Proof.

Let $w$ be a $2$D word of size $(2,n)$. The number of HV-palindromes in $w$ is the sum of horizontal palindromes in $w$ and HV-palindromes of size $(2,t),\;t\geq 1,$ in $w$. It can be observed that the HV-palindromes of size $(2,t),\;t\geq 1,$ are of the form $p\ominus p$, where $p$ is a $1$D palindrome. Let the number of the HV-palindromes of size $(2,t),\;t\geq 1,$ in $w$ be $k$. Then, there are $k$ palindromes of the form $p_{i}\ominus p_{i}$ where $1\leq i\leq k$. This implies there are $k$ common horizontal palindromes $p_{i}$ in both the rows of $w$ and hence, these $k$ horizontal palindromes should be counted only once. Thus, as only one horizontal palindrome can be created on the concatenation of a letter, then the maximum number of horizontal palindromes in $w$ is $\leq 2n-k$. Hence, the total number of HV-palindromes in $w$ $\leq 2n-k+k=2n$. ∎

We now give an example of a word of size $(2,n)$ that achieves this bound. Let $w=(a\ominus a)^{n\obar}$. It has $n$ horizontal palindromes: $a^{i}$ for $1\leq i\leq n$ and $n$ HV- palindromes: $(a\ominus a)^{j\obar}$ of size $(2,j)$ for $1\leq j\leq n$. We deduce the following from Theorem 5.1.

** Corollary 5.2****.**

The maximum number of HV-palindromes in any $2$D word of size $(m,2)$ is $2m,\;m\geq 1$.

We recall the following from [23].

** Lemma 5.3****.**

Let $w$ be a $2D$ word of size $(m,n)$ for $m,n\geq 2$. Then, the column concatenation of $w$ and $(x_{1}\ominus x_{2}\ominus\cdots\ominus x_{m})$, where $x_{i}\;\in\Sigma$ for each $i$ creates at most one distinct palindrome of size $(m,t)$ for some $t\geq 1$. Hence, there can be at most $\sum_{i=1}^{m}(m-i+1)=\frac{m(m+1)}{2}$ new palindromes created on the concatenation of a column to a word in $\Sigma^{m\times n}$. Note that, this also holds true in the case of HV-palindromes.

As $P_{2d}(w)=P_{2d}(w^{T})$, we deduce the following result from Lemma 5.3.

** Corollary 5.4****.**

Let $w$ be a $2D$ word of size $(m,n)$ for $m,n\geq 2$. Then, the row concatenation of $w$ and $(x_{1}\obar x_{2}\obar\cdots\obar x_{n})$, where $x_{i}\;\in\Sigma$ for each $i$ creates at most one distinct palindrome of size $(t,n)$ for some $t\geq 1$. The result also holds true in the case of HV-palindromes.

We generalize Theorem 5.1 to words of larger sizes.

** Theorem 5.5****.**

The upper bound of the number of HV-palindromes in any $2$D word of size $(m,n)$ for $m,\;n\geq 2$ is

[TABLE]

Proof.

We prove the result by induction on $n$. The base case for $n=2$ is clear from Corollary 5.2. Assume the result to be true for $n=k-1$. Let $w$ be a word of size $(m,k)$. Then by induction, the number of HV-palindromes in the prefix of size $(m,k-1)$ is bounded above by $a_{k-1}$. On the concatenation of the $k^{th}$ column, new palindromes of size $(i,t),\;t\geq 1$ for $1\leq i\leq m$ can be created.

Consider any prefix of $w$ of size $(i,k)$ say $R_{i}$ for $1\leq i\leq m$. As a suffix of these $R_{i}$, by Lemma 5.3, only one extra HV-palindrome of size $(r,t)$ for each $r$ such that $1\leq r\leq i$ can be created. We observe that as a suffix of each of these $R_{i}$, an HV-palindrome of size $(r,t)$ and an HV-palindrome of size $(i-r+1,t)$ for $t\geq 1$ and $1\leq r\leq\lceil\frac{i}{2}\rceil$ can not be both newly formed as then the HV-palindrome with less number of rows will be present as the sub-word of the other one. Hence, there can be at most $\sum_{i=1}^{m}\lceil\frac{i}{2}\rceil$ new HV-palindromes formed on the concatenation of the $k^{th}$ column. Thus, the maximum number of HV-palindromes in any $2$D word of size $(k,n)$ is

[TABLE]

Solving the recurrence relation for even and odd values of $m$ and with initial condition $a_{2}=2m$, we have

[TABLE]

Hence, the result holds for $m=k$ and thus, the result follows by induction for $m\geq 2$. ∎

Table $1$ depicts the maximum number of distinct non-empty HV-palindromic sub-arrays in any binary word of size $(m,n)$ for larger values of $m$ and $n$ obtained by a computer program along with our obtained upper bound.

It can be observed that the upper bound obtained in Theorem 5.5 is close to the actual values.

5.2. Palindromic sub-arrays in a 2D palindrome

It was proved in [1] that the maximum number of palindromes in a $1$D word $w$ is $|w|$ and $a^{|w|}$ is an example of a palindrome that achieves it.

In this section find the maximum number of palindromes in a $2$D palindrome and the maximum number of HV-palindromes in a HV-palindrome of size $(m,n)$. We recall the following result from [23].

** Theorem 5.6****.**

In a word $w$ of size $(2,n)$,

1. (1)

the maximum number of palindromes in $w$ is $2n+\lfloor\frac{n}{2}\rfloor-1$. 2. (2)

if $w$ is a palindrome, then the maximum number of palindromes in $w$ is $2n$.

We first prove the following.

** Lemma 5.7****.**

The maximum number of palindromes in a palindrome of size $(3,n)$ is $3n+\lfloor\frac{n}{2}\rfloor-1$.

Proof.

If $w$ is a palindrome of size $(3,n)$, then $w=w_{1}\ominus w_{2}\ominus w_{1}^{R}$, where $w_{2}=w_{2}^{R}$. By Theorem 5.6, there are $2n+\lfloor\frac{n}{2}\rfloor-1$ palindromes in $w_{1}\ominus w_{2}$. Since, $w$ is obtained by concatenating $w_{1}^{R}$ to $w_{1}\ominus w_{2}$ as the last row, the only new palindromes that are obtained are palindromes of size $(3,t)$, $t\geq 1$. Thus, we just count the palindromes formed by concatenating the last row. By Corollary 5.4, at most $n$ such palindromes can be created. Hence, $P_{2d}(w)\leq n+2n+\lfloor\frac{n}{2}\rfloor-1\;=\;3n+\lfloor\frac{n}{2}\rfloor-1$. ∎

Using Theorem 5.6 and Lemma 5.7, we have the following result for the maximum number of palindromes in a palindrome of size $(m,n)$.

** Theorem 5.8****.**

The upper bound on the number of palindromes in a $2$D palindrome/HV-palindrome of size $(m,n)$ for $m,\;n\geq 2$ is $a_{n}$ which is one of the following.

[TABLE]

Proof.

Let $w$ be a $2$D palindrome of size $(m,n)$. We prove the result using induction on $n$. The cases when $n=2$ and $n=3$ are clear by taking transpose of the words in Theorem 5.6 and Lemma 5.7 respectively. Assume the result to be true for $n=k-1$. Let $w$ be a palindrome of size $(m,k)$. Then, $w$ is of the form $w_{1}\obar v\obar w_{1}^{R}$, where $w_{1}$ is of size $(m,1)$ and $v$ is a palindrome of size $(m,k-2)$. Note that as $w$ is a palindrome and $v=v^{R}$, the palindromes of size $(i,t),\;1\leq t\leq k-1$ for $1\leq i\leq m$ formed by concatenation of $w_{1}$ and $v$ are also formed on concatenation of $v^{R}$ and $w_{1}^{R}$. Hence, the number of palindromes in $w$ is the sum of palindromes in $v$, the palindromes formed on concatenation of $w_{1}$ to $v$ and the palindromes of size $(i,n)$ formed on concatenation of $w_{1}^{R}$ to $w_{1}\obar v$. By Lemma 5.3, at most $\frac{m(m+1)}{2}$ palindromes are formed on concatenation of $w_{1}$ to $v$. We just have to count the number of palindromes of size $(i,n)$ formed by concatenation of $w_{1}^{R}$ to $w_{1}\obar v$. As $w$ is a palindrome, the $i^{th}$ row of $w$ is the reverse of the $m-i+1^{th}$ row of $w$, so, there can be at most $\lceil\frac{i}{2}\rceil$ distinct palindromes formed of size $(m-i+1,n)$ for $1\leq i\leq m$. Hence,

[TABLE]

Solving the recurrence relation, with initial conditions of $n=2$ and $n=3$ for even and odd values of $k$, we get the result. ∎

We now find the maximum number of HV-palindromes in an HV-palindrome of size $(m,n)$. We first observe the following result by Theorem 5.1 and Corollary 5.4.

** Lemma 5.9****.**

The maximum number of HV-palindromes in a HV-palindrome of size $(3,n)$ is $3n$. The word $a^{n}\ominus b^{n}\ominus a^{n}$ proves that the bound is tight.

** Corollary 5.10****.**

The upper bound on the number of HV-palindromes/palindromes in a HV-palindrome of size $(m,n)$ for $m,\;n\geq 2$ is $a_{n}$ which is one of the following.

[TABLE]

Proof.

Following the proofs of Theorems 5.5 and 5.8, the word $w$ of size $(k,n)$ is of the form $w_{1}\obar v\obar w_{1}$, where $v$ is an HV-palindrome of size $(k-2,n)$, there can be at most $\sum_{l=1}^{m}\lceil\frac{l}{2}\rceil$ palindromes added by concatenating $w_{1}$ to $v$. We just have to count the palindromes of size $(i,n)$ for $1\leq i\leq m$ formed in $w_{1}\obar v\obar w_{1}$. We get the following recurrence relation

[TABLE]

which gives the result. ∎

Note that the bounds in Corollary 5.10 are equal to that of Theorem 5.5 for even values of $n$ and are less than that of Theorem 5.5 for odd values of $n$.

5.3. On the least number of HV-palindromes

In this section, we find the least number of non-empty distinct palindromes in a $2$D infinite word $w$ with $|Alph(w)|=q$.

A $1$D infinite word is an infinite sequence of symbols. It was proved in [9] that there are at least $8$ palindromes in an infinite $1$D word. A $2$D infinite word is an array with infinite rows and columns. We recall the following result on the least number of $2$D palindromes in an infinite $2$D word from [24].

** Theorem 5.11****.**

The least number of $2$D palindromes in an infinite $2$D word is

[TABLE]

We have the following result for HV-palindromes.

** Theorem 5.12****.**

The least number of HV-palindromes in an infinite $2$D word is

[TABLE]

Proof.

Let $w$ be a $2$D infinite word with $|Alph(w)|=q$. We have the following.

• •

Case 1: If $|Alph(w)|=1$, then $w=(x^{\infty\ominus})^{\infty\obar}$, where $x\in\Sigma$. This word has infinite HV-palindromes: $(x^{i\ominus})^{j\obar}$ for $i,\;j\geq 1$.

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Case 2: If $|Alph(w)|=2$, then let $w$ be a binary word on $\Sigma=\{a,b\}$. It was shown in [9], that any finite $1$D binary word of length greater than $8$, has at least $8$ palindromic factors. All these palindromes are HV-palindromes. Since every infinite $1$D word must have at least $8$ HV-palindromes, every row and column of $w$ has at least $8$ HV-palindromes. The only palindromes that can be common to both are the trivial palindromes i.e. $a$ and $b$. Thus, $w$ has at least $6$ horizontal and $6$ vertical non-trivial HV-palindromes. Thus, any $2$D infinite binary word $w$ has at least $8+8-2=14$ HV-palindromes. We give an example of the word that achieves the bound. Let $u_{1}=ababba$ and $u_{i+1}$ be the $1$-cyclic shift of $u_{i}$ for $1\leq i\leq 5$ and $v=u_{1}\ominus u_{2}\ominus\cdots\ominus u_{6}$. Note that, for the given $v$, $(v^{2\ominus})^{2\obar}$ has exactly $14$ HV-palindromes: $\{p,\;p^{T}:p\in A\}$ where $A=\{a,b,aa,bb,aba,bab,abba,baab\}$. As there is no palindrome of size $(i,1)$ or $(1,j)$ for $i,\;j\geq 5$ in $(v^{2\ominus})^{2\obar}$, thus, there are only $14$ HV-palindromes in $(v^{\infty\ominus})^{\infty\obar}$ which is the required word.

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Case 3: If $|Alph(w)|=q$, where $q\geq 3$, then there are at least $q$ trivial HV-palindromes. We give a word with exactly $q$ HV-palindromes. Let $Alph(w)=\{a_{1},a_{2},\ldots,a_{q}\}$ and

[TABLE]

Then, the word $w=(u^{\infty\ominus})^{\infty\obar}$ has $q$ HV-palindromes: $\{a_{1},a_{2},\ldots,a_{q}\}$

∎

As the palindromes when $q\geq 3$ in the above theorem are all trivial, we find the least number of HV-palindromes in an infinite $2$D word $w$ with at least one non-trivial HV-palindrome such that $|Alph(w)|\geq 3$.

** Theorem 5.13****.**

The least number of HV-palindromes in an infinite $2$D word $w$ with $|Alph(w)|=q$ that has at least one non-trivial HV-palindrome is \begin{cases}5,&\text{if q=3},\\ q+1,&\text{if q>3}.\end{cases}

Proof.

Let $w$ be a $2$D infinite word with $|Alph(w)|=q$. We have the following cases.

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Case 1: If $|Alph(w)|=3$, then consider an infinite $2$D word with exactly one non-trivial HV-palindrome. We observe that this non-trivial HV-palindrome should occur as a sub-word and is of one of the forms $xx,xyx$ or their transpose where $x,y\in\Sigma$. If $|Alph(w)|=3$ i.e $\Sigma=\{a,b,c\}$, then we cannot construct a sub-word of size $(3,3)$ with the above as prefixes with only one non-trivial HV-palindrome. Hence, there is no infinite $2$D word with exactly one non-trivial HV-palindrome when $|Alph(w)|=3$. Thus, we construct an infinite $2$D word $w$ with exactly two non-trivial HV-palindromes such that $|Alph(w)|=3$. Let $w=\left(a(abc)^{\obar\infty}\right)\ominus{\left(v^{\obar\infty}\right)}^{\ominus\infty}$ for $v=bca\ominus abc\ominus cab$. It has only two non-trivial HV-palindromes $aa$ and $a\ominus b\ominus a$. Thus, the least number of HV-palindromes in an infinite $2$D word $w$ with $|Alph(w)|=3$ that has at least one non-trivial HV-palindrome is $5$.

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Case 2: If $|Alph(w)|=q$, for $q\geq 4$, then the least number of HV-palindromes in an infinite $2$D word $w$ with at least one non-trivial HV-palindrome should be greater than or equal to $q+1$. We give the existence of such a word with exactly $q+1$ HV-palindromes. Let $w=(u^{\infty\ominus})^{\infty\obar}$ for

[TABLE]

Here, $w$ has one non-trivial HV-palindrome $a_{1}\ominus a_{1}$ along with trivial palindromes.

∎

6. Conclusions

HV-palindromes is a special class of $2$D palindromes in which every row and column is a $1$D palindrome. We witnessed certain important combinatorial properties by investigating the structure of an HV-palindrome. We have an affirmative answer to the conjecture proposed in [3] for the maximum number of HV-palindromes in a word of size $(2,n)$ and a generalization for a word of size $(m,n),\;m\geq 2$. We also analyzed the least number of HV-palindromes in an infinite $2$D word. In future, it will be interesting to study and compare the properties of $1$D palindromes and HV-palindromes/$2$D palindromes.