Regularity of k-Abelian Equivalence Classes of Fixed Cardinality

Abstract

Two words u and v are said to be k-Abelian equivalent if, for each word x of length at most k, the number of occurrences of x as a factor of u is the same as for v. In this note we continue the analysis of k-Abelian equivalence classes. In particular, we show that, for any fixed integer $r\ge 1$, the language of words representing equivalence classes of cardinality r is regular.

1 Introduction

The k-Abelian equivalence, originally introduced in [9], is an equivalence relation in between equality and Abelian equivalence of words. It identifies words u and v which contain all words of length at most k equally many times as factors. For $k=1$ it coincides with the Abelian equivalence. Obviously, if two words u and v are k-Abelian equivalent, in symbols $u\sim _{k}v$, they are of equal length and, moreover, if they are so for each $k\ge 1$, then they are equal as words.

The k-Abelian equivalence defines in a natural way a complexity measure for languages, as well as for infinite words. Such a research was initiated in [12], and later continued, e.g., in [4]. Other topics of the k-Abelian equivalence such as k-Abelian repetitions, k-Abelian palindromicity and k-Abelian singletons were studied in [8, 10, 11, 14], respectively.

A characterization of the k-Abelian equivalence in terms of rewriting was obtained in [11]. This was based on the so-called k-switching lemma. This turned out to be quite an interesting approach. It allowed, see [2, 3], to show that the union of singleton classes is a regular language. Similarly, the set of all lexicographically least (or, equivalently, greatest) representatives constitutes a regular set. All these proofs are constructive, that is, given $k\in \mathbb {N}$, the above regular sets can be algorithmically found, in principle. In practice, this is not true since the size of the automaton grows at least exponentially in k. It follows that the sequence of the number of singletons of length n is a rational, or in fact, an $\mathbb {N}$-rational sequence. Similarly, the sequence of numbers of minimal elements in equivalence classes of words of length n is $\mathbb {N}$-rational, in other words, the number of equivalence classes of words of length n defines an $\mathbb {N}$-rational sequence. This latter sequence can be viewed as the complexity function of the k-Abelian equivalence. For a few small values of k and alphabet size m, the sequences are computed in [2], and are included in the On-Line Encyclopedia of Integer Sequences [17] as the sequences A289657 and A289658.

From the above it was concluded in [2] that also the union of two-element equivalence classes constitutes a regular set. This is based on the closure properties of regular sets and the above-mentioned three regular sets: the singletons, the set of minimal, and maximal elements in the equivalence classes. We recall this proof in Sect. 3. Interestingly, this approach does not extend to larger equivalence classes. The reasons for that are elaborated at the beginning of Sect. 3.

In [2], the main usage of the tool of the regular languages was to show that the number of equivalence classes of length n is asymptotic to a certain polynomial. However, by using the ideas presented in [2], we are able to prove our main theorem. For each $r\ge 1$, the union of the k-Abelian equivalence classes of cardinality r is a regular set. This is the main context of this note.

This note is arranged as follows. In Sect. 2 we lay down the basic terminology of combinatorics on words, automata theory, and graph theory needed in the remainder of the text. We also recall relevant results from the literature. In Sect. 3 we prove our main result. We also discuss an approach suggested in [2] and show that it fails. We then conclude with straightforward implications of the main result and further discussion in Sect. 4.

2 Preliminaries and Notation

We set some basic terminology and notions from the literature of combinatorics on words, for more on this topic we refer the reader to [13]. A finite set $\varSigma$ of symbols is called an alphabet. The set of finite sequences, or words, over $\varSigma$ is denoted by $\varSigma ^*$ and the set of non-empty words is denoted by $\varSigma ^+$. The empty word is denoted by $\varepsilon$. We let |w| denote the length of a word (as a sequence) $w\in \varSigma ^*$. By convention, we set $|\varepsilon | = 0$. The set of words of length n over the alphabet $\varSigma$ is denoted by $\varSigma ^n$.

We index the letters of a given word starting from 1. For a word $w = a_1a_2\cdots a_{n}\in \varSigma ^*$ and indices $1\leqslant i \leqslant j \leqslant n$, we call the word $a_i\cdots a_j$, denoted by w[i, j], a factor of w. For $i>j$ we set $w[i,j] = \varepsilon$. Similarly, for $i<j$ we let w[i, j) denote the factor $a_i\cdots a_{j-1}$, and we set $w[i,j) = \varepsilon$ when $i\geqslant j$. We let w[i..] (resp., w[..i]) denote the factor w[i, n] (resp., w[1, i]) for brevity. For any i, the factor w[..i] (resp., w[i..]) is called a prefix (resp., suffix) of w. We say that a word $x\in \varSigma ^*$ occurs at position i in w if the word w[i..] has x as a prefix. For $u\in \varSigma ^+$ we let $|w|_u$ denote the number of occurrences of u as a factor of w. The set of factors of a word w is denoted by F(w), and we set $F_n(w) = F(w)\cap \varSigma ^n$. We call u a complete return to x in w if $u\in F(w)$ such that $|u|_x = 2$ and x occurs as both a prefix and a suffix of u. The set of complete first returns to x in u is denoted by $\mathfrak {R}_w(x)$.

We also need a few basic properties of regular languages. Regular expressions over an alphabet $\varSigma$ are the finite expressions constructed recursively as follows. The symbol $\emptyset$, and each $a\in \varSigma \cup \{\varepsilon \}$ are expressions. If E and $E'$ are expressions then so are $E\,\cdot \,E'$, $E\,+\,E'$, and $E^*$. Each expression E defines a language, denoted by L(E) as follows: Each $a\in \varSigma \cup \{\varepsilon \}$ defines the singleton language $L(a)=\{a\}$ and $\emptyset$ defines the empty language. For expressions E and $E'$, the expressions $E\cdot E'$, $E+E'$ and $E^*$ define the languages $L(E)\cdot L(E')$, $L(E)\cup L(E')$, and $\bigcup _{n\ge 0}L(E)^n$, respectively.

A deterministic finite automaton (DFA) $\mathcal {A}$ over $\varSigma$ is a tuple $(Q,q_0,\delta ,F)$, where Q is a finite set of states, $q_0$ is the initial state, $\delta$ is a partial function $\delta :Q\times \varSigma \rightarrow Q$ called the transition function, and $F\subseteq Q$ is the set of final states. Given a word $w = a_1\cdots a_n$, the automaton operates on w using $\delta$ starting from $q_0$ by the rule $\delta (q,au) = \delta (\delta (q,a),u)$ for all $u\in \varSigma ^+$. If $\delta (q_0,w) \in F$ we say that $\mathcal {A}$ accepts w. We let $L(\mathcal {A})$ denote the language recognized by $\mathcal {A}$; $L(\mathcal {A}) = \{w\in \varSigma ^*\mid \mathcal {A}$ accepts $w\}$.

The languages defined by regular expressions (or recognized by finite automata) are exactly the regular languages, in fact, these two models are equivalent. Another equivalent model for regular languages considered here are nondeterministic finite automata (NFA), in which case the transition function may be multi-valued. We refer to [7] for this knowledge, and on more of equivalent models and closure properties of regular languages (closure under complementation, taking a morphic pre-image, etc.). In addition to classical language-theoretic notions, we use the theory of languages with multiplicities. This counts how many times a word occurs in a language. This leads to the theory of $\mathbb {N}$-rational sets. Using the terminology of [5, 16], a multiset over $\varSigma ^*$ is called $\mathbb {N}$-rational if it is obtained from finite multisets by applying finitely many times the rational operations product, union, and taking quasi-inverses, i.e., iteration restricted to $\varepsilon$-free languages. Equivalently, an $\mathbb {N}$-rational set equals the set of multiplicities of distinct ways an NFA accepts each word. Further, a unary $\mathbb {N}$-rational set is referred to as an $\mathbb {N}$-rational sequence. We refer to [5, 16] for more on this topic. For a language $L\subseteq \varSigma ^*$, the generating function $G_{L}(x)$ of L is defined as the formal power series $G_L(x) = \sum _{n\ge 0}\#(L\cap \varSigma ^n)x^n$. The basic result we need is (see [5, 16]):

Proposition 1

Let L be a regular language. The sequence $(\#(L\cap \varSigma ^n))_{n\ge 0}$, is an $\mathbb {N}$-rational sequence. Consequently, the generating function $G_L$ is a rational function.

When speaking of the generating function for a language $L\subseteq \varSigma ^*$, we mean the generating function for the function $\ell _L$ defined by $\ell _L(n) = \#(L\cap \varSigma ^n)$.

We now turn to the main notion of this paper, k-Abelian equivalence. We recall some results from the literature needed in the remainder of the paper.

Definition 1

The words $u,v\in \varSigma ^*$ are k-Abelian equivalent, $u\sim _{k}v$ in symbols, if $|u|_x = |v|_x$ for all $x\in \varSigma ^+$ with $|x|\leqslant k$.

The relation $\sim _{k}$ is clearly an equivalence relation; we let $[u]_k$ denote the k-Abelian equivalence class represented by u.

In [11], k-Abelian equivalence is characterized in terms of rewriting, namely by k-switching. For this we define the following. Let $k \geqslant 1$ and let $u \in \varSigma ^*$. Suppose that there exist $x,y\in \varSigma ^{k-1}$, not necessarily distinct, and indices i, j, l and m, with $1\le i<j \leqslant l<m\le n$, such that x has positions i and l in u, and y has positions j and m in u. In other words, we have

$$\begin{aligned} u = u{[1,i)} \cdot u{[i,j)} \cdot u{[j,l)} \cdot u{[l,m)} \cdot u{[m..]}, \end{aligned}$$

where the words u[i..] and u[l..] both begin with x, and the words u[j..] and u[m..] both begin with y. Furthermore, u[i, j), $u[l,m)\ne \varepsilon$, but we allow $l = j$, in which case $y = x$ and $u[j,l) = \varepsilon$. We define a k-switching on u, denoted by $S_{u,k}(i,j,l,m)$, as

$$\begin{aligned} S_{u,k}(i,j,l,m) = u[1,i) \cdot u[l,m) \cdot u[j,l) \cdot u[i,j) \cdot u[m..]. \end{aligned}$$

(1)

An illustration of a k-switching can be found in Fig. 1.

Fig. 1.

Illustration of a k-switching. Here $v=S_{k,u}(i,j,l,m)$; the white rectangles symbolize $x\in \varSigma ^{k-1}$ and the black rectangles symbolize $y\in \varSigma ^{k-1}$.

Let us define a relation $R_k$ of $\varSigma ^*$ by $u R_k v$ if and only if v is obtained from u by a k-switching. Now $R_k$ is clearly symmetric, so that the reflexive and transitive closure $R_k^*$ of $R_k$ is an equivalence relation on $\varSigma ^*$. This gives quite a different characterization of the relation $\sim _{k}$ in terms of rewriting, see [11] for more details:

Proposition 2

For $u,v\in \varSigma ^*$, we have $u\sim _k v$ if and only if $u R_k^* v$.

In order to recall the following result, we need some notation. Let denote a total order on $\varSigma$ and the corresponding lexicographic order on $\varSigma ^*$. We define the language operation by . We shall often omit from the prefix to avoid cluttering the text, as this does not usually concern us. Whenever is omitted, the reader should consider some fixed (but arbitrary) total order .

The characterization in Proposition 2 allows to conclude, see [2]:

Proposition 3

The language is regular for any lexicographic order . In other words, the language of lexicographically least representatives with respect to of the k-Abelian equivalence classes over $\varSigma$ is regular.

Example 1

In [2], minimal DFAs are constructed for the languages $\text {Min}_{k}(\varSigma ^*)$ for small values of k and small alphabet $\varSigma$. The motivation for this is to compute explicit formulae for the number of k-Abelian equivalence classes for a given length. We recall the number of states in the constructed automata. In the case of a binary alphabet, we have minimal DFAs with 10, 49, and 936 states (including the cases for $k=2$, 3, and 4, respectively). For the ternary alphabet and $k=2$, the minimal DFA has 66 states. The automata grow quite fast with respect to k, so in practice such automata seem to be intractable to compute.

Another result, more in the spirit of classical formal language theory, is shown in [2] (for a detailed proof, see [3]):

Proposition 4

Regular languages are closed under the language operation $R_k$ defined by $R_k(L) = \{v\in \varSigma ^*\mid \exists u \in L:uR_k v\}$.

Note that we overload the symbol $R_k$; we use $R_k$ either as a relation or as a many-valued function, and it will always be clear from context.

We need some terminology and notation of directed graphs with loops and multiple labeled edges. For a graph $G=(V,E)$, we let V(G) denote the set of vertices, and $E(G)\subseteq V\times V$ the set of edges of G. For an edge $e \in E(G)$ from x to y, we call the vertex x the tail of e, denoted by $\mathop {\mathrm {tail}}(e)$, and y the head of e, denoted by $\mathop {\mathrm {head}}(e)$. The number of edges from x to y is denoted by $m_G(x,y)$. A sequence $W=(e_i)_{i=1}^t$ of edges satisfying $\mathop {\mathrm {head}}(e_i)=\mathop {\mathrm {tail}}(e_{i+1})$ for each $i\in [1,t-1]$ is called a walk (in G). We set $\mathop {\mathrm {tail}}(W)=\mathop {\mathrm {tail}}(e_1)$ and $\mathop {\mathrm {head}}(W)=\mathop {\mathrm {head}}(e_t)$. Further, we let |W| denote the length t of W. We call W a path if $\mathop {\mathrm {tail}}(e_i)\ne \mathop {\mathrm {tail}}(e_j)$ when $i\ne j$, and $\mathop {\mathrm {head}}(e_t)\ne \mathop {\mathrm {tail}}(e_i)$ for all $i\in [1,t]$. In other words, a path P does not visit any vertex twice. If the walk $(e_i)_{i=1}^{t-1}$ is a path and $e_0$ is an edge such that $\mathop {\mathrm {tail}}(e_0) = \mathop {\mathrm {head}}(e_{t-1})$ and $\mathop {\mathrm {head}}(e_0) = \mathop {\mathrm {tail}}(e_1)$, then we call the walk $(e_i)_{i=0}^{t-1}$ a cycle. We index the edges of a cycle starting from 0 for notational reasons. We consider also loops as cycles. Finally, W is called an Eulerian walk if W traverses each edge of G exactly once.

The concatenation $W\cdot W'$ of walks W and $W'$ satisfying $\mathop {\mathrm {head}}(W)=\mathop {\mathrm {tail}}(W')$ is defined in a natural way. For an empty walk W, we define $W\cdot W' = W'\cdot W = W'$. Note here that a cycle C can be concatenated with itself arbitrarily many times. We say that a walk W is a repetition of a cycle if we may write $W = C^r$ (C concatenated r times) for some $r\ge 1$.

In this note we make use of de Bruijn graphs (see [1] and references therein) defined as follows. For any $k\geqslant 1$ and alphabet $\varSigma$, the de Bruijn graph $dB_{\varSigma }(k)$ of order k over $\varSigma$ is defined as a directed graph for which the set of vertices equals $\varSigma ^{k}$. For each word $z\in \varSigma ^{k+1}$ we have an edge $(z[..k],z[2..]))\in dB(k)$. In other words, we have $(x,y)\in E(dB_{\varSigma }(k))$ if and only if there exists a letter $a\in \varSigma$ such that the word $xa\in \varSigma ^{k+1}$ ends with y. In this case (x, y) is denoted by (x, a), a being the label $\text {label}((x,y))$ of the edge. We shall often omit $\varSigma$ from the subscript, as it is usually clear from the context. For a walk $W=(e_i)_{i=1}^t$ in dB(k), we set $\text {label}{W}=\text {label}(e_1)\cdots \text {label}(e_t)$.

We note that any word $u=a_1\cdots a_n$, where $n\ge k$ and $a_i\in \varSigma$ for each $i\in [1,n]$, defines the walk $W_u = (e_i)_{i=1}^{n-k}$ in dB(k). (Here $e_i = (u[i,i+k),a_{i+k})$, $i\in [1,n-k]$.) Conversely, any walk $W_u = ((x_i,a_i))_{i=1}^{t}$ in dB(k) defines the word $\mathop {\mathrm {tail}}(W_0)\cdot \text {label}(W) = x_1\cdot a_1a_2 \cdots a_t\in \varSigma ^{k+t}$. Thus a (long enough) word $u\in \varSigma ^*$ should be considered as a walk in dB(k) and vice versa.

The authors of [12] observed a connection between k-Abelian equivalence and Eulerian paths in multigraph versions of de Bruijn graphs and we overview it here. Let $f \in \mathbb {N}^{\varSigma ^k}$ be an arbitrary vector. We define $G_f = (V,E)$ as follows. We set V as the set of words $x\in \varSigma ^{k-1}$ such that x is a prefix or a suffix of a word $z\in \varSigma ^k$ for which $f[z]>0$. For each $z\in \varSigma ^k$ with $f[z]>0$, we have the edge $(z[..k-1],z[2..])$ with multiplicity f[z]. Now $G_f$ is a subgraph of $dB(k-1)$ equipped with weights on edges. Note that, if $f[z]=|w|_z$ for all $z\in \varSigma ^*$ for some $w\in \varSigma ^*$ (this being the case we set $f=f_w$), the graph $G_{f}$ is the Rauzy graph of w of order $k-1$ (see [15]) equipped with weights on edges.

In the following, for $u,v \in \varSigma ^{k-1}$, we let $\varSigma (u,v) = u \varSigma ^*\, \cap \, \varSigma ^*v$ and $\varSigma (u,v,n) = \varSigma (u,v)\cap \varSigma ^n$.

Lemma 1

(Karhumäki et al. [12, Lemma 2.12]). For a vector $f \in \mathbb {N}^{\varSigma ^k}$ and words $u,v \in \varSigma ^{k-1}$, the following are equivalent:

• 1. There exists a word $w\in \varSigma (u,v)$ such that $f = f_w$;

• 2. $G_f$ has an Eulerian path starting from u and ending at v;

• 3. The underlying graph of $G_f$ is connected, and $d^-(s)=d^+(s)$ for every vertex s, except that if $u\ne v$, then $d^-(u) = d^+(u)-1$ and $d^-(v)=d^+(v)+1$.

The following corollary is immediate, as noted in [11].

Corollary 1

For a word $w\in \varSigma (u,v)$ and $k\ge 1$, we have that $w' \sim _k w$ if and only if the walk $W_{w'}$ is an Eulerian path from u to v in $G_{w}$.

Example 2

Let $u\in \varSigma ^*$ and $x\in F_{k-1}(u)$ such that $|u|_x\ge 3$. We may then write $W_u = W_1W_2W_3W_4$ for some walks $W_i$ with $\mathop {\mathrm {head}}(W_i)=x=\mathop {\mathrm {tail}}(W_{i+1})$ for each $i=1,\ldots ,3$. Then, by the above corollary, we have $u\sim _{k}v$, where v is defined by the walk $W_v = W_1W_3W_2W_4$. Indeed, $W_v$ is well-defined due to the choice of the extremal vertices of the walks $W_i$ and the same edges are traversed equally many times as in $W_u$.

Continuing this line of thought, a formula for computing the size of a k-Abelian equivalence class represented by a given word is obtained in [11]. In the following, a rooted spanning tree with root v of a graph G is a spanning tree of G for which all edges are directed towards the root vertex v.

Proposition 5

Let $k\ge 1$ and $w\in \varSigma (u,v)$ for some $u,v\in \varSigma ^{k-1}$. Then

$$\begin{aligned} \#[w]_k = \kappa _v \prod _{x\in V(G_w)} \frac{(|w|_x-1)!}{\prod _{a\in \varSigma }|w|_{xa}!}, \end{aligned}$$

(2)

where $\kappa _v$ is the number of rooted spanning trees with root v in $G_{w}$.

3 The Regularity of Classes of Constant Cardinality

Our main goal is to analyze the language $L_{r,k,\varSigma }$ of words w over $\varSigma$ satisfying $\#[w]_k = r$, or more formally, $L_{r,k,\varSigma }=\{w\in \varSigma ^*\mid \#[w]_k = r\}$. We shall often omit $\varSigma$ from the subscript when there is no danger of confusion.

The language $L_{1,k}$ consists of words representing singleton classes. They are thus uniquely defined by the frequencies of the factors of length k together with the suffix of length $k-1$. The number of such words of length n is considered extensively in [11]. In [2], $L_{1,k}$ is shown to be regular. Indeed, the complement of $L_{1,k,\varSigma }$ is defined by the regular expression

(3)

We remark that a crude upper bound on the number of states in the minimal DFA recognizing $L_{1,k}$ can be obtained from the above regular expression using well-known conversions between various models of regular languages (see, [6, 7]). Indeed, e.g., Glushkov’s algorithm outputs an equivalent NFA of $n+1$ states, given a regular expression of n occurrences of alphabet symbols. The determinization of an n-state NFA can, in the worst case, give a DFA with $2^n$ states. Observe that the minimal DFA of a language and its complement have the same number of states.

The first few exact values for $\varSigma =\{a,b\}$ are as follows. The minimal DFAs recognizing $L_{1,k}$ for $\varSigma = \{a,b\}$ contain 15, 87, and 1011 states for $k=2$, 3, and 4, respectively. These values include the garbage state. The automata for $k=2$ and 3 are presented in [2] (garbage state omitted), and we propose the value for $k=4$ without proof. For a ternary $\varSigma$, we have 84 states for $k=2$. We recall the minimal DFA of $L_{1,2,\{a,b\}}$ in Fig. 2. For more on this automaton and for automata for other values of k and alphabets, we refer the reader to [2].

Fig. 2.

The minimal DFA recognizing $L_{1,2,\{a,b\}}$. The garbage state is not illustrated. All other states are accepting.

Not only the languages $L_{1,k}$ are regular, but so are the languages $L_{2,k}$ as shown in [2]. The proof goes as follows. Recall that is regular for any lexicographic order $\lhd$. Let $\lhd ^R$ be the reversal of $\lhd$, that is, $b\lhd ^R a$ if and only if $a\lhd b$. After a brief consideration, it becomes clear that

is regular since all the language operations, including $R_k$, preserve regularity. It follows that is regular since $L_{1,k,\varSigma }$ is regular.

The main result of this paper is the generalization of the above to all $r\in \mathbb {N}$:

Theorem 1

For any $k,r\ge 1$ and alphabet $\varSigma$, the language $L_{r,k}$ is regular.

The approach of removing (in a regular way) one element of each class at a time does not seem to extend to the languages $L_{r,k}$, $r\ge 3$. The approach of proving the above theorem as suggested in [2, end of Sect. 6], fails as we will shortly show. The method described is of similar flavour as in the proof of the regularity of $L_{2,k,\varSigma }$, namely, by setting $K_{i+1} = K_i\setminus \text {Min}_{k}(K_i)$, $K_0=\varSigma ^*$, we obtain that $\varSigma ^*\setminus R_k^r(K_r) = \cup _{i\le r}L_{r,k}$ for each $r\in \mathbb {N}$. If $\text {Min}_{k}$ preserved regularity, the above theorem would follow since a finite sequence of regularity-preserving operations would be used.

The following example shows that, unfortunately, this approach does not work, as $\text {Min}_{k}$ does not preserve regularity.

Example 3

Let $k\ge 1$ and $L = (ab^k)^* \cup ab^{k-1}b^*(ab^{k-1})^*$. It is straightforward to check, e.g., using Corollary 1, that

$$\begin{aligned} \text {Min}_{k}(L) = (ab^k)^* \cup \{ab^{k-1}b^r(ab^{k-1})^s\mid r\ne s+1\}. \end{aligned}$$

It follows that $L\setminus \text {Min}_{k}(L) = \{(ab^{k-1})b^{r+1}(ab^{k-1})^r\mid r\ge 1\}$. The language $h^{-1}(L\setminus \text {Min}_{k}(L))$, where $a\mapsto ab^{k-1}$, $b\mapsto b$, equals $\{abb^ra^r \mid r\ge 1\}$ which is not regular. Since all other operations preserve regularity, we conclude that $\text {Min}_{k}$ does not preserve regularity.

As a conclusion, to prove Theorem 1 we need a new approach. We do this via a characterization of the lexicographically least representatives of k-Abelian equivalence classes given in [2].

3.1 The Proof of Theorem 1

Before turning to the formal proof, we sketch the main ingredients. Our first observation (Lemma 3) states that there exists a constant $\mathcal {B}_{k,r}$ (depending on k and r) such that, for any factor x having at least two distinct returns in u, the total number of occurrences of x in $u\in L_{r,k}$ is bounded by $\mathcal {B}_{k,r}$. We then turn to the language of minimal representatives and show that $\text {Min}_{k}(L_{r,k})$ is regular (Theorem 2). For the proof, we make an observation concerning factors x occurring more than $\mathcal {B}_{k,r}$ times in $u\in \text {Min}_{k}(L_{r,k})$ (Lemma 4). The main implication is that corresponding to u and a factor x occurring more than $\mathcal {B}_{k,r}$ times, we may construct a regular language $L_x = zy^*z'$ such that $u\in L_x \subseteq \text {Min}_{k}(L_{r,k})$. In the proof of Theorem 2, this observation is applied to all such factors x to obtain, for each $u\in \text {Min}_{k}(L_{r,k})$, a regular expression $L_u = z_0y_1^*z_1\cdots y_t^*z_t\subseteq \text {Min}_{k}(L_{r,k})$. This implies that $\text {Min}_{k}(L_{r,k})$ is a (possibly infinite) union of regular expressions. To conclude the proof we show that there are actually only finitely many distinct regular expressions in the union with the help of Lemma 5. Finally, Theorem 1 follows from Theorem 2 by applying the regularity-preserving language operation $R_k$ on $\text {Min}_{k}(L_{r,k})$ finitely many times.

We express the above lemmas via de Bruijn graphs and walks within. To this end we first recall some terminology from [2]. We say that a cycle $C=(d_j)_{j=0}^{s-1}$ occurs along the walk W if W can be written as the concatenation $W = W_1\cdot (d_{r+j \pmod s})_{j=0}^{s-1} \cdot W_2$ for some $r\in [0,s-1]$ and some (possibly empty) walks $W_1$, $W_2$. We say that W enters C via the vertex $\mathop {\mathrm {tail}}(d_r)$ if $W_1$ is either empty or $d_{r-1 \pmod s}$ is not the last edge of $W_1$. In this case we say that W enters C at position $|W_1|+1$. We say that W leaves C via the vertex $\mathop {\mathrm {head}}(d_{r+s-1 \pmod s})$ if $W_2$ is empty or $d_r$ is not the first edge of $W_2$. In this case we say that W leaves C at position $|W_1C|$.

Example 4

Consider the de Bruijn graph $dB_{\{a,b\}}(2)$. The walk $W_u=(e_i)_{i=1}^{10}$, defined by the word $u=aaaabaabaaba$, has two distinct cycles occurring along it, namely the loop $C_1 = (aa,a)$ and the cycle $C_2=((aa,b),(ab,a),(ba,a))$. The walk $W_u$ enters $C_1$ at position 1 and leaves $C_1$ at position 3 (both via the vertex aa), and does not enter $C_1$ later on. Further, $W_u$ enters the cycle $C_2$ at position 2 (via the vertex aa) and W leaves the cycle at position 10 (via the vertex ba). We may write $W = C_1^2\cdot C_2^2 \cdot ((aa,b),(ab,a))$.

On the other hand, the walk $W_{uaa}$ in dB(2) defined by the word uaa is not cycle-deterministic, as we may write $W_{uaa} = W_u\cdot ((ba,a),(aa,a)) = C_1^2\,\cdot \,C_2^3\,\cdot \, C_1$, whence $W_{uaa}$ enters the cycle $C_1$ at positions 1 and 12. The cycle $C_2$ is now left at position 11.

We recall a lemma we need in future considerations.

Lemma 2

(Cassaigne et al. [2, Lemma 5.17]). Let $W_u$ be a walk in $dB(k-1)$ defined by $u\in \text {Min}_{k}(\varSigma ^*)$. Let $C=(d_j)_{j=0}^{s-1}$ be a cycle occurring along $W_u$. Suppose further that we may write $W_u = W_0\cdot C^r \cdot W_1$ for some walks $W_1$, $W_2$, $r\ge 2$. Then, for any $t\geqslant 0$, the word $u_t$ corresponding to the walk $W_0 \cdot C^{t} \cdot W_1$ is in $\text {Min}_{k}(\varSigma ^*)$.

We first need a couple of observations about the properties of lexicographically least representatives of equivalence classes.

Lemma 3

Let $k,r\ge 1$. There exists an integer $\mathcal {B}_{k,r}$ such that, for each $u\in L_{r,k}$, if $\#\mathfrak {R}_u(x)\ge 2$ for some $x\in F_{k-1}(u)$, then $|u|_x\le \mathcal {B}_{k,r}$.

Proof

Let $u\in L_{r,k}$ and assume $\#\mathfrak {R}_u(x)\ge 2$ for some $x\in F_{k-1}(u)$. Let us write $W_u$ in terms of complete first returns of x in u;

$$\begin{aligned} W_u = W_0W_1\cdots W_{|u|_x-1}W_{|u|_x}, \end{aligned}$$

where $\mathop {\mathrm {tail}}(W_i)=\mathop {\mathrm {head}}(W_i)=x$ for all $i=1,\ldots ,|u|_x-1$. Observe now that each walk $W_i$, $i=1,\ldots ,|u|_x-1$, corresponds to the complete first return to x in u. Furthermore, $W_0$ and $W_{|u|_x}$ do not contain the vertex x anywhere else other than what is implied above. Now, for any permutation $\sigma$ of $[1,|u|_x)$, we have that $u\sim _{k}v_{\sigma }$, where $v_{\sigma }$ is defined by the walk $W_{\sigma } = W_0W_{\sigma (1)}\cdots W_{\sigma (|u|_x-1)}W_{|u|_x}$ (compare to Example 2). The number of distinct words obtained by this method is $\left( {\begin{array}{c}|u|_x-1\\ m_1,\ldots ,m_k\end{array}}\right) =\frac{(|u|_x-1)!}{m_1!\cdots m_k!}$, where $k=\#\mathfrak {R}_x(u)$ and $(m_i)_i = (|u|_y)_{y\in \mathfrak {R}_u(x)}$. This is the number of distinct permutations of words $y\in \mathfrak {R}_u(x)$ with multiplicities $|u|_y$, $y\in \mathfrak {R}(x)$. (To see that two words obtained from distinct permutations are distinct, consider their prefixes and recall the definition of a complete first return word.) We now have, by assumption, $k\ge 2$ whence $\left( {\begin{array}{c}|u|_x-1\\ m_1,\ldots ,m_k\end{array}}\right) \ge |u|_x-1$. This implies $r = \#[u]_k\ge |u|_x-1$, or in other words, $|u|_x\le r+1$.    $\square$

Remark 1

For $r\ge 2$, we have $\mathcal {B}_{k,r}\ge 2$. Indeed, we have $u = a^{k+r-2}ba^{k-1}\in L_{r,k}$ and $\#\mathfrak {R}_u(a^{k-1}) = 2$.

In the case of $r=1$, similar ideas were considered in [11]. Indeed, there it is shown that, for any $u\in L_{1,k,\varSigma }$, we have $\#\mathfrak {R}_u(x)\le 1$ for all $x\in \varSigma ^{k-1}$. In fact a characterization of words in $L_{1,k,\varSigma }$ is obtained in terms of a slight generalization of our notion of return words.

Lemma 4

Let $u \in \text {Min}_{k}(L_{r,k})$ and assume $|u|_x>\mathcal {B}_{k,r}$. Then we may write $W_u = W \cdot C^{|u|_x-1}\cdot W'$ for some cycle C with $\mathop {\mathrm {tail}}(C)=x$.

Proof

Assume that $|u|_x>\mathcal {B}_{k,r}$ for some $x\in \varSigma ^{k-1}$. Since $|u|_x > \mathcal {B}_{k,r}$, we have $\mathfrak {R}_{u}(x)=1$ by the above lemma. We may write $W_u$ in terms of y, the unique complete first return to x in u; $W_u = W\cdot W_y^{|u|_x-1}\cdot W'$, where $\mathop {\mathrm {head}}(W)=\mathop {\mathrm {tail}}(W_y)=\mathop {\mathrm {head}}(W_y)=\mathop {\mathrm {tail}}(W')=x$. We claim that $W_y$ is a cycle. Assume the converse; there then exists a vertex z in $W_y$ such that there are two distinct edges both with tail z along $W_u$. It now follows that $\#\mathfrak {R}_u(z)\ge 2$ and $|u|_z > \mathcal {B}_{k,r}$, a contradiction.    $\square$

We now prove another lemma which already hints towards regular properties of our language.

Lemma 5

Let $r\ge 2$. Let then $u_s$, for each $s\ge 0$, denote the word defined by the walk $W(s) = W\,\cdot \,C^s\,\cdot \,W'$ (in $dB(k-1)$) for some cycle C. Then $u_s\in \text {Min}_{k}(L_{r,k})$ for some $s\ge \mathcal {B}_{k,r}$ if and only if $u_s\in \text {Min}_{k}(L_{r,k})$ for all $s\in \mathbb {N}$.

Proof

The other implication is immediate, so assume $u_s\in \text {Min}_{k}(L_{r,k})$ for some $s\ge \mathcal {B}_{k,r}$. The fact that $u_s\in \text {Min}_{k}(\varSigma ^*)$ for all $s\in \mathbb {N}$ follows by Lemma 2, so it is enough to show that $u_s\in L_{r,k}$ for all $s\in \mathbb {N}$ (recall that for $r\ge 2$, we have $\mathcal {B}_{k,r}\ge 2$ by Remark 1). Without loss of generality, we may assume that W(s) enters $C=(e_i)_{i=0}^{l-1}$ ($|C|=l\ge 1$) at position $|W|+1$ and that s is maximal, i.e., W(s) leaves C before position $|W|+(s+1)|C|$ and, further, that W(s) leaves C via vertex $y = \mathop {\mathrm {tail}}(e_o)$, $0\le o \le l-1$.

Observe now that $|u_s|_x \ge s$ for all $x\in V(C)$. It follows by the above lemma that for each vertex $x \in V(C)$ we have $\#\mathfrak {R}_{u_s}(x) = 1$ (if there were another complete first return to x in u for some $x\in V(C)$, we would have $|u_s|_x>\mathcal {B}_{k,r}$, a contradiction). Consequently, by the maximality of s, $|u_s|_{\mathop {\mathrm {tail}}(e_i)}= s+1$ for all $i\in [0,o]$, and $|u_s|_{\mathop {\mathrm {tail}}(e_i)}=s$ for all $i=(o,l)$. Further, each (except possibly the last) occurrence of x is followed by the same letter $a_x$ in $u_s$. Moreover, the only vertex $y\in V(C)$ followed by a letter $b\ne a_y$ in u is the vertex $y=\mathop {\mathrm {tail}}(e_o)$ via which W(s) leaves C (the only exception is that $W'$ is a subpath of C).

Consider the graph $G_{s}=G_{u_s}$ in light of Proposition 5. Let $\kappa _s$ denote $\kappa _{\mathop {\mathrm {head}}(W(s))}$ for each $s\ge 0$. By the above observations we conclude that any rooted spanning tree with root $\mathop {\mathrm {head}}(W(s))$ of $G_{s}$ contains one of the (multiple copies of the) edge $e_i$ for each $i=[0,l)\setminus \{o\}$, and the edge $(y,b)\notin E(C)$ (unless $\mathop {\mathrm {head}}(W(s))=y$ whence no edge from y exists in such a tree). Let us compute $\kappa _s$ in terms of $\kappa _0$ and s. Adding s copies to an edge $e_i$, $0\le i < o$, to $G_0$ increases the number of trees $(s+1)$-fold, as each tree must contain exactly one copy of this edge and there are $s+1$ to choose from. For the remainder of the vertices $z\in V(C)\setminus y$, any tree in $G_{s}$ must contain some copy of the path $(e_j)_{j=o+1}^{l-1}$ which connects to a copy of a tree defined by $G_0$. Given s copies of each edge along this path, there are altogether $s^{l-o-1}$ choices for the path. We conclude that $\kappa _s = \kappa _0 \cdot (s+1)^{o}s^{l-o-1}$. This may be expressed as $\kappa _s = \kappa _0\cdot \prod _{x\in V(C)\setminus y}(s+|u_0|_x)$. Further, we observe that, in the product $\prod _{x\in F_{k-1}(u_s)}\frac{(|u_s|_x-1)!}{\prod _{a\in \varSigma }|u_s|_{|xa|}!}$, the only values that vary according to s are certain values corresponding to the vertices of C. In particular, $|u_s|_x=|u_s|_{xa_x} = s+ |u_s|_{x_0}\in \{s,s+1\}$ for each $x\in V(C)\setminus y$, $|u_s|_y = s+1$, and $|u_s|_{ya_y}=s$. Recall that $|u_s|_{yb}=0$ or 1 depending on whether $\mathop {\mathrm {head}}(W(s))=y$ or not. Plugging these values in (2), we find that the following ratio equals 1 for any $s\ge 1$:

$$\begin{aligned} \frac{\#[u_s]_k}{\#[u_0]_k} = \prod _{x\in V(C)\setminus y}(s+|u_0|_x)\cdot \prod _{x\in V(C)\setminus y}\frac{1}{(s+|u_0|_x)} = 1. \end{aligned}$$

Thus, for any choice of s, the obtained word $u_s$ has $\#[u_s]=r$. The claim follows.    $\square$

We are now in the position to prove the key result, from which our main result follows.

Theorem 2

The language $\text {Min}_{k}(L_{r,k})$ is regular.

Proof

We claim that $\text {Min}_{k}(L_{r,k})$ is a finite union of languages defined by regular expressions of the form $z_0y_1^*z_1\cdots y_t^*z_t$.

Consider now a word $u \in \text {Min}_{k}(L_{r,k})$ and write

$$\begin{aligned} W_u = W_0 C_1^{s_1} W_1\cdots C_t^{s_t}W_t \end{aligned}$$

for some paths $W_i$, $i=0,\ldots ,t$, and some repetitions $C_i^{s_i}$ of cycles $C_i$, $i=1,\ldots ,t$, such that $W_u$ enters cycle $C_i$ at position $|W_0|+\sum _{j=1}^{i-1}|C_{j}^{s_{j}}W_j|+1$ for all $1\le i \le t$ and leaves $C_i$ before entering $C_{i+1}$. Now u may be written as

$$\begin{aligned} u = \mathop {\mathrm {tail}}(W_0)\cdot \text {label}(W_0C_1^{s_1} W_1\cdots C_t^{s_t}W_t) = z_0y_1^{s_1}z_1\cdots y_t^{s_t}z_t, \end{aligned}$$

where $z_0 = \mathop {\mathrm {tail}}(W_0)\cdot \text {label}(W_0)$, $y_i = \text {label}(C_i)$, and $z_i=\text {label}(W_i)$ for $1\le i\le t$. The above lemma asserts that, if $s_i\ge \mathcal {B}_{k,r}$, then

$$\begin{aligned} L(z_0y_1^{s_1}z_1\cdots y_i^*z_i\cdots y_t^{s_t}z_t) \subseteq L_{r,k}. \end{aligned}$$

(4)

By repeating the above, we may replace all exponents $s_j$ satisfying $s_j \ge \mathcal {B}_{k,r}$ with $*$ in (4).

Let L be the union of all the languages obtained as above from words $u\in \text {Min}_{k}(L_{r,k})$ satisfying $|u|_x \le \mathcal {B}_{k,r}+1$ for all $x\in \varSigma ^{k-1}$. These words are bounded in length, so that the union is finite. Clearly $L\subseteq \text {Min}_{k}(L_{r,k})$ by the above observation. We claim that $\text {Min}_{k}(L_{r,k})\subseteq L$.

Indeed, let $u\in \text {Min}_{k}(L_{r,k})$. If $|u|_x > B_{k,r}+1$, Lemma 4 ensures that we have $W_u= W_0W_y^{|u|_x-1}W_1$ for y being the unique complete first return to x in u. Further $W_y$ is a cycle. If $W_u$ does not enter $W_y$ at position $|W_0|+1$, we may extend the cycle to the left and right to obtain $W_0'W_{y'}^{t}W_1'$, where $t\in \{|u|_x-1,|u|_x\}$. By the above lemma we may reduce the number of repetitions of $W_y'$ to obtain a word $u'$ for which $|u'|_x\le \mathcal {B}_{k,r}+1$ and u is in the language defined by $u'$ as in (4). If $|u'|_{x'}> \mathcal {B}_{k,r}+1$ for some $x'\in \varSigma ^{k-1}$, we may repeat the above for $u'$ to obtain a word $u''$ having $|u''|_{x'}\le \mathcal {B}_{k,r}+1$ and such that u and $u'$ are in the language defined by $u''$ as in (4). This can be continued until we obtain a word v such that $|v|_x\le \mathcal {B}_{k,r}+1$ for all $x\in \varSigma ^{k-1}$ and u is contained in the language defined by v as in (4). We thus have $u\in L$, which concludes the proof.    $\square$

Proof

(of Theorem 1). The language $\text {Min}_{k}(L_{r,k})$ is regular. Since the operation $R_k$ preserves regularity by Proposition 4, by applying finitely many iterations of $R_k$, we have that $L_{r,k} = R_k^{r}(\text {Min}_{k}(L_{r,k}))$ is regular.    $\square$

4 Conclusions

In this note, we continued to analyze the structure of the k-Abelian equivalence classes, in particular in the framework of regularity. In [2] we concluded, as a consequence of the k-switching lemma of [11], that the set of singleton classes is a regular language. Therein, this was extended to the union of all two-element classes as well. This was based, on one hand, on the regularity of the union of lexicographically least representatives of the equivalence classes and, on the other hand, on strong closure properties of the family of regular languages.

The approach does not extend, at least immediately, and indeed fails for the first attempt, to the union of larger (fixed-size) classes. To show that also these classes are regular, we developed new techniques. This is the content of this note.

All these regular languages are algorithmically constructable. However, in practice, this can be done only in very restricted cases. Indeed, the analysis of the size of the automata of the mentioned regular languages remains for future research.

Once the regularity of the above languages is established, the well-known techniques in rational power series allow to determine the corresponding enumeration functions. This was discussed in [2]. In the current work, Theorem 2 implies the following result:

Corollary 2

For each $k\ge 1$, the sequence of numbers of k-Abelian equivalence classes with cardinality r of length n is a rational sequence.

In this spirit, the first few sequences of minimal elements were shown in the On-Line Encyclopedia of Integer Sequences [17]. In more detail, the sequences

$$\begin{aligned} \mathcal {P}_{k,m}(n) = \#\{[w]_k \mid |w|=n\} \end{aligned}$$

were determined for $k=2,3$ in the binary alphabet. Similarly, a first few sequences considered in this note, that is, of the sequences

$$\begin{aligned} \mathcal {S}_{r,k,m}(n) = \#\{[w]_k\mid |w|=n, \#[w]_k = r\} \end{aligned}$$

might be worth including in the encyclopedia.