Abstract
Prefix normal words are binary words in which each prefix has at least the same number of 
s as any factor of the same length. Firstly introduced in 2011, the problem of determining the index (amount of equivalence classes for a given word length) of the prefix normal equivalence relation is still open. In this paper, we investigate two aspects of the problem, namely prefix normal palindromes and so-called collapsing words (extending the notion of critical words). We prove characterizations for both the palindromes and the collapsing words and show their connection. Based on this, we show that still open problems regarding prefix normal words can be split into certain subproblems.
Introduction
Two words are called abelian equivalent if the amount of each letter is identical in both words, e.g. rotor and torro are abelian equivalent albeit banana and ananas are not. Abelian equivalence has been studied with various generalisations and specifications such as abelian-complexity, k-abelian equivalence, avoidability of (k-)abelian powers and much more (cf. e.g., [6, 10, 11, 13, 17, 22–24]). The number of occurrences of each letter is captured in the Parikh vector (also known as Parikh image or Parikh mapping) [21]: given a lexicographical order on the alphabet, the 
component of this vector is the amount of the 
letter of the alphabet in a given word. Parikh vectors have been studied in [12, 16, 19] and are generalised to Parikh matrices for saving more information about the word than just the amount of letters (cf. eg., [20, 25]).
A recent generalisation of abelian equivalence, for words over the binary alphabet 
, is prefix normal equivalence (pn-equivalence) [14]. Two binary words are pn-equivalent if their maximal numbers of 
s in any factor of length n are equal for all 
. Burcsi et al. [5] showed that this relation is indeed an equivalence relation and moreover that each class contains exactly one uniquely determined representative - called a prefix normal word. A word w is said to be prefix normal if the prefix of w of any length has at least the number of 
s as any of w’s factors of the same length. For instance, the word 
is prefix normal but 
is not, witnessed by the fact that 
is a factor but not a prefix. Both words are pn-equivalent. In addition to being representatives of the pne-classes, prefix normal words are also of interest since they are connected to Lyndon words, in the sense that every prefix normal word is a pre-necklace [14]. Furthermore, as shown in [14], the indexed jumbled pattern matching problem (see e.g. [2, 4, 18]) is connected to prefix normal forms: if the prefix normal forms are given, the indexed jumbled pattern matching problem can be solved in linear time 
of the word length n. The best known algorithm for this problem has a run-time of 
(see [7]). Consequently there is also an interest in prefix normal forms from an algorithmic point of view. An algorithm for the computation of all prefix normal words of length n in run-time 
per word is given in [8]. Balister and Gerke [1] showed that the number of prefix normal words of length n is 
and the class of a given prefix normal word contains at most 
elements. A closed formula for the number of prefix normal words is still unknown. In “OEIS” [15] the number of prefix normal words of length n (A194850), a list of binary prefix normal words (A238109), and the maximum size of a class of binary words of length n having the same prefix normal form (A238110), can be found. An extension to infinite words is presented in [9].
Our Contribution. In this work we investigate two conspicuities mentioned in [3, 14]: palindromes and extension-critical words. Generalising the result of [3] we prove that prefix normal palindromes (pnPal) play a special role since they are not pn-equivalent to any other word. Since not all palindromes are prefix normal, as witnessed by 
, determining the number of pnPals is an (unsolved) sub-problem. We show that solving this sub-problem brings us closer to determining the index, i.e. number of equivalence classes w.r.t. a given word length, of the pn-equivalence relation. Moreover we give a characterisation based on the maximum-ones function for pnPals. The notion of extension-critical words is based on an iterative approach: compute the prefix normal words of length 
based on the prefix normal words of length n. A prefix normal word w is called extension-critical if 
is not prefix normal. For instance, the word 
is prefix normal but 
is not and thus 
is called extension-critical. This means that all non-extension-critical words contribute to the class of prefix normal words of the next word-length. We investigate the set of extension-critical words by introducing an equivalence relation collapse, grouping all extensional-critical words that are pn-equivalent w.r.t. length 
. Finally we prove that (prefix normal) palindromes and the collapsing relation (extensional-critical words) are related. In contrast to [14] we work with suffix-normal words (least representatives) instead of prefix-normal words. It follows from Lemma 1 that both notions lead to the same results.
Structure of the Paper. In Sect. 2, the basic definitions and notions are presented. In Sect. 3, we present the results on pnPals. Finally, in Sect. 4, the iterative approach based on collapsing words is shown. This includes a lower bound and an upper bound for the number of prefix normal words, based on pnPals and the collapsing relation. Due to space restrictions all proofs are in the appendix.
Preliminaries
Let 
denote the set of natural numbers starting with 1, and let 
. Define 
, for 
, and set 
.
An alphabet is a finite set 
, the set of all finite words over 
is denoted by 
, and the empty word by 
. Let 
be the free semigroup for the free monoid 
. Let w[i] denote the 
letter of 
that is 
or 
. The length of a word 
is denoted by |w| and let 
. Set 
for 
. Set 
for all 
. The number of occurrences of a letter 
in 
is denoted by 
. For a given word 
the reversal of w is defined by 
. A word 
is a factor of 
if 
holds for some words 
. If 
then u is called a prefix of w and a suffix if 
. Let 
denote the sets of all factors, prefixes, and suffixes respectively. Define 
and 
are defined accordingly. Notice that 
for all 
. The powers of 
are recursively defined by 
, 
for 
.
Following [14], we only consider binary alphabets, namely 
with the fixed lexicographic order induced by 
on 
. In analogy to binary numbers we call a word 
odd if 
and even otherwise.
For a function 
for 
and an arbitrary alphabet 
the concatenation of the images defines a finite word 
. Since 
is bijective, we will identify 
with f and use in both cases f (as long as it is clear from the context). This definition allows us to access f’s reversed function
easily by 
.
Definition 1
The maximum-ones functions is defined for a word 
by 
giving for each 
the maximal number of 
s occuring in a factor of length k. Likewise the prefix-ones and suffix-ones functions are defined by 
and 
.
Definition 2
Two words 
are called prefix-normal equivalent (pn-equivalent, 
) if 
holds and v’s equivalence class is denoted by 
. A word 
is called prefix (suffix) normal iff 
(
resp.) holds. Let 
denote the maximal-one sum of a 
.
Remark 1
Notice that 
for all 
. By 
and 
follows immediately that a word 
is prefix normal iff its reversal is suffix normal.
Fici and Lipták [14] showed that for each word 
there exists exactly one 
that is prefix normal - the prefix normal form of w. We introduce the concept of least representative, which is the lexicographically smallest element of a class and thus also unique. As mentioned in [5] palindromes play a special role. Immediately by 
for 
, we have 
, i.e. palindromes are the only words that can be prefix and suffix normal. Recall that not all palindromes are prefix normal witnessed by 
.
Definition 3
A palindrome is called prefix normal palindrome (pnPal) if it is prefix normal. Let 
denote the set of all prefix normal palindromes of length 
and set 
. Let 
be the set of all palindromes of length 
.
Table 1.
Prefix normal palindromes (pnPals).
| Word length | Prefix normal palindromes | # prefix normal words |
|---|---|---|
| 1 |
|
2 |
| 2 |
|
3 |
| 3 |
|
5 |
| 4 |
|
8 |
| 5 |
|
14 |
| 6 |
|
23 |
| 7 |
|
41 |
| 8 |
|
70 |
Properties of the Least-Representatives
Before we present specific properties of the least representatives (LR) for a given word length, we mention some useful properties of the maximum-ones, prefix-ones, and suffix-ones functions (for the basic properties we refer to [5, 14] and the references therein). Since we are investigating only words of a specific length, we fix 
. Beyond the relation 
the mappings 
and 
are determinable from each other. Counting the 
s in a suffix of length i and adding the 
s in the corresponding prefix of length 
of a word w, gives the overall amount of 
s of w, namely
 |
For suffix (resp. prefix) normal words this leads to 
resp. 
witnessing the fact 
for palindromes (since both equation hold). Before we show that indeed pnPals form a singleton class w.r.t. 
, we need the relation between the lexicographical order and prefix and suffix normality.
Lemma 1
The prefix normal form of a class is the lexicographically largest element in the class and the suffix-normal of a class is a LR.
Lemma 1 implies that a word being prefix and suffix normal forms a singleton class w.r.t. 
. As mentioned 
only holds for palindromes.
Proposition 1
For a word 
it holds that 
iff 
.
The general part of this section is concluded by a somewhat artificial equation which is nevertheless useful for pnPals : by 
with 
for 
and 
we get
 |
The rest of the section will cover properties of the LRs of a class.
Remark 2
For completeness, we mention that 
is the only even LR w.r.t. 
and the only pnPal starting with 
. Moreover, 
is the largest LR. As we show later in the paper 
and 
are of minor interest in the recursive process due to their speciality.
The following lemma is an extension of [5, Lemma 1] for the suffix-one function by relating the prefix and the suffix of the word 
for a least representative. Intuitively the suffix normality implies that the 
s are more at the end of the word w rather than at the beginning: consider for instance 
for 
. The associated word w cannot be suffix normal since the suffix of length two has only one 
(
) but by 
, and 
we get that within two letters two 
s are present and consequently 
. Thus, a word w is only least representative if the amount of 
s at the end of 
does not exceed the amount of 
s at the beginning of 
.
Lemma 2
Let 
be a LR. Then we have
 |
The remaining part of this section presents results for prefix normal palindromes. Notice that for 
with 
with 
is not necessarily a pnPal; consider for instance 
with 
. The following lemma shows a result for prefix normal palindromes which is folklore for palindromes substituting 
by 
or 
.
Lemma 3
For 
with 
we have
 |
In the following we give a characterisation of when a palindrome w is prefix normal depending on its maximum-ones function 
and a derived function 
. In particular we observe that 
if and only if w is a prefix normal palindrome. Intuitively 
captures the progress of 
in reverse order. This is an intriguing result because it shows that properties regarding prefix and suffix normality can be observed when 
are considered in their serialised representation.
Definition 4
For 
define 
by 
with the extension 
of f and 
. Define 
and 
analogously.
Example 1
Consider the pnPal 
with 
. Then 
is 43221 and we have 
. On the other hand for 
we have 
and 
and 
and thus 
.
The following lemma shows a connection between the reversed prefix-ones function and the suffix-ones function that holds for all palindromes.
Lemma 4
For 
we have 
.
By Lemma 4 we get 
since 
for a palindrome w. As advocated earlier, our main theorem of this part (Theorem 1) gives a characterisation of pnPals. The theorem allows us to decide if a word is a pnPal by only looking at the maximum-ones-function, thus a comparison of all factors is not required.
Theorem 1
Let 
. Then w is a pnPal if and only if 
.
Table 2 presents the amount of pnPals up to length 30 These results support the conjecture in [5] that there is a different behaviour for even and odd length of the word.
Table 2.
Number of pnPals. [15] (A308465)
| i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| # | 2 | 2 | 3 | 3 | 5 | 4 | 8 | 7 | 12 | 11 | 21 | 18 | 36 | 31 | 57 |
| i | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| # | 55 | 104 | 91 | 182 | 166 | 308 | 292 | 562 | 512 | 1009 | 928 | 1755 | 1697 | 3247 | 2972 |
Recursive Construction of Prefix Normal Classes
In this section we investigate how to generate LRs of length 
using the LRs of length n. This is similar to the work of Fici and Lipták [14] except they investigated appending a letter to prefix normal words while we explore the behaviour on prepending letters to LRs. Consider the words 
and 
, both being (different) LRs of length 4. Prepending a 
to them leads to 
and 
which are pn-equivalent. We say that v and w
collapse and denote it by 
. Hence for determining the index of 
based on the least representatives of length 
, only the least representative of one class matters.
Definition 5
Two words 
collapse if 
holds. This is denoted by 
.
Prepending a 
to a non LR will never lead to a LR. Therefore It is sufficient to only look at LRs. Since collapsing is an equivalence relation, denote the equivalence class w.r.t. 
of a word 
by 
. Next, we present some general results regarding the connections between the LRs of lengths n and 
. As mentioned in Remark 2, 
and 
are for all 
LRs. This implies that they do not have to be considered in the recursive process.
Remark 3
By [14] a word 
is prefix-normal if w is prefix-normal. Consequently we know that if a word 
is suffix normal, 
is suffix normal as well. This leads in accordance to the naïve upper bound of 
to a naïve lower bound of 
for 
.
Remark 4
The maximum-ones functions for 
and 
are equal on all 
and 
since the factor determining the maximal number of 
’s is independent of the leading 
. Prepending 
to a word w may result in a difference between 
and 
, but notice that since only one 
is prepended, we always have 
for all 
. In both cases we have 
for 
and 
and 
as well as 
.
Firstly we improve the naïve upper bound to 
by proving that only LRs in 
can become LRs in 
by prepending 
or 
.
Proposition 2
Let 
not be LR. Neither 
nor 
are LRs in 
.
By Proposition 1 prefix (and thus suffix) normal palindromes form a singleton class. This implies immediately that a word 
such that 
is a prefix normal palindrome, does not collapse with any other 
. The next lemma shows that even prepending once a 
and once a 
to different words leads only to equivalent words in one case.
Lemma 5
Let 
be different LRs. Then 
if and only if 
and 
.
By Lemma 5 and Remark 3 it suffices to investigate the collapsing relation on prepanding 
s. The following proposition characterises the LR 
among the elements 
for all LRs 
with 
for 
.
Proposition 3
Let 
be a LR. Then 
is a LR if and only if 
holds for 
and 
.
Corollary 1
Let 
. Then 
for 
and 
. Moreover 
for 
and 
.
This characterization is unfortunately not convenient for determining either the number of LRs of length 
from the ones from length n or the collapsing LRs of length n. For a given word w, the maximum-ones function 
has to be determined, 
to be extended by 
, and finally the associated word - under the assumption 
has to be checked for being suffix normal. For instance, given 
leads to 
, and is extended to 
. This would correspond to 
which is not suffix normal and thus w is not extendable to a new LR. The following two lemmata reduce the amount of LRs that needs to be checked for extensibility.
Lemma 6
Let 
be a LR such that 
is a LR as well. Then for all LRs 
collapsing with 
holds for all 
, i.e. all other LRs have a smaller maximal-one sum.
Corollary 2
If 
and 
are LRs with 
and 
then 
.
Remark 5
By Corollary 2 the lexicographically smallest LR w among the collapsing leads to the LR of 
. Thus if w is a LR not collapsing with any lexicographically smaller word then 
is LR.
Before we present the theorem characterizing exactly the collapsing words for a given word w, we show a symmetry-property of the LRs which are not extendable to LRs, i.e. a property of words which collapse.
Lemma 7
Let 
be a LR. Then 
for some 
iff 
.
By [5, Lemma 10] a word 
is prefix normal if and only if 
for all 
. The following theorem extends this result for determining the collapsing words 
for a given word w.
Theorem 2
Let 
be a LR and 
with 
. Let moreover 
for all 
with 
. Then 
iff
for all 
,
implies 
,
Theorem 2 allows us to construct the equivalence classes w.r.t. the least representatives of the previous length but more tests than necessary have to be performed: Consider, for instance 
which is a smallest LR of length 17 not collapsing with any lexicographically smaller LR. For w we have 
where the dots just act as separators between letters. Thus we know for any 
collapsing with w, that 
and 
. The constraints 
and 
implies 
. First the check that 
is impossible excludes 
. Since no collapsing word can have a factor of length 2 with only one 
, a band in which the possible values range can be defined by the unique greatest collapsing word 
. It is not surprising that this word is connected with the prefix normal form. The following two lemmata define the band in which the possible collapsing words 
are.
Lemma 8
Let 
be a LR with 
for all 
with 
. Set 
. Then 
and for all LRs 
with 
and all 
, thus 
.
Notice that 
is not necessarily a LR in 
witnessed by the word of the last example. For w we get 
with 
and 
violating the symmetry property given in Lemma 7. The following lemma alters 
into a LR which represents still the lower limit of the band.
Lemma 9
Let 
be a LR such that 
is also a LR. Let 
with 
, and I the set of all 
with
 |
and 
for all 
. Then 
defined such that 
for all 
and 
(
resp.) for all 
holds, collapses with w.
Remark 6
Lemma 9 applied to 
gives the lower limit of the band. Let 
denote the output of this application for a given 
according to Lemma 9.
Continuing with the example, we firstly determine 
for 
. We get with 
Since for all collapsing 
we have 
is determined for 
. Since the value for 5 determines the one for 13 there are only two possibilities, namely 
and 
and 
and 
. Notice that the words 
corresponding to the generated words 
are not necessarily LRs of the shorter length as witnessed by the one with 
and 
. In this example this leads to at most three words being not only in the class but also in the list of former representatives. Thus we are able to produce an upper bound for the cardinality of the class. Notice that in any case we only have to test the first half of 
’s positions by Lemma 7. This leads to the following definition.
Table 3.
f for 
.
| i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
 |
1 | 2 | 3 | 4 | 5 | 5 | 6 | 7 | 8 | 8 | 8 | 9 | 10 | 10 | 11 | 12 | 13 |
 |
1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 7 | 7 | 8 | 9 | 9 | 10 | 11 | 12 | 13 |
 |
1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 7 | 8 | 8 | 9 | 9 | 10 | 11 | 12 | 13 |
Definition 6
Let 
be the Hamming-distance. The palindromic distance
is defined by 
. Define the palindromic prefix length
by 
.
The palindromic distance gives the minimal number of positions in which a bit has to be flipped for obtaining a palindrome. Thus, 
for all palindromes w, and, for instance, 
since the first half of w and the reverse of the second half mismatch in two positions. The palindromic prefix length determines the length of w’s longest prefix being a palindrome. For instance 
and 
. Since a LR w determines the upper limit of the band and 
the lower limit, the palindromic distance of 
is in relation to the positions of 
in which collapsing words may differ from w.
Theorem 3
If 
and 
are both LRs then 
.
For an algorithmic approach to determine the LRs of length n, we want to point out that the search for collapsing words can also be reduced using the palindromic prefix length. Let 
be the LRs of length 
. For each w we keep track of 
. For each 
we check firstly if 
since in this case the prepended 
leads to a palindrome. Only if this is not the case, 
needs to be determined. All collapsing words computed within the band of 
and 
are deleted in 
.
In the remaining part of the section we investigate the set 
w.r.t. 
for 
. This leads to a second calculation for an upper bound and a refinement for determining the LRs of 
faster.
Lemma 10
If 
then 
is not a LR but 
is a LR.
Remark 7
By Lemma 10 follows that all words 
collapse with a smaller LR. Thus, for all 
, an upper bound for 
is given by 
.
For a closed recursive calculation of the upper bound in Remark 7, the exact number 
is needed. Unfortunately we are not able to determine 
for arbitrary 
. The following results show relations between prefix normal palindromes of different lengths. For instance, if 
then 1w1 is a prefix normal palindrome as well. The importance of the pnPals is witnessed by the following estimation.
Theorem 4
For all 
and 
we have
 |
The following results only consider pnPals that are different from 
and 
. Notice for these special palindromes that 
, 
for an appropriate 
but 
.
Lemma 11
If 
then neither ww nor 
are prefix normal palindromes.
Lemma 12
Let 
with 
. If 
is also a prefix normal palindrome then 
or 
for some 
and 
.
A characterisation for 
being a pnPal is more complicated. By 
follows that a block of 
s contains at most the number of 
s of the previous block. But if such a block contains strictly less 
s the number of 
s in between can increase by the same amount the number of 
s decreased.
Lemma 13
Let 
. If 
is also a prefix normal palindrome then 
.
Lemmas 11, 12, and 13 indicate that a characterization of prefix normal palindromes based on smaller ones is hard to determine.
Conclusion
Based on the work in [14], we investigated prefix normal palindromes in Sect. 3 and gave a characterisation based on the maximum-ones function. At the end of Sect. 4 results for a recursive approach to determine prefix normal palindromes are given. These results show that easy connections between prefix normal palindromes of different lengths cannot be expected. By introducing the collapsing relation we were able to partition the set of extension-critical words introduced in [14]. This leads to a characterization of collapsing words which can be extended to an algorithm determining the corresponding equivalence classes. Moreover we have shown that palindromes and the collapsing classes are related.
The concrete values for prefix normal palindromes and the index of the collapsing relation remain an open problem as well as the cardinality of the equivalence classes w.r.t. the collapsing relation. Further investigations of the prefix normal palindromes and the collapsing classes lead directly to the index of the prefix equivalence.
Acknowledgments
We would like to thank Florin Manea for helpful discussions and advice.
Contributor Information
Alberto Leporati, Email: alberto.leporati@unimib.it.
Carlos Martín-Vide, Email: carlos.martin@urv.cat.
Dana Shapira, Email: shapird@g.ariel.ac.il.
Claudio Zandron, Email: zandron@disco.unimib.it.
Pamela Fleischmann, Email: fpa@informatik.uni-kiel.de.
Mitja Kulczynski, Email: mku@informatik.uni-kiel.de.
Dirk Nowotka, Email: dn@informatik.uni-kiel.de.
Danny Bøgsted Poulsen, Email: dannybpoulsen@cs.aau.dk.
References
- 1.Balister P, Gerke S. The asymptotic number of prefix normal words. J. Comb. Theory. 2019;784:75–80. doi: 10.1016/j.tcs.2019.03.036. [DOI] [Google Scholar]
- 2.Burcsi P, Cicalese F, Fici G, Lipták Z. Algorithms for jumbled pattern matching in strings. Int. J. Found. CS. 2012;23(2):357–374. doi: 10.1142/S0129054112400175. [DOI] [Google Scholar]
- 3.Burcsi P, Fici G, Lipták Z, Ruskey F, Sawada J. On combinatorial generation of prefix normal words. In: Kulikov AS, Kuznetsov SO, Pevzner P, editors. Combinatorial Pattern Matching; Cham: Springer; 2014. pp. 60–69. [Google Scholar]
- 4.Burcsi P, Fici G, Lipták Z, Ruskey F, Sawada J. Normal, abby normal, prefix normal. In: Ferro A, Luccio F, Widmayer P, editors. Fun with Algorithms. Cham: Springer; 2014. pp. 74–88. [Google Scholar]
- 5.Burcsi P, Fici G, Lipták Z, Ruskey F, Sawada J. On prefix normal words and prefix normal forms. TCS. 2017;659:1–13. doi: 10.1016/j.tcs.2016.10.015. [DOI] [Google Scholar]
- 6.Cassaigne J, Richomme G, Saari K, Zamboni LQ. Avoiding Abelian powers in binary words with bounded Abelian complexity. Int. J. Found. CS. 2011;22(04):905–920. doi: 10.1142/S0129054111008489. [DOI] [Google Scholar]
- 7.Chan, T.M., Lewenstein, M.: Clustered integer 3SUM via additive combinatorics. In: 47th ACM Symposium on TOC, pp. 31–40. ACM (2015)
- 8.Cicalese F, Lipták Z, Rossi M. Bubble-flip—a new generation algorithm for prefix normal words. In: Klein ST, Martín-Vide C, Shapira D, editors. Language and Automata Theory and Applications; Cham: Springer; 2018. pp. 207–219. [Google Scholar]
- 9.Cicalese, F., Lipták, Z., Rossi, M.: On infinite prefix normal words. In: Proceedings of the SOFSEM, pp. 122–135 (2019)
- 10.Coven EM, Hedlund GA. Sequences with minimal block growth. TCS. 1973;7(2):138–153. [Google Scholar]
- 11.Currie J, Rampersad N. Recurrent words with constant Abelian complexity. Adv. Appl. Math. 2011;47(1):116–124. doi: 10.1016/j.aam.2010.05.001. [DOI] [Google Scholar]
- 12.Dassow J. Parikh mapping and iteration. In: Calude CS, PĂun G, Rozenberg G, Salomaa A, editors. Multiset Processing; Heidelberg: Springer; 2001. pp. 85–101. [Google Scholar]
- 13.Ehlers T, Manea F, Mercas R, Nowotka D. k-Abelian pattern matching. J. Discrete Algorithms. 2015;34:37–48. doi: 10.1016/j.jda.2015.05.004. [DOI] [Google Scholar]
- 14.Fici G, Lipták Z. On prefix normal words. In: Mauri G, Leporati A, editors. Developments in Language Theory; Heidelberg: Springer; 2011. pp. 228–238. [Google Scholar]
- 15.OEIS Foundation Inc.: The on-line encyclopedia of integer sequencess (2019). http://oeis.org/
- 16.Karhumäki J. Generalized Parikh mappings and homomorphisms. Inf. Control. 1980;47(3):155–165. doi: 10.1016/S0019-9958(80)90493-3. [DOI] [Google Scholar]
- 17.Keränen V. Abelian squares are avoidable on 4 letters. In: Kuich W, editor. Automata, Languages and Programming; Heidelberg: Springer; 1992. pp. 41–52. [Google Scholar]
- 18.Lee L-K, Lewenstein M, Zhang Q. Parikh matching in the streaming model. In: Calderón-Benavides L, González-Caro C, Chávez E, Ziviani N, editors. String Processing and Information Retrieval; Heidelberg: Springer; 2012. pp. 336–341. [Google Scholar]
- 19.Mateescu, A., Salomaa, A., Salomaa, K., Yu, S.: On an extension of the Parikh mapping, 06 September 2000. http://citeseer.ist.psu.edu/440186.html
- 20.Mateescu A, Salomaa A, Yu S. Subword histories and Parikh matrices. J. Comput. Syst. Sci. 2004;68(1):1–21. doi: 10.1016/j.jcss.2003.04.001. [DOI] [Google Scholar]
- 21.Parikh RJ. On context-free languages. J. ACM. 1966;13:570–581. doi: 10.1145/321356.321364. [DOI] [Google Scholar]
- 22.Puzynina S, Zamboni LQ. Abelian returns in Sturmian words. J. Comb. Theory. 2013;120(2):390–408. doi: 10.1016/j.jcta.2012.09.002. [DOI] [Google Scholar]
- 23.Richomme G, Saari K, Zamboni LQ. Abelian complexity of minimal subshifts. J. Lond. Math. Soc. 2010;83(1):79–95. doi: 10.1112/jlms/jdq063. [DOI] [Google Scholar]
- 24.Richomme G, Saari K, Zamboni LQ. Balance and Abelian complexity of the Tribonacci word. Adv. Appl. Math. 2010;45(2):212–231. doi: 10.1016/j.aam.2010.01.006. [DOI] [Google Scholar]
- 25.Salomaa A. Connections between subwords and certain matrix mappings. TCS. 2005;340(2):188–203. doi: 10.1016/j.tcs.2005.03.024. [DOI] [Google Scholar]