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Extremal Pattern-Avoiding Words [article]

Natalya Ter-Saakov, Emily Zhang
2020 arXiv   pre-print
In this paper, we determine the number of extremal XY_1XY_2X... XY_tX-avoiding words on a k-letter alphabet.  ...  We also derive a lower bound on the shortest possible length of an extremal square-free word on a k-letter alphabet that grows exponentially in k.  ...  As such, the only extremal XY X-avoiding words on a two-letter alphabet are aabb and bbaa. Having looked at specific cases, we consider the k-letter alphabet A = {a 1 , a 2 , . . . , a k }.  ... 
arXiv:2009.10186v1 fatcat:wjhfogz4jbheppds7zae4ca5ea

Containing all permutations [article]

Michael Engen, Vincent Vatter
2020 arXiv   pre-print
Numerous versions of the question "what is the shortest object containing all permutations of a given length?"  ...  We survey and synthesize these questions and their partial answers, introduce infinitely more related questions, and then establish an improved upper bound for one of these questions.  ...  We are additionally grateful to Jay Pantone for his assistance in verifying that no permutation of length 16 or less contains all permutations of length 6 as subsequences.  ... 
arXiv:1810.08252v4 fatcat:zx2nto72ibeypk2hmzpojiqrk4

Transposition Rearrangement: Linear Algorithm for Length-Cost Model

Łukasz Mikulski
2009 Annales UMCS Informatica  
Primary structures of DNA or proteins are represented by such sequences (also called words or strings). In the paper a linear algorithm, computing the distance between two words, is presented.  ...  The model operates with transpositions of single letters. The cost of a single transposition is equal to the distance which transposed letter has to cover.  ...  These strings are the Parikh equivalent (both of them consist of two letters a, one letter b and one letter c).  ... 
doi:10.2478/v10065-009-0001-4 fatcat:2ksdhaxogbbn7m2kyxrurdsosu

Evolutionary search techniques for the Lyndon factorization of biosequences

Amanda Clare, Jacqueline W. Daykin, Thomas Mills, Christine Zarges
2019 Proceedings of the Genetic and Evolutionary Computation Conference Companion on - GECCO '19  
Here we investigate the impact of permuting the alphabet ordering on the resulting factorization and demonstrate significant variations in the numbers of factors obtained.  ...  A non-empty string x over an ordered alphabet is said to be a Lyndon word if it is alphabetically smaller than all of its cyclic rotations.  ...  Two strings are considered Abelian equivalent if one can be turned into the other by permuting its letters; in other words, if the two strings have the same Parikh vector.  ... 
doi:10.1145/3319619.3326872 dblp:conf/gecco/ClareDMZ19 fatcat:jgrg7qijvndvrlkz6ekf7vl4gy

Words over an ordered alphabet and suffix permutations

Jean-Pierre Duval, Arnaud Lefebvre
2002 RAIRO - Theoretical Informatics and Applications  
Given an ordered alphabet and a permutation, according to the lexicographic order, on the set of suffixes of a word w, we present in this article a linear time and space method to determine whether a word  ...  Finally, we note that this work can lead to a method for generating a Lyndon word randomly in linear time or for computing the set of Lyndon words of length n.  ...  We assume the words with k different letters to be given with the first k letters of the alphabet. This leads to consider the minimum size of the alphabet required to build words for a given σ.  ... 
doi:10.1051/ita:2002012 fatcat:u45ukr5muvbelnbbxaf7iw6s6e

A Multiset Version of Even-Odd Permutations Identity [article]

Hossein Teimoori Faal
2022 arXiv   pre-print
In this paper, we give a new bijective proof of a multiset analogue of even-odd permutations identity.  ...  This multiset version is equivalent to the original coin arrangements lemma which is a key combinatorial lemma in the Sherman's Proof of a conjecture of Feynman about an identity on paths in planar graphs  ...  Proposition 4 . 1 . 41 Let Σ be a finite alphabet of k letters.  ... 
arXiv:2206.01291v1 fatcat:qx2f4lcrjvc3pnoi6wgty76hue

Left-to-right maxima in words and multiset permutations [article]

Amy Myers, Herb Wilf
2007 arXiv   pre-print
We extend classical theorems of Renyi by finding the distributions of the numbers of both weak and strong left-to-right maxima (a.k.a. outstanding elements) in words over a given alphabet and in permutations  ...  of a given multiset.  ...  In this paper we will explore the contexts in which w is a permutation, a multiset permutation, or a word over some finite alphabet.  ... 
arXiv:math/0701078v1 fatcat:ila3eyginzduhfvrxmzrugupyu

On the combinatorics of suffix arrays

Gregory Kucherov, Lilla Tóthmérész, Stéphane Vialette
2013 Information Processing Letters  
We present a bijective characterization of suffix array permutations obtained from a characterization of Burrows-Wheeler arrays given in [1] .  ...  Our characterization of suffix arrays is the first based on their relationship with Burrows-Wheeler permutations.  ...  n over an alphabet of size k, that have at least one occurrence of each letter and have π as their suffix array, (iii) Count the number of permutations π ∈ S n that are suffix arrays of some word over  ... 
doi:10.1016/j.ipl.2013.09.009 fatcat:2udknaaxhzfmfhi22dvgghggaq

Retrograde codes and bounded synchronization delay

Hollie L. Buchanan, Michael E. Mays
1992 Information and Computation  
Retrograde codes are a subclass of comma-free codes in which the dictionary of code words excludes not only overlaps of code words but also reversals of overlaps.  ...  We give a general upper bound on the maximum size of a retrograde comma-free distionary, provide a construction for a bounded synchronization delay retrograde code which attains this maximum size, and  ...  a word as being equivalent to a cyclic permutation of its letters or a reversal of a cyclic permutation of its letters.  ... 
doi:10.1016/0890-5401(92)90054-j fatcat:55jgnzaymbdfhbrjjqhfqfv5hi

Parikh-friendly permutations and uniformly parikh-friendly words

Wen Chean Teh
2020 The Australasian Journal of Combinatorics  
In this study, we show not only that every permutation is Parikh-friendly, but also that there exists a single word that witnesses the Parikh-friendliness of every permutation on a given ordered alphabet  ...  In fact we introduce a relativized version of a Parikh-friendly permutation. As a result, words that are uniformly Parikh-friendly in a wider sense are effectively constructed.  ...  Note that each letter occurs as the first letter in one of the three words.  ... 
dblp:journals/ajc/Teh20 fatcat:xberhuxon5g4njwwioku3fl4b4

Short witnesses for Parikh-friendly permutations

Jamie Simpson
2020 The Australasian Journal of Combinatorics  
showed that for every permutation π of the ordered alphabet A = {a 1 , a 2 , . . . , a n } there exists a word w ∈ A * , in which each letter of A appears at least once, such that w and π(w) have the same  ...  The letters in x come from some alphabet.  ...  Therefore, if there is a consecutive triple in one of these words which is not in the other it must contain b 1 , and since we have assumed b 1 is the lexicographically least letter in the permutation  ... 
dblp:journals/ajc/Simpson20 fatcat:pyzczwf4mjdyvnd7xvpwkkewf4

Rationality for subclasses of 321-avoiding permutations

Michael Albert, Robert Brignall, Nik Ruškuc, Vincent Vatter
2019 European journal of combinatorics (Print)  
letter of the word w).  ...  Given a set of letters X and a word w we denote by w| X the projection 92 of w onto X, i.e., the subword of w formed by its letters in X. Finally, we denote the empty word 93 by . 94 2.  ...  with input alphabet Σ and output alphabet Σ 638 which marks precisely one letter of its input.  ... 
doi:10.1016/j.ejc.2019.01.001 fatcat:5ngbbewgabcbzkpcn2ryrblxl4

Shirshov's theorem and ω-permutability of semigroups

J Justin, G Pirillo
1991 Advances in Mathematics  
The semigroup S is right (resp. left, resp. two-sided) o-permutable if in each word s over the alphabet S which is right (resp. left, resp. two-sided) infinite there exists a factor, say s(m) . ..s(n).  ...  In the same way there exist a letter a1 and infinitely many words u such that u(r,-1, r,+ l)=a,a,a,.  ... 
doi:10.1016/0001-8708(91)90069-j fatcat:trvw4ayit5gx5b4yqgi27rghry

Rationality for subclasses of 321-avoiding permutations [article]

Michael H. Albert, Robert Brignall, Nik Ruškuc, Vincent Vatter
2019 arXiv   pre-print
To do so we show that any such class is in bijective correspondence with a regular language.  ...  We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well quasi-ordered has a rational generating function.  ...  Significant inspiration for this research came from the work of Lozin [24] , who proved that while the class of bipartite inversion graphs (the inversion graphs of 321-avoiding permutations) has unbounded  ... 
arXiv:1602.00672v3 fatcat:u3upjvt7bbg4tmdncpefkyylyq

On the combinatorics of suffix arrays [article]

Gregory Kucherov, Lilla Tóthmérész, Stéphane Vialette
2012 arXiv   pre-print
We show that the characterization of suffix arrays for a special case of binary alphabet given in [2] easily follows from our characterization.  ...  We prove several combinatorial properties of suffix arrays, including a characterization of suffix arrays through a bijection with a certain well-defined class of permutations.  ...  n over an alphabet of size k, that have at least one occurrence of each letter and have π as their suffix array, (iii) Count the number of permutations π ∈ S n that are suffix arrays of some word over  ... 
arXiv:1206.3877v1 fatcat:xhdvxdkoubdqvp5wjkb7kc2pbm
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