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Extremal Pattern-Avoiding Words
[article]

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 }. ...

##
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Containing all permutations
[article]

2020
*
arXiv
*
pre-print

Numerous versions of the question "what is the shortest object containing all

arXiv:1810.08252v4
fatcat:zx2nto72ibeypk2hmzpojiqrk4
*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. ...##
###
Transposition Rearrangement: Linear Algorithm for Length-Cost Model

2009
*
Annales UMCS Informatica
*

Primary structures of DNA or proteins are represented by such sequences (also called

doi:10.2478/v10065-009-0001-4
fatcat:2ksdhaxogbbn7m2kyxrurdsosu
*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). ...##
###
Evolutionary search techniques for the Lyndon factorization of biosequences

2019
*
Proceedings of the Genetic and Evolutionary Computation Conference Companion on - GECCO '19
*

Here we investigate the impact of

doi:10.1145/3319619.3326872
dblp:conf/gecco/ClareDMZ19
fatcat:jgrg7qijvndvrlkz6ekf7vl4gy
*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. ...##
###
Words over an ordered alphabet and suffix permutations

2002
*
RAIRO - Theoretical Informatics and Applications
*

Given an ordered

doi:10.1051/ita:2002012
fatcat:u45ukr5muvbelnbbxaf7iw6s6e
*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. ...##
###
A Multiset Version of Even-Odd Permutations Identity
[article]

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*. ...

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

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)

arXiv:math/0701078v1
fatcat:ila3eyginzduhfvrxmzrugupyu
*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*. ...##
###
On the combinatorics of suffix arrays

2013
*
Information Processing Letters
*

We present

doi:10.1016/j.ipl.2013.09.009
fatcat:2udknaaxhzfmfhi22dvgghggaq
*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 ...##
###
Retrograde codes and bounded synchronization delay

1992
*
Information and Computation
*

Retrograde codes are

doi:10.1016/0890-5401(92)90054-j
fatcat:55jgnzaymbdfhbrjjqhfqfv5hi
*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*. ...##
###
Parikh-friendly permutations and uniformly parikh-friendly words

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*. ...

##
###
Short witnesses for Parikh-friendly permutations

2020
*
The Australasian Journal of Combinatorics
*

showed that for every

dblp:journals/ajc/Simpson20
fatcat:pyzczwf4mjdyvnd7xvpwkkewf4
*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*...##
###
Rationality for subclasses of 321-avoiding permutations

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. ...

##
###
Shirshov's theorem and ω-permutability of semigroups

1991
*
Advances in Mathematics
*

The semigroup S is right (resp. left, resp.

doi:10.1016/0001-8708(91)90069-j
fatcat:trvw4ayit5gx5b4yqgi27rghry
*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*,. ...##
###
Rationality for subclasses of 321-avoiding permutations
[article]

2019
*
arXiv
*
pre-print

To do so we show that any such class is

arXiv:1602.00672v3
fatcat:u3upjvt7bbg4tmdncpefkyylyq
*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 ...##
###
On the combinatorics of suffix arrays
[article]

2012
*
arXiv
*
pre-print

We show that the characterization of suffix arrays for

arXiv:1206.3877v1
fatcat:xhdvxdkoubdqvp5wjkb7kc2pbm
*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 ...
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