FINE AND WILF'S THEOREM FOR PARTIAL WORDS WITH ARBITRARILY MANY WEAK PERIODS

2010 ◽  
Vol 21 (05) ◽  
pp. 705-722 ◽  
Author(s):  
F. BLANCHET-SADRI ◽  
TAKTIN OEY ◽  
TIMOTHY D. RANKIN
Keyword(s):  
Arbitrary Number ◽  
Finite Sequences ◽  
Partial Word ◽  
Partial Words

Fine and Wilf's well-known theorem states that any word having periods p,q and length at least p+q- gcd (p,q) also has gcd (p,q) as a period. Moreover, the length p+q- gcd (p,q) is critical since counterexamples can be provided for shorter words. This result has since been extended to partial words, or finite sequences that may contain some "holes." More precisely, any partial word u with H holes having weak periods p,q and length at least the so-denoted lH(p,q) also has strong period gcd (p,q) provided u is not (H,(p,q))-special. This extension was done for one hole by Berstel and Boasson (where the class of (1,(p,q))-special partial words is empty), and for an arbitrary number of holes by Blanchet-Sadri. In this paper, we further extend these results, allowing an arbitrary number of weak periods. In addition to speciality, the concepts of intractable period sets and interference between periods play a role.

CYCLIC PARTIAL WORDS

2020 ◽  
Vol 9 (11) ◽  
pp. 9219-9230
Author(s):  
R.K. Kumari ◽  
R. Arulprakasam ◽  
R. Perumal ◽  
V.R. Dare
Keyword(s):  
Dna Sequencing ◽  
Partial Word ◽  
Partial Words ◽  
Cyclic Words

Partial words are linear words with holes. Cyclic words are derived from linear words by linking its first letter after the last one. Both partial words and cyclic words have wide applications in DNA sequencing. In this paper we introduce cyclic partial words and discuss their periodicity and certain properties. We also establish representation of a cyclic partial word using trees.

scholarly journals Pattern Avoidance in Partial Permutations

10.37236/512 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Anders Claesson ◽  
Vít Jelínek ◽  
Eva Jelínková ◽  
Sergey Kitaev
Keyword(s):  
Equivalence Class ◽  
Arbitrary Number ◽  
Pattern Avoidance ◽  
Permutation Patterns ◽  
Wilf Equivalence ◽  
Partial Words

Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A partial permutation of length $n$ with $k$ holes is a sequence of symbols $\pi=\pi_1\pi_2\dotsb\pi_n$ in which each of the symbols from the set $\{1,2,\dotsc,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ are "holes". We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length $k$ correspond to a Wilf-type equivalence class with respect to partial permutations with $(k-2)$ holes. Lastly, we enumerate the partial permutations of length $n$ with $k$ holes avoiding a given pattern of length at most four, for each $n\ge k\ge 1$.

scholarly journals Counting Bordered Partial Words by Critical Positions

10.37236/625 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Emily Allen ◽  
F. Blanchet-Sadri ◽  
Cameron Byrum ◽  
Mihai Cucuringu ◽  
Robert Mercaş
Keyword(s):  
Combinatorial Property ◽  
Finite Alphabet ◽  
Theoretical Problem ◽  
Fixed Size ◽  
Fixed Length ◽  
Word Sequence ◽  
Partial Word ◽  
Recursive Formulas ◽  
Partial Words

A partial word, sequence over a finite alphabet that may have some undefined positions or holes, is bordered if one of its proper prefixes is compatible with one of its suffixes. The number theoretical problem of enumerating all bordered full words (the ones without holes) of a fixed length $n$ over an alphabet of a fixed size $k$ is well known. It turns out that all borders of a full word are simple, and so every bordered full word has a unique minimal border no longer than half its length. Counting bordered partial words having $h$ holes with the parameters $k, n$ is made extremely more difficult by the failure of that combinatorial property since there is now the possibility of a minimal border that is nonsimple. Here, we give recursive formulas based on our approach of the so-called simple and nonsimple critical positions.

Square-Free Partial Words with Many Wildcards

2018 ◽  
Vol 29 (05) ◽  
pp. 845-860
Author(s):  
Daniil Gasnikov ◽  
Arseny M. Shur
Keyword(s):  
Natural Generalization ◽  
Interesting Phenomenon ◽  
Maximal Density ◽  
Classical Notion ◽  
Partial Word ◽  
Partial Words ◽  
Free Word

We contribute to the study of square-free words. The classical notion of a square-free word has a natural generalization to partial words, studied in several papers since 2008. We prove that the maximal density of wildcards in the ternary infinite square-free partial word is surprisingly big: [Formula: see text]. Further we show that the density of wildcards in a finitary infinite square-free partial words is at most [Formula: see text] and this bound is reached by a quaternary word. We demonstrate that partial square-free words can be viewed as “usual” square-free words with some letters replaced by wildcards and introduce the corresponding characteristic of infinite square-free words, called flexibility. The flexibility is estimated for some important words and classes of words; an interesting phenomenon is the existence of “rigid” square-free words, having no room for wildcards at all.

ALGORITHMIC COMBINATORICS ON PARTIAL WORDS

2012 ◽  
Vol 23 (06) ◽  
pp. 1189-1206 ◽  
Author(s):  
F. BLANCHET-SADRI
Keyword(s):  
Molecular Biology ◽  
Data Compression ◽  
Pattern Avoidance ◽  
Finite Alphabet ◽  
Open Problems ◽  
The Past ◽  
Partial Word ◽  
De Bruijn ◽  
Subword Complexity ◽  
Partial Words

Algorithmic combinatorics on partial words, or sequences of symbols over a finite alphabet that may have some do-not-know symbols or holes, has been developing in the past few years. Applications can be found, for instance, in molecular biology for the sequencing and analysis of DNA, in bio-inspired computing where partial words have been considered for identifying good encodings for DNA computations, and in data compression. In this paper, we focus on two areas of algorithmic combinatorics on partial words, namely, pattern avoidance and subword complexity. We discuss recent contributions as well as a number of open problems. In relation to pattern avoidance, we classify all binary patterns with respect to partial word avoidability, we classify all unary patterns with respect to hole sparsity, and we discuss avoiding abelian powers in partial words. In relation to subword complexity, we generate and count minimal Sturmian partial words, we construct de Bruijn partial words, and we construct partial words with subword complexities not achievable by full words (those without holes).

scholarly journals Pattern avoidance in partial permutations (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Anders Claesson ◽  
Vít Jelínek ◽  
Eva Jelínková ◽  
Sergey Kitaev
Keyword(s):  
Equivalence Class ◽  
Arbitrary Number ◽  
Pattern Avoidance ◽  
Permutation Patterns ◽  
Nous Montrons ◽  
International Audience ◽  
Wilf Equivalence ◽  
Partial Words

International audience Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A $\textit{partial permutation of length n with k holes}$ is a sequence of symbols $\pi = \pi_1 \pi_2 \cdots \pi_n$ in which each of the symbols from the set $\{1,2,\ldots,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ are "holes''. We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length $k$ correspond to a Wilf-type equivalence class with respect to partial permutations with $(k-2)$ holes. Lastly, we enumerate the partial permutations of length $n$ with $k$ holes avoiding a given pattern of length at most four, for each $n \geq k \geq 1$. Nous introduisons un concept de permutations partielles. $\textit{Une permutation partielle de longueur n avec k trous}$ est une suite finie de symboles $\pi = \pi_1 \pi_2 \cdots \pi_n$ dans laquelle chaque nombre de l'ensemble $\{1,2,\ldots,n-k\}$ apparaît précisément une fois, tandis que les $k$ autres symboles de $\pi$ sont des "trous''. Nous introduisons l'étude des permutations partielles à motifs exclus et nous montrons que la plupart des résultats sur l'équivalence de Wilf peuvent être généralisés aux permutations partielles avec un nombre arbitraire de trous. De plus, nous montrons que les permutations de Baxter d'une longueur donnée $k$ forment une classe d'équivalence du type Wilf par rapport aux permutations partielles avec $(k-2)$ trous. Enfin, nous présentons l'énumération des permutations partielles de longueur $n$ avec $k$ trous qui évitent un motif de longueur $\ell \leq 4$, pour chaque $n \geq k \geq 1$.

Kinematic analysis and deformation of a planar lattice with an arbitrary number of panels

2020 ◽  
pp. 15-19
Author(s):  
M.N. Kirsanov
Keyword(s):  
Exact Solution ◽  
Kinematic Analysis ◽  
Arbitrary Number ◽  
Design Parameters ◽  
Lattice Deformation ◽  
Planar Lattice ◽  
Force Loading

Formulae are obtained for calculating the deformations of a statically determinate lattice under the action of two types of loads in its plane, depending on the number of panels located along one side of the lattice. Two options for fixing the lattice are analyzed. Cases of kinematic variability of the structure are found. The distribution of forces in the rods of the lattice is shown. The dependences of the force loading of some rods on the design parameters are obtained. Keywords: truss, lattice, deformation, exact solution, deflection, induction, Maple system. [email protected]

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

scholarly journals Free fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Marius de Leeuw ◽  
Chiara Paletta ◽  
Anton Pribytok ◽  
Ana L. Retore ◽  
Alessandro Torrielli
Keyword(s):  
Spectral Problem ◽  
Transfer Matrix ◽  
Arbitrary Number ◽  
Free Fermion ◽  
Nearest Neighbour ◽  
Bogoliubov Transformation ◽  
Free Fermions ◽  
New Models ◽  
Lower Dimensional

Abstract In this paper we first demonstrate explicitly that the new models of integrable nearest-neighbour Hamiltonians recently introduced in PRL 125 (2020) 031604 [36] satisfy the so-called free fermion condition. This both implies that all these models are amenable to reformulations as free fermion theories, and establishes the universality of this condition. We explicitly recast the transfer matrix in free fermion form for arbitrary number of sites in the 6-vertex sector, and on two sites in the 8-vertex sector, using a Bogoliubov transformation. We then put this observation to use in lower-dimensional instances of AdS/CFT integrable R-matrices, specifically pure Ramond-Ramond massless and massive AdS3, mixed-flux relativistic AdS3 and massless AdS2. We also attack the class of models akin to AdS5 with our free fermion machinery. In all cases we use the free fermion realisation to greatly simplify and reinterpret a wealth of known results, and to provide a very suggestive reformulation of the spectral problem in all these situations.

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