scholarly journals Patterns in Inversion Sequences I

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Sylvie Corteel ◽  
Megan A. Martinez ◽  
Carla D. Savage ◽  
Michael Weselcouch
Keyword(s):  
Computer Science ◽  
Fibonacci Numbers ◽  
Natural Extension ◽  
Great Depth ◽  
Permutation Patterns ◽  
Bell Numbers ◽  
Combinatorial Structures ◽  
Vast Array

Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying interpretation that relates a vast array of combinatorial structures. In this paper, we introduce the notion of patterns in inversion sequences. A sequence $(e_1,e_2,\ldots,e_n)$ is an inversion sequence if $0 \leq e_i<i$ for all $i \in [n]$. Inversion sequences of length $n$ are in bijection with permutations of length $n$; an inversion sequence can be obtained from any permutation $\pi=\pi_1\pi_2\ldots \pi_n$ by setting $e_i = |\{j \ | \ j < i \ {\rm and} \ \pi_j > \pi_i \}|$. This correspondence makes it a natural extension to study patterns in inversion sequences much in the same way that patterns have been studied in permutations. This paper, the first of two on patterns in inversion sequences, focuses on the enumeration of inversion sequences that avoid words of length three. Our results connect patterns in inversion sequences to a number of well-known numerical sequences including Fibonacci numbers, Bell numbers, Schr\"oder numbers, and Euler up/down numbers.

scholarly journals Statistical Distributions and $q$-Analogues of $k$-Fibonacci Numbers

10.37236/2550 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Adam M Goyt ◽  
Brady L Keller ◽  
Jonathan E Rue
Keyword(s):  
Fibonacci Numbers ◽  
Natural Extension ◽  
Statistical Distributions ◽  
Set Partition ◽  
Set Partitions ◽  
Partition Statistics

We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most k.  The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics.  We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers.  This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations.  In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles.  We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities.

Weak Separation Problem for Tree Languages

2020 ◽  
Vol 31 (05) ◽  
pp. 583-593
Author(s):  
Saeid Alirezazadeh ◽  
Khadijeh Alibabaei
Keyword(s):  
Computer Science ◽  
Natural Extension ◽  
Classical Problem ◽  
Finite Alphabet ◽  
Direct Products ◽  
Separation Problem ◽  
Finite Algebras ◽  
Tree Languages ◽  
Weak Separation ◽  
Recognizable Language

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. By looking at the syntactic congruence for monoids and as the natural extension in the case of forest algebras, we could define a version of syntactic congruence of a subset of the free forest algebra, not just a forest language. Let [Formula: see text] be a finite alphabet and [Formula: see text] be a pseudovariety of finite forest algebras. A language [Formula: see text] is [Formula: see text]-recognizable if its syntactic forest algebra belongs to [Formula: see text]. Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class. Suppose that a forest language [Formula: see text] and a forest [Formula: see text] are given. We want to find if there exists any proof for that [Formula: see text] does not belong to [Formula: see text] just by using [Formula: see text]-recognizable languages, i.e. given such [Formula: see text] and [Formula: see text], if there exists a [Formula: see text]-recognizable language [Formula: see text] which contains [Formula: see text] and does not contain [Formula: see text]. In this paper, we present how one can use profinite forest algebra to separate a forest language and a forest term and also to separate two forest languages.

scholarly journals Ascent sequences and Fibonacci numbers

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 703-712
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Functional Equation ◽  
Kernel Method ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Pattern Avoidance ◽  
Combinatorial Structures ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations ◽  
Difficult Cases ◽  
Combinatorial Methods

An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets and a variety of other combinatorial structures. Let Fn denote the Fibonacci sequence given by the recurrence Fn = Fn-1 + Fn-2 if n ? 2, with F0 = 0 and F1 = 1. In this paper, we draw connections between ascent sequences and the Fibonacci numbers by showing that several pattern-avoidance classes of ascent sequences are enumerated by either Fn+1 or F2n-1. We make use of both algebraic and combinatorial methods to establish our results. In one of the apparently more difficult cases, we make use of the kernel method to solve a functional equation and thus determine the distribution of some statistics on the avoidance class in question. In two other cases, we adapt the scanning-elements algorithm, a technique which has been used in the enumeration of certain classes of pattern-avoiding permutations, to the comparable problem concerning pattern-avoiding ascent sequences.

scholarly journals $2\times 2$ monotone grid classes are finitely based

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Michael Albert ◽  
Robert Brignall
Keyword(s):  
Computer Science ◽  
General Result ◽  
Theoretical Computer Science ◽  
Discrete Mathematics ◽  
Finite Collection ◽  
Permutation Patterns ◽  
Finitely Based ◽  
Special Issue ◽  
Theoretical Computer

In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalized) grid classes having two monotone cells in the same row. Comment: 10 pages, 5 figures. To appear in Discrete Mathematics and Theoretical Computer Science, special issue for Permutation Patterns 2015

scholarly journals On the structure of periodic complex Horadam orbits

2016 ◽  
Vol 32 (1) ◽  
pp. 29-36
Author(s):  
OVIDIU D. BAGDASAR ◽  
◽  
PETER J. LARCOMBE ◽  
ASHIQ ANJUM ◽  
◽  
...  
Keyword(s):  
Random Number ◽  
Fibonacci Numbers ◽  
Natural Extension ◽  
Fibonacci Sequence ◽  
Random Number Generators ◽  
Geometric Patterns ◽  
Recurrent Sequences ◽  
Star Polygons ◽  
Parameter Values

Numerous geometric patterns identified in nature, art or science can be generated from recurrent sequences, such as for example certain fractals or Fermat’s spiral. Fibonacci numbers in particular have been used to design search techniques, pseudo random-number generators and data structures. Complex Horadam sequences are a natural extension of Fibonacci sequence to complex numbers, involving four parameters (two initial values and two in the defining recursion), therefore successive sequence terms can be visualized in the complex plane. Here, a classification of the periodic orbits is proposed, based on divisibility relations between orders of generators (roots of the characteristic polynomial). Regular star polygons, bipartite graphs and multi-symmetric patterns can be recovered for selected parameter values. Some applications are also suggested.

A Pagination Method for Indexes in Metric Databases

2009 ◽  
pp. 728-736
Author(s):  
Ana Valeria Villegas ◽  
Carina Mabel Ruano ◽  
Norma Edith Herrera
Keyword(s):  
Computer Science ◽  
Similarity Search ◽  
Natural Extension ◽  
Data Types ◽  
Vast Number ◽  
Proximity Search ◽  
Metric Databases

Searching for database elements that are close or similar to a given query element is a problem that has a vast number of applications in many branches of computer science, and it is known as proximity search or similarity search. This type of search is a natural extension of the exact searching due to the fact that the databases have included the ability to store new data types such as images, sound, text, and so forth.

scholarly journals Combinatorial Identities with Generalized Higher-order Genocchi Sequences

2019 ◽  
Vol 12 (2) ◽  
pp. 605-621
Author(s):  
Tian Hao ◽  
Wuyungaowa Bao
Keyword(s):  
Probabilistic Method ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Higher Order ◽  
Harmonic Numbers ◽  
Bell Numbers ◽  
Genocchi Numbers ◽  
Genocchi Polynomials ◽  
Cauchy Numbers ◽  
The Moment

In this paper, we make use of the probabilistic method to calculate the moment representation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characteriza tions and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established. 

The Broad Applications of the Scanning Electron Microscope

1969 ◽  
Vol 27 ◽  
pp. 20-21
Author(s):  
C. T. Nightingale ◽  
S. E. Summers ◽  
T. P. Turnbull
Keyword(s):  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
Light Microscopy ◽  
Surface Topography ◽  
Great Depth ◽  
Resolving Power ◽  
Depth Of Field ◽  
Pictorial Representations ◽  
Scanning Electron

The ease of operation of the scanning electron microscope has insured its wide application in medicine and industry. The micrographs are pictorial representations of surface topography obtained directly from the specimen. The need to replicate is eliminated. The great depth of field and the high resolving power provide far more information than light microscopy.

Visualization of Internal Skin Structures with the Scanning Electron Microscope

1969 ◽  
Vol 27 ◽  
pp. 46-47
Author(s):  
Emil Bernstein
Keyword(s):  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
Great Depth ◽  
Hair Follicles ◽  
Depth Of Field ◽  
Pig Skin ◽  
Realistic Picture ◽  
Main Components ◽  
Scanning Electron

An interesting method for examining structures in g. pig skin has been developed. By modifying an existing technique for splitting skin into its two main components—epidermis and dermis—we can in effect create new surfaces which can be examined with the scanning electron microscope (SEM). Although this method is not offered as a complete substitute for sectioning, it provides the investigator with a means for examining certain structures such as hair follicles and glands intact. The great depth of field of the SEM complements the technique so that a very “realistic” picture of the organ is obtained.

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