scholarly journals Two Permutation Classes related to the Bubble Sort Operator

10.37236/2556 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Marilena Barnabei ◽  
Flavio Bonetti ◽  
Matteo Silimbani
Keyword(s):  
Fixed Point ◽  
Generating Function ◽  
Sorting Algorithm ◽  
Sigma 1 ◽  
Set Of Permutations

We introduce the Dual Bubble Sort operator $\hat{B}$ (a sorting algorithm such that, if $\sigma=\alpha\,1\,\beta$ is a permutation, then $\hat{B}(\sigma)=1\,\alpha\,\hat{B} (\beta)$) and consider the set of permutations sorted by the composition $\hat{B}B$, where $B$ is the classical Bubble Sort operator. We show that this set is a permutation class and we determine the generating function of the descent and fixed point distributions over this class. Afterwards, we characterize the same distributions over the set of permutations that are sorted by both $\hat{B}^2$ and $B^2$.

A fixed-point property for Galton–Watson processes with generation dependence

1981 ◽  
Vol 18 (02) ◽  
pp. 514-519
Author(s):  
Dean H. Fearn
Keyword(s):  
Fixed Point ◽  
Generating Function ◽  
Probability Generating Function ◽  
Fixed Point Property ◽  
Point Property

Conditions for the non-sure extinction of Galton-Watson processes with generation dependence are obtained. Also a condition is given for such processes to have a strictly increasing probability generating function.

The Translation Planes of Dempwolff

1981 ◽  
Vol 33 (5) ◽  
pp. 1060-1073 ◽  
Author(s):  
N. L. Johnson
Keyword(s):  
Fixed Point ◽  
Vector Space ◽  
Translation Plane ◽  
Affine Plane ◽  
Translation Planes ◽  
Linear Transformations ◽  
Set Of Permutations ◽  
Transitive Set ◽  
Set Of Points

In [2], Dempwolff constructs three translation planes of order 16 using sharply 2-transitive sets of permutations in S16. That is, if acting on Λ is a sharply 2-transitive set of permutations then an affine plane of order n may be defined as follows: The set of points = {(x, y)|x, y ∊ Λ} and the lines = {(x, y)|y = xg for fixed }, {(x, y)|x = c}, {(x, y)|y = c} for c ∊ Λ.Let V be a vector space of dimension k over F ≅ GF(pr). A translation plane may be defined by finding a set M of pkr – 1 linear transformations such that xy–l is fixed point free on for all x ≠ y in M.Notice that if we allow V to act on itself then MV is a sharply 2-transitive set on V if and only if xy–1 is fixed point free on for all x ≠ y in M.

A fixed-point property for Galton–Watson processes with generation dependence

1981 ◽  
Vol 18 (2) ◽  
pp. 514-519 ◽  
Author(s):  
Dean H. Fearn
Keyword(s):  
Fixed Point ◽  
Generating Function ◽  
Probability Generating Function ◽  
Fixed Point Property ◽  
Point Property

Conditions for the non-sure extinction of Galton-Watson processes with generation dependence are obtained. Also a condition is given for such processes to have a strictly increasing probability generating function.

Fixed points of a generalized smoothing transformation and applications to the branching random walk

1998 ◽  
Vol 30 (01) ◽  
pp. 85-112 ◽  
Author(s):  
Quansheng Liu
Keyword(s):  
Fixed Point ◽  
Random Walk ◽  
Fixed Points ◽  
Branching Processes ◽  
Random Variables ◽  
Initial Distribution ◽  
Branching Random Walk ◽  
Independent Random Variables ◽  
Sorting Algorithm ◽  
Hausdorff Measures

Let {Ai:i≥ 1} be a sequence of non-negative random variables and letMbe the class of all probability measures on [0,∞]. Define a transformationTonMby lettingTμ be the distribution of ∑i=1∞AiZi, where theZiare independent random variables with distribution μ, which are also independent of {Ai}. Under first moment assumptions imposed on {Ai}, we determine exactly whenThas a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + ε, ε > 0, we findallthe fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (byT) with an appropriate initial distribution; convergence to the trivial fixed points δ0and δ∞is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.

scholarly journals Wilf Classes of Pairs of Permutations of Length 4

10.37236/1922 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian Le
Keyword(s):  
Generating Function ◽  
Equivalence Classes ◽  
Order Relations ◽  
Set Of Permutations ◽  
Wilf Equivalence

$S_n(\pi_1,\pi_2,\dots, \pi_r)$ denotes the set of permutations of length $n$ that have no subsequence with the same order relations as any of the $\pi_i$. In this paper we show that $|S_n(1342,2143)|=|S_n(3142,2341)|$ and $|S_n(1342,3124)|=|S_n(1243,2134)|$. These two facts complete the classification of Wilf-equivalence classes for pairs of permutations of length four. In both instances we exhibit bijections between the sets using the idea of a "block", and in the former we find a generating function for $|S_n(1342,2143)|$.

scholarly journals POSITIVE SOLUTIONS IN AN ANNULUS FOR NONLINEAR DIFFERENTIAL EQUATIONS ON A MEASURE CHAIN

1999 ◽  
Vol 30 (3) ◽  
pp. 231-240
Author(s):  
CHUAN-JEN CHYAN ◽  
JOHNNY HENDERSON ◽  
HUI-CHUN LO
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Positive Solutions ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Positive Function ◽  
Existence Of Positive Solutions ◽  
Sigma 1

We study the existence of positive solutions of the second order differential equation in an annulus on a measure chain, $u^{\Delta\Delta}(t) + f(u(\sigma(t))) = 0$, $t \in [0, 1]$, satisfying the boundary conditions, $\alpha y(0)-\beta y^\Delta(0)=0$ and $\gamma y(\sigma(1))+\delta y^\Delta ((1))=0$, where $f$ is a positive function and $f(x)$ is sublinear (respectively supcrlinear) at $x = 0$ and is superlinear (respectively sublinear) at $x = \infty$· The methods involve applications of a fixed point theorem for operators on a cone in a Banach space.

A simple sorting algorithm for compositions

2020 ◽  
Vol 12 (06) ◽  
pp. 2050079
Author(s):  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Sorting Algorithm ◽  
Arbitrary Composition ◽  
Transport Industry ◽  
Before And After ◽  
Underlying Geometry ◽  
Number Of Cells ◽  
Natural Way

We provide a particular measure for the degree to which an arbitrary composition deviates from increasing sorted order. The application of such a measure to the transport industry is given in the introduction. In order to obtain this measure, we define a statistic called the number of pushes in an arbitrary composition (which is required to produce sorted order) and obtain a generating function for this. The concept of a push is a geometrical one and leads naturally to several dependant concepts which are investigated. These are the number of cells which do not move in the pushing process and the number of cells that coincide before and after the pushing process (a number not less than those that do not move). The concept of a push leads to combining certain single pushes in a natural way which we define as a frictionless push. A generating function for these is also developed. The underlying geometry of the process also leads naturally to counting the largest first component of arbitrary compositions that are already in a sorted order. We provide a generating function for this.

scholarly journals On the Number of Walks in a Triangular Domain

10.37236/4125 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Paul R.G. Mortimer ◽  
Thomas Prellberg
Keyword(s):  
Fixed Point ◽  
Explicit Formula ◽  
Generating Function ◽  
Triangular Lattice ◽  
Triangular Domain ◽  
Finite Height ◽  
Central Result ◽  
Motzkin Paths

We consider walks on a triangular domain that is a subset of the triangular lattice. We then specialise this by dividing the lattice into two directed sublattices with different weights. Our central result is an explicit formula for the generating function of walks starting at a fixed point in this domain and ending anywhere within the domain. Intriguingly, the specialisation of this formula to walks starting in a fixed corner of the triangle shows that these are equinumerous to two-coloured Motzkin paths, and two-coloured three-candidate Ballot paths, in a strip of finite height.

scholarly journals Some Remarks on the Joint Distribution of Descents and Inverse Descents

10.37236/2135 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Mirkó Visontai
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Nonnegative Integer ◽  
Joint Distribution ◽  
Type B ◽  
Integer Coefficients ◽  
Set Of Permutations ◽  
Combinatorial Model

We study the joint distribution of descents and inverse descents over the set of permutations of $n$ letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients in question but are unable to settle the conjecture.  We examine generalizations of the conjecture and obtain a type $B$ analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model for the joint distribution of descents and inverse descents in terms of statistics on inversion sequences.

scholarly journals Enumeration of derangements with descents in prescribed positions

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Niklas Eriksen ◽  
Ragnar Freij ◽  
Johan Wästlund
Keyword(s):  
Fixed Point ◽  
Generating Function ◽  
Combinatorial Proof ◽  
Explicit Formulas ◽  
International Audience ◽  
Special Case

International audience We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated.

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