scholarly journals Enumeration of Derangements with Descents in Prescribed Positions

10.37236/121 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Niklas Eriksen ◽  
Ragnar Freij ◽  
Johan Wästlund
Keyword(s):  
Fixed Point ◽  
Generating Function ◽  
Combinatorial Proof ◽  
Explicit Formulas ◽  
Special Case

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.

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.

scholarly journals The Generating Function of Ternary Trees and Continued Fractions

10.37236/1079 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ira M. Gessel ◽  
Guoce Xin
Keyword(s):  
Generating Function ◽  
Continued Fraction ◽  
Hypergeometric Series ◽  
Combinatorial Proof ◽  
Hankel Determinant ◽  
Simple Method ◽  
Alternating Sign Matrices ◽  
Hankel Determinants ◽  
Fraction Method ◽  
Special Case

Michael Somos conjectured a relation between Hankel determinants whose entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of $3$, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

scholarly journals A Combinatorial Proof of an Identity for the Divisor Generating Function

10.37236/2153 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Masanori Ando
Keyword(s):  
Generating Function ◽  
Combinatorial Proof ◽  
Divisor Function

In this paper, we give combinatorial proofs and new generalizations of $q$-series identities of Dilcher and Uchimura related to divisor function. Some interesting combinatorial results related to partition and arm-length are also presented.

scholarly journals New fixed point theorems for orthogonal $F_m$-contractions in incomplete $m$-metric spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 405-412
Author(s):  
M. Mehmood ◽  
H. Isik ◽  
F. Uddin ◽  
A. Shoaib
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Special Case

In this paper, we introduce the concept of orthogonal $m$-metric spaces and by using $F_m$-contraction in orthogonal $m$-metric spaces, we give the concept of orthogonal $F_m$-contraction (briefly, $\bot_{F_m}$-contraction) and investigate fixed point results for such mappings. Many existing results in the literature appear to be special case of results proved in this paper. An example to support our main results is also mentioned.

A GENERATING FUNCTION APPROACH TO THE SURFACE AREAS OF SOME INTERCONNECTION NETWORKS

2009 ◽  
Vol 10 (03) ◽  
pp. 189-204 ◽  
Author(s):  
EDDIE CHENG ◽  
KE QIU ◽  
ZHIZHANG SHEN
Keyword(s):  
Generating Function ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Direct Consequence ◽  
Alternating Group ◽  
Function Technique ◽  
Explicit Formulas ◽  
Star Graphs ◽  
Surface Areas ◽  
Whitney Number

An important and interesting parameter of an interconnection network is the number of vertices of a specific distance from a specific vertex. This is known as the surface area or the Whitney number of the second kind. In this paper, we give explicit formulas for the surface areas of the (n, k)-star graphs and the arrangement graphs via the generating function technique. As a direct consequence, these formulas will also provide such explicit formulas for the star graphs, the alternating group graphs and the split-stars since these graphs are related to the (n, k)-star graphs and the arrangement graphs. In addition, we derive the average distances for these graphs.

Bounds on the extinction time distribution of a branching process

1974 ◽  
Vol 6 (2) ◽  
pp. 322-335 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Time Distribution ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Expected Time ◽  
Time To Extinction ◽  
Special Case ◽  
Better Than

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) < ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

scholarly journals Enumerating (2+2)-free posets by the number of minimal elements and other statistics

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel
Keyword(s):  
Generating Function ◽  
Minimal Elements ◽  
Nous Montrons ◽  
International Audience ◽  
Ascent Sequences ◽  
Special Case

International audience A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+2)-free posets: $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. We extend this result by finding the generating function for (2+2)-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form $P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$ where $p_n,k$ equals the number of (2+2)-free posets of size $n$ with $k$ minimal elements. Un poset sera dit (2+2)-libre s'il ne contient aucun sous-poset isomorphe à 2+2, l'union disjointe de deux chaînes à deux éléments. Dans un article récent, Bousquet-Mélou et al. ont trouvé, à l'aide de "suites de montées'', la fonction génératrice des nombres de posets (2+2)-libres: c'est $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. Nous étendons ce résultat en trouvant la fonction génératrice des posets (\textrm2+2)-libres rendant compte de quatre statistiques, dont le nombre d'éléments minimaux du poset. Nous montrons aussi que lorsqu'on ne s'intéresse qu'au nombre d'éléments minimaux, notre fonction génératrice assez compliquée peut être simplifiée en$P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$, où $p_n,k$ est le nombre de posets (2+2)-libres de taille $n$ avec $k$ éléments minimaux.

scholarly journals A Study of Properties and Applications of Gamma Distribution

2021 ◽  
Vol 4 (2) ◽  
pp. 52-65
Author(s):  
Eric U. ◽  
Oti M.O.O. ◽  
Francis C.E.
Keyword(s):  
Probability Distribution ◽  
Gamma Distribution ◽  
Generating Function ◽  
Real Life ◽  
Moment Generating Function ◽  
Reliability Theory ◽  
Burn Out ◽  
Electrical Devices ◽  
The Moment ◽  
Special Case

The gamma distribution is one of the continuous distributions; the distributions are very versatile and give useful presentations of many physical situations. They are perhaps the most applied statistical distribution in the area of reliability. In this paper, we present the study of properties and applications of gamma distribution to real life situations such as fitting the gamma distribution into data, burn-out time of electrical devices and reliability theory. The study employs the moment generating function approach and the special case of gamma distribution to show that the gamma distribution is a legitimate continuous probability distribution showing its characteristics.

scholarly journals Iterated nonexpansive mappings in Hilbert spaces

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Bozena Piatek
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Fixed Point Theory ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Normal Structure ◽  
Point Theory ◽  
Affirmative Answer ◽  
Weakly Compact ◽  
Special Case

AbstractIn [T. Dominguez Benavides and E. Llorens-Fuster, Iterated nonexpansive mappings, J. Fixed Point Theory Appl. 20 (2018), no. 3, Paper No. 104, 18 pp.], the authors raised the question about the existence of a fixed point free continuous INEA mapping T defined on a closed convex and bounded subset (or on a weakly compact convex subset) of a Banach space with normal structure. Our main goal is to give the affirmative answer to this problem in the very special case of a Hilbert space.

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