scholarly journals A Note on Naturally Embedded Ternary Trees

10.37236/629 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Markus Kuba
Keyword(s):  
Generating Function ◽  
Counting Problem ◽  
Unique Path ◽  
External Node ◽  
Enumeration Problems ◽  
Ternary Tree ◽  
The Right

In this note we consider ternary trees naturally embedded in the plane in a deterministic way. The root has position zero, or in other words label zero, and the three children of a node with position $j\in\mathbb{Z}$ have positions $j-1$, $j$, and $j+1$. We derive the generating function of embedded ternary trees where all internal nodes have labels less than or equal to $j$, with $j\in\mathbb{N}$. Furthermore, we study the generating function of the number of ternary trees of size $n$ with a given number of internal nodes with label $j$. Moreover, we discuss generalizations of this counting problem to several labels at the same time. We also study a refinement of the depth of the external node of rank $s$, with $0\le s\le 2n$, by keeping track of the left, center, and right steps on the unique path from the root to the external node. The $2n+1$ external nodes of a ternary tree are ranked from the left to the right according to an inorder traversal of the tree. Finally, we discuss generalizations of the considered enumeration problems to embedded $d$-ary trees.

scholarly journals On The Improvement of Combinatorial Mathematics Teaching From Generating Function

2017 ◽  
Vol 10 (2) ◽  
pp. 104
Author(s):  
Cheng Ji
Keyword(s):  
Generating Function ◽  
Mathematics Teaching ◽  
Teaching Improvement ◽  
Counting Problem ◽  
Combination Number ◽  
Combinatorial Counting

Generating function is an effective method to solve combinatorial counting problem, but it most likely to be neglected in combinatorial mathematics teaching. In this paper, we provides a demonstration for combinatorial mathematics teaching improvement by using the generating function solving combination number and the sum of preceding  terms among sequence of numbers.

A note on the equilibrium M/G/1 queue length

1988 ◽  
Vol 25 (1) ◽  
pp. 228-231 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Infinite Divisibility ◽  
Queue Length Distribution ◽  
Service Distribution ◽  
Bulk Arrival ◽  
Finite Sum ◽  
The Right

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.

scholarly journals Infinite Systems of Functional Equations and Gaussian Limiting Distributions

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Bernhard Gittenberger ◽  
Johannes F. Morgenbesser
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Combinatorial Enumeration ◽  
Limiting Distributions ◽  
Infinite Dimensional ◽  
Infinite Systems ◽  
Enumeration Problems ◽  
International Audience

International audience In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution.

scholarly journals Impulse Propagation in Compositions and Words

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Margaret Archibald ◽  
Aubrey Blecher ◽  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Finite Alphabet ◽  
Impulse Propagation ◽  
The Right

We consider compositions of n represented as bargraphs and subject these to repeated impulses which start from the left at the top level and destroy horizontally connected parts. This is repeated while moving to the right first and then downwards to the next row and the statistic of interest is the number of impulses needed to annihilate the whole composition. We achieve this by conceptualizing a generating function that tracks compositions as well as the number of impulses used. This conceptualization is repeated for words (over a finite alphabet) represented by bargraphs.

A note on the equilibrium M/G/1 queue length

1988 ◽  
Vol 25 (01) ◽  
pp. 228-231 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Infinite Divisibility ◽  
Queue Length Distribution ◽  
Service Distribution ◽  
Bulk Arrival ◽  
Finite Sum ◽  
The Right

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.

A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v.It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1]k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant.This result is then used to give a new proof of the exponential limit law.

A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

1978 ◽  
Vol 15 (02) ◽  
pp. 235-242
Author(s):  
Martin I. Goldstein
Keyword(s):  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Uniform Limit ◽  
Second Moment ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
The Right

Let Z(t) ··· (Z 1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v. It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1] k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant. This result is then used to give a new proof of the exponential limit law.

scholarly journals A Fourier-Neumann series and its application to the reduction of triple cosine series

1973 ◽  
Vol 14 (2) ◽  
pp. 198-201 ◽  
Author(s):  
C. J. Tranter ◽  
J. C. Cooke
Keyword(s):  
Generating Function ◽  
Neumann Series ◽  
Definite Integral ◽  
Cosine Series ◽  
Jacobi Expansion ◽  
Image Position ◽  
The Right

The Jacobi expansionis well known and easily obtained from the generating function of the Besselcoefficients. The sum of the series on the right of equation (1) when sin (n+½)x is replaced by cos (n+½)x cannot be found in this way but it can be expressed in terms of a definite integral as shown below. The result so obtained is useful in reducing certain triple cosine series to dual series and so simplifying the solution given by one of us for such series in an earlier paper [1].

Right-angled Artin groups, polyhedral products and the -generating function

2021 ◽  
pp. 1-25
Author(s):  
Jorge Aguilar-Guzmán ◽  
Jesús González ◽  
John Oprea
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Base Space ◽  
Topological Complexity ◽  
Product Spaces ◽  
Polyhedral Product ◽  
Right Angled Artin Groups ◽  
The Right ◽  
The Relationship ◽  
Mac Lane

For a graph $\Gamma$ , let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$ . We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$ , we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$ . Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

scholarly journals Wall-Crossings for Hassett Descendant Potentials

2020 ◽  
Author(s):  
Vance Blankers ◽  
Renzo Cavalieri
Keyword(s):  
Generating Function ◽  
Intersection Theory ◽  
Polynomial Functions ◽  
Intersection Numbers ◽  
Transformation Of Variables ◽  
The Right

Abstract This paper solves the combinatorics relating the intersection theory of $\psi $-classes of Hassett spaces to that of $\overline{{\mathcal{M}}}_{g,n}$. A generating function for intersection numbers of $\psi $-classes on all Hassett spaces is obtained from the Gromov–Witten potential of a point via a non-invertible transformation of variables. When restricting to diagonal weights, the changes of variables are invertible and explicitly described as polynomial functions. Finally, the comparison of potentials is extended to the level of cycles: the pinwheel cycle potential, a generating function for tautological classes of rational tail type on $\overline{{\mathcal{M}}}_{g,n}$ is the right instrument to describe the pull-back to $\overline{{\mathcal{M}}}_{g,n}$ of all monomials of $\psi $-classes on Hassett spaces.

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