A generalization of the Whitney rank generating function

1993 ◽  
Vol 113 (2) ◽  
pp. 267-280 ◽  
Author(s):  
G. E. Farr
Keyword(s):  
Generating Function ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Natural Generalization ◽  
Chromatic Polynomial ◽  
Weight Enumerator ◽  
Hadamard Transform ◽  
Percolation Probability ◽  
Special Cases

AbstractThe Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.

scholarly journals Cycle Index, Weight Enumerator, and Tutte Polynomial

10.37236/1663 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Peter J. Cameron
Keyword(s):  
Permutation Group ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Weight Enumerator ◽  
Permutation Groups ◽  
Cycle Index ◽  
Transitive Groups ◽  
Index Weight

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a group from a code, the matroid differs only trivially from that which arises directly from the code. In this case, the Tutte polynomial of the code specialises to the weight enumerator (by Greene's Theorem), and hence also to the cycle index. However, in another subclass of IBIS groups, the base-transitive groups, the Tutte polynomial can be derived from the cycle index but not vice versa. I propose a polynomial for IBIS groups which generalises both Tutte polynomial and cycle index.

scholarly journals Flexible models for overdispersed and underdispersed count data

2021 ◽  
Author(s):  
Dexter Cahoy ◽  
Elvira Di Nardo ◽  
Federico Polito
Keyword(s):  
Poisson Distribution ◽  
Count Data ◽  
Hypergeometric Functions ◽  
Natural Generalization ◽  
Model Parameters ◽  
Probability Models ◽  
Limiting Behavior ◽  
Poisson Models ◽  
Special Cases ◽  
Flexible Models

AbstractWithin the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD’s) to allow both overdispersion and underdispersion. Similarly to Kemp’s generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD’s related to a generalization of Mittag–Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

scholarly journals Theory of Generalized Canonical Transformations for Birkhoff Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yi Zhang
Keyword(s):  
Canonical Transformation ◽  
Natural Generalization ◽  
Analytical Mechanics ◽  
Hamiltonian Mechanics ◽  
Canonical Transformations ◽  
Transformation Theory ◽  
Special Cases ◽  
Criterion Equation ◽  
Important Means ◽  
Transformation Formulas

Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and criterion of the generalized canonical transformation for the Birkhoff system are established. Secondly, based on the criterion equation and considering the generating functions of different forms, six generalized canonical transformation formulas are derived. As special cases, the canonical transformation formulas of classical Hamilton’s equations are given. At the end of the paper, two examples are given to illustrate the application of the results.

scholarly journals Note on Dual Symmetric Functions

1931 ◽  
Vol 2 (3) ◽  
pp. 164-167 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Generating Function ◽  
Symmetric Functions ◽  
General Theorem ◽  
Special Cases ◽  
Dual Forms

In an earlier paper, which this note is intended to supplement and in some respects improve, the writer gave a general theorem of duality relating to isobaric determinants with elements Cr and Hr, the elementary and the complete homogeneous symmetric functions of a set of variables. The result was shewn to include as special cases the dual forms of “bi-alternant” symmetric functions given by Jacobi and Naegelsbach, as well as two equivalent forms of isobaric determinant used by MacMahon as a generating function in an important problem of permutations.

A Tutte Polynomial for Maps

2018 ◽  
Vol 27 (6) ◽  
pp. 913-945 ◽  
Author(s):  
ANDREW GOODALL ◽  
THOMAS KRAJEWSKI ◽  
GUUS REGTS ◽  
LLUÍS VENA
Keyword(s):  
Finite Group ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Polynomial Invariant ◽  
Flow Polynomial ◽  
Orientable Surfaces

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

scholarly journals Some relations of interpolated multiple zeta values

2017 ◽  
Vol 28 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Zhonghua Li ◽  
Chen Qin
Keyword(s):  
Generating Function ◽  
Hypergeometric Functions ◽  
Multiple Zeta Values ◽  
Fixed Weight ◽  
Multiple Zeta ◽  
Special Cases ◽  
Zeta Values

In this paper, the extended double shuffle relations for interpolated multiple zeta values (MZVs) are established. As an application, Hoffman’s relations for interpolated MZVs are proved. Furthermore, a generating function for sums of interpolated MZVs of fixed weight, depth and height is represented by hypergeometric functions, and we discuss some special cases.

scholarly journals GIX/MY/1 systems with resident server and generally distributed arrival and service groups

1996 ◽  
Vol 9 (2) ◽  
pp. 159-170
Author(s):  
Alexander Dukhovny
Keyword(s):  
Group Size ◽  
Generating Function ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Queueing Process ◽  
Imbedded Markov Chain ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Service Groups ◽  
Steady State Probabilities

Considered are bulk systems of GI/M/1 type in which the server stands by when it is idle, waits for the first group to arrive if the queue is empty, takes customers up to its capacity and is not available when busy. Distributions of arrival group size and server's capacity are not restricted. The queueing process is analyzed via an augmented imbedded Markov chain. In the general case, the generating function of the steady-state probabilities of the chain is found as a solution of a Riemann boundary value problem. This function is proven to be rational when the generating function of the arrival group size is rational, in which case the solution is given in terms of roots of a characteristic equation. A necessary and sufficient condition of ergodicity is proven in the general case. Several special cases are studied in detail: single arrivals, geometric arrivals, bounded arrivals, and an arrival group with a geometric tail.

scholarly journals Minimal Overlapping Patterns in Colored Permutations

10.37236/2021 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Adrian Duane ◽  
Jeffrey Remmel
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Minimal Length ◽  
Special Cases ◽  
Maximum Packings ◽  
Colored Permutations

A pattern $P$ of length $j$ has the minimal overlapping property if two consecutive occurrences of the pattern can overlap in at most one place, namely, at the end of the first consecutive occurrence of the pattern and at the start of the second consecutive occurrence of the pattern. For patterns $P$ which have the minimal overlapping property, we derive a general formula for the generating function for the number of consecutive occurrences of $P$ in words, permutations and $k$-colored permutations in terms of the number of maximum packings of $P$ which are patterns of minimal length which has $n$ consecutive occurrences of the pattern $P$. Our results have as special cases several results which have appeared in the literature. Another consequence of our results is to prove a conjecture of Elizalde that two permutations $\alpha$ and $\beta$ of size $j$ which have the minimal overlapping property are strongly $c$-Wilf equivalent if $\alpha$ and $\beta$ have the same first and last elements.

scholarly journals The Zeta-G Class: Some Computational and Analytical Aspects

2020 ◽  
Vol 42 ◽  
pp. e111
Author(s):  
Ana Carla Percontini ◽  
Frank Gomes-Silva ◽  
Gauss Moutinho Crdeiro ◽  
Pedro Rafael Marinho
Keyword(s):  
Maximum Likelihood ◽  
Generating Function ◽  
Real Data ◽  
Data Set ◽  
R Language ◽  
New Class ◽  
Mathematical Properties ◽  
Special Cases ◽  
Computational Aspects

We define a new class of distributions with one extra shapeparameter including some special cases. We provide numerical and computational aspects of the new class. We proposefunctions using the \textsf{R} language to fit any distribution in this family to a data set. In addition, such functions are implemented efficientlyusing the library \textsf{Rcpp} that enables the incorporation of the codes \textsf{C++} in \textsf{R} automatically. Some examples are presentedfor using the implemented routines in practice. We derive some mathematical properties of this class including explicit expressionsfor the moments, generating function and mean deviations. We discuss the estimation of the model parametersby maximum likelihood and provide an application to a real data set.

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