Binomial Identities Research Articles

Proofs of Some Binomial Identities Using the Method of Last Squares

Proofs of Some Binomial Identities Using the Method of Last Squares

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set { 1 , 2 , … , n } , the local direction of each edge ( i j ) is from i to j if i < j . For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence λ = 1 e 1 2 e 2 … of a tree on the vertex set { 1 , 2 , … , n } is a partition of n − 1 . We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a q-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.

Decomposing socioeconomic inequality for binary health outcomes: an improved estimation that does not vary by choice of reference group

BackgroundDecomposition of concentration indices yields useful information regarding the relative importance of various determinants of inequitable health outcomes. But the two estimation approaches to decomposition in current use are not suitable for binary outcomes.FindingsThe paper compares three estimation approaches for decomposition of inequality concentration indices: Ordinary Least Squares (OLS), probit, and the Generalized Linear Model (GLM) binomial distribution and identity link. Data are from the Thai Health and Welfare Survey 2003. The OLS estimates do not take into account the binary nature of the outcome and the probit estimates depend on the choice of reference groups, whereas the GLM binomial identity approach has neither of these problems.ConclusionsThe GLM with binomial distribution and identity link allows the inequality decomposition model to hold, and produces valid estimates of determinants that do not vary according to choice of reference groups. This GLM approach is readily available in standard statistical packages.

Counting $k$-Marked Durfee Symbols

An alternative characterization of $k$-marked Durfee symbols defined by Andrews is given. Some identities involving generating functions of $k$-marked Durfee symbols are proven combinatorially by considering the symbols not individually, but in equivalence classes. Also, a related binomial coefficient identity is obtained in the course.

N-Colour even self-inverse compositions

An n-colour even self-inverse composition is defined as an n-colour selfinverse composition with even parts. In this paper, we get generating functions, explicit formulas and recurrence formulas for n-colour even self-inverse compositions. One new binomial identity is also obtained.

Binomial symmetries inspired by Bruckman's problem

The partial fraction decomposition method is employed to establish two general algebraic identities, which contain consequently several binomial identities and their q-analogues as special cases. 2010 Mathematics Subject Classifications. Primary 05A10; Secondary 05A30. .

An Explicit Evaluation of the Gosper Sum

AbstractIn this paper, we give a new method to derive a binomial series identity discovered by J. M. Borwein and R. Girgensohn.

The Euler Series Transformation and the Binomial Identities of Ljunggren, Munarini and Simons

AbstractWe point out that the curious identity of Simons follows immediately from Euler's series transformation formula and also from an identity due to Ljunggren. We also mention its relation to Legendre's polynomials. At the end we use the generalized Euler series transformation to obtain two recent binomial identities of Munarini.

Generalizations of Milne’s U(n+1)q-binomial theorems

In this paper, we use the generalized Andrews–Askey integral and Milne’s U ( n + 1 ) q -binomial theorems to derive some multiple extensions of the q -binomial identities.

Discovering Binomial Identities with PascGaloisJE

We describe exercises in which students use PascGaloisJE to formulate conjectures about certain binomial identities which hold when the binomial coefficients are interpreted as elements in the cyclic group of integers modulo a prime integer p. In addition to having an appealing visual component, these exercises are open-ended and expose students to the investigative nature of mathematical discovery.

Counting Defective Parking Functions

Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if $k$ drivers fail to park, we have a defective parking function of defect $k$. Let ${\rm cp}(n,m,k)$ be the number of such functions. In this paper, we establish a recurrence relation for the numbers ${\rm cp}(n,m,k)$, and express this as an equation for a three-variable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel's binomial identity. Finally, we compute the asymptotics of ${\rm cp}(n,m,k)$. In particular, for the case $m=n$, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of $n$, is the Rayleigh distribution. On the other hand, in the case $m=\omega(n)$, the probability that all spaces are occupied tends asymptotically to one.

MacWilliams Identity for Codes with the Rank Metric

The MacWilliams identity, which relates the weight distribution of a code to the weight distribution of its dual code, is useful in determining the weight distribution of codes. In this paper, we derive the MacWilliams identity for linear codes with the rank metric, and our identity has a different form than that by Delsarte. Using our MacWilliams identity, we also derive related identities for rank metric codes. These identities parallel the binomial and power moment identities derived for codes with the Hamming metric.

Paint It Black—A Combinatorial Yawp

Have you experienced a “mathematical yawp” lately? (Not sure you want to answer until you know what one is?) Well, the phrase “mathematical yawp” was coined by Francis Su in his James R. Leitzel Lecture at the 2006 MathFest. In essence, a mathematical yawp is one of those “light bulb” or “aha!” moments when a mathematician comes to an understanding of a topic so moving that it is accompanied by a yelp of joy or disbelief. By specialization, a combinatorial yawp is one of those moments achieved while counting. Combinatorial proofs are appreciated for the elegance and/or simplicity of their arguments (see [2]). However, the true (and frequently underappreciated) beauty lies in their power to generalize results. Understanding the components of a mathematical identity in a concrete counting context provides the first clue for exploring natural extensions. Investigating and stretching the role of each parameter in turn, leads to different generalizations—ones that might not be connected without the combinatorial insight. Our yawp occurred while exploring Problem # 11220, proposed by David Beckwith, from the April 2006 issue of the American Mathematical Monthly [1], the innocuous-looking alternating binomial identity below.

Identities involving rational sums by inversion and partial fraction decomposition

Identities appearing recently in: J. L. Diaz-Barrero, J. Gibergans-Baguena P. G. Popescu: Some identities involving rational sums. Appl. Anal. Discrete Math., 1 (2007), 397-402, are treated by inverting them; the resulting sums are evaluated using partial fraction decomposition, following Wenchang Chu: A binomial coefficient identity associated with Beukers’ conjecture on Apery numbers. Electron. J. Combin., 11 (1): Note 15, 3 pp. (electronic), 2004. This approach produces a general formula, not only special cases.

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well.

Lattice paths and [formula omitted]-colour compositions

Extending the ‘walks’ of van Lint and Wilson, we introduce a new kind of weighted lattice paths and show that the number of lattice paths with weight ν + m - 1 ( 0 ⩽ m ⩽ ν - 1 ) equals the number of n-colour compositions of ν . Two new binomial identities with their combinatorial meaning are also obtained.

91.13 A new proof of a curious identity

We give a short WZ-proof of a binomial coefficient identity due to Zhi-Wei Sun. In [3], [2], and [1] the identity m ∑ i=0 (x+m+ 1)(−1) ( x+ y + i m− i )( y + 2i i ) − m ∑ i=0 ( x+ i m− i ) (−4) = (x−m) ( x m ) (1) was proved using generating functions, double recursions and the concept of Riordan Arrays respectively. Here we use the WZ-method to give yet another proof. First we divide both sides of (1) by (x + m + 1) and then try to write the second indefinite sum on the LHS of (1) without the running index m under the summation sign. This is automated in EKHAD in the procedure zeillim and can be run by the command: zeillim(SUMMAND , i,m,M, 0, 0). But we include here the mathematics behind zeillim for the sake of clarity. Consider the indefinite sum

Distribution of Runs and Longest Runs

Exact distributions of run statistics are traditionally obtained using combinatorial methods, which, under certain situations, become very tedious. Run distributions of multiple object systems, although appearing frequently in applications from various fields, such as computational biology, are not commonly used, due in part to the lack of easy-to-use formulas. In this article, a method for evaluating partition functions of lattice models in the field of statistical mechanics is used to develop a systematic method to study various run statistics in multiple object systems. By using particular generating functions for the specified situation under study, many new distributions can be obtained in a unified and coherent way. The method makes it possible to manipulate formulas of run statistics by using binomial identities to obtain more general, yet simpler formulas. To illustrate the applications of the general method, the distributions of the total number of runs and the longest runs are investigated. Novel and general explicit formulas are derived for the distribution and moments of the total number of runs, and simple explicit formulas are derived for the distributions of the longest runs. In addition, some classical run statistics are recovered and generalized in the same unified way. As examples of applications to biological sequence analysis, the run statistics developed using the general method are applied to several protein sequences to examine their global and local features.

Exhaustive Sampling and Related Binomial Identities

There are many situations that involve repeated sampling from the same set of observations. For example, suppose a professor has a test bank of 100 questions for a particular course and randomly chooses 25 of these questions for the final exam each semester. A persistent but not very talented student repeats the course several times. Obviously, the student has no chance of having seen all the questions before taking the course four times. What is the probability that the student will have seen all the questions after k repetitions? That is, what is the probability that the entire test bank will have been exhausted after k repetitions? A more practical example involves drug testing. Suppose, for example that a bicycle race has 100 contestants and consists of several stages where random samples of 20 contestants are taken at each stage and screened for banned substances. If the race has 10 stages, what is the probability that each contestant will be tested for banned substances at least once?

N-Colour self-inverse compositions

MacMahon’s definition of self-inverse composition is extended ton-colour self-inverse composition. This introduces four new sequences which satisfy the same recurrence relation with different initial conditions like the famous Fibonacci and Lucas sequences. For these new sequences explicit formulas, recurrence relations, generating functions and a summation formula are obtained. Two new binomial identities with combinatorial meaning are also given.