scholarly journals Counting $k$-Marked Durfee Symbols

10.37236/528 ◽  
2010 ◽  
Vol 18 (1) ◽  
Author(s):  
Kağan Kurşungöz
Keyword(s):  
Generating Functions ◽  
Binomial Coefficient ◽  
Equivalence Classes ◽  
Alternative Characterization ◽  
Binomial Coefficient Identity

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.

scholarly journals Combinatorial Proof of a Curious $q$-Binomial Coefficient Identity

10.37236/462 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Binomial Coefficient ◽  
Combinatorial Proof ◽  
Binomial Coefficient Identity

Using the Algorithm Z developed by Zeilberger, we give a combinatorial proof of the following $q$-binomial coefficient identity $$ \sum_{k=0}^m(-1)^{m-k}{m\brack k}{n+k\brack a}(-xq^a;q)_{n+k-a}q^{{k+1\choose 2}-mk+{a\choose 2}} $$ $$=\sum_{k=0}^n{n\brack k}{m+k\brack a}x^{m+k-a}q^{mn+{k\choose 2}}, $$ which was obtained by Hou and Zeng [European J. Combin. 28 (2007), 214–227].

scholarly journals A General Theory of Wilf-Equivalence for Catalan Structures

10.37236/5629 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
General Theory ◽  
Equivalence Relation ◽  
Generating Functions ◽  
Asymptotic Estimate ◽  
Equivalence Classes ◽  
Catalan Number ◽  
Combinatorial Structures ◽  
Dyck Paths ◽  
Significant Interest ◽  
Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

Classification Trees as Proxies

2015 ◽  
Vol 2 (2) ◽  
pp. 31-44 ◽  
Author(s):  
Anthony Scime ◽  
Nilay Saiya ◽  
Gregg R. Murray ◽  
Steven J. Jurek
Keyword(s):  
Data Analysis ◽  
Classification Trees ◽  
Original Model ◽  
Classification Model ◽  
Model Based ◽  
Alternative Characterization

In data analysis, when data are unattainable, it is common to select a closely related attribute as a proxy. But sometimes substitution of one attribute for another is not sufficient to satisfy the needs of the analysis. In these cases, a classification model based on one dataset can be investigated as a possible proxy for another closely related domain's dataset. If the model's structure is sufficient to classify data from the related domain, the model can be used as a proxy tree. Such a proxy tree also provides an alternative characterization of the related domain. Just as important, if the original model does not successfully classify the related domain data the domains are not as closely related as believed. This paper presents a methodology for evaluating datasets as proxies along with three cases that demonstrate the methodology and the three types of results.

scholarly journals PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS

2015 ◽  
Vol 80 (2) ◽  
pp. 433-449 ◽  
Author(s):  
KEVIN WOODS
Keyword(s):  
Generating Functions ◽  
Counting Function ◽  
Order Theory ◽  
Characteristic Functions ◽  
Presburger Arithmetic ◽  
First Order ◽  
First Order Theory ◽  
Rational Generating Function ◽  
Rational Generating Functions

AbstractPresburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= (p1, . . . ,pn) are a subset of the free variables in a Presburger formula, we can define a counting functiong(p) to be the number of solutions to the formula, for a givenp. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

scholarly journals Structural Transformation, the Mismeasurement of Productivity Growth, and the Cost Disease of Services

2014 ◽  
Vol 104 (11) ◽  
pp. 3635-3667 ◽  
Author(s):  
Alwyn Young
Keyword(s):  
Productivity Growth ◽  
Developed Economies ◽  
Goods And Services ◽  
Employment Share ◽  
Alternative Characterization ◽  
The Difference ◽  
The Cost ◽  
Relative Productivity ◽  
Absolute Advantage

If workers self-select into industries based upon their relative productivity in different tasks, and comparative advantage is aligned with absolute advantage, then the average efficacy of a sector's workforce will be negatively correlated with its employment share. This might explain the difference in the reported productivity growth of contracting goods and expanding services. Instrumenting with defense expenditures, I find the elasticity of worker efficacy with respect to employment shares is substantially negative, albeit estimated imprecisely. The estimates suggest that the view that goods and services have similar productivity growth rates is a plausible alternative characterization of growth in developed economies. (JEL E23, E24, H56, J24, O41, O47)

A New Study of Parikh Matrices Restricted to Terms

2020 ◽  
Vol 31 (05) ◽  
pp. 621-638
Author(s):  
Zi Jing Chern ◽  
K. G. Subramanian ◽  
Azhana Ahmad ◽  
Wen Chean Teh
Keyword(s):  
Equivalence Classes ◽  
Graph Distance ◽  
Rewriting Rules

Parikh matrices as an extension of Parikh vectors are useful tools in arithmetizing words by numbers. This paper presents a further study of Parikh matrices by restricting the corresponding words to terms formed over a signature. Some [Formula: see text]-equivalence preserving rewriting rules for such terms are introduced. A characterization of terms that are only [Formula: see text]-equivalent to themselves is studied for binary signatures. Graphs associated to the equivalence classes of [Formula: see text]-equivalent terms are studied with respect to graph distance. Finally, the preservation of [Formula: see text]-equivalence under the term self-shuffle operator is studied.

Sign in / Sign up

Export Citation Format

Share Document