• Title, Summary, Keyword: Combinatorics

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The Origin of Combinatorics (조합수학의 유래)

  • Ree, Sang-Wook;Koh, Young-Mee
    • Journal for History of Mathematics
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    • v.20 no.4
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    • pp.61-70
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    • 2007
  • Combinatorics, often called the 21 st century mathematics, has turned out a very important subject for the present information era. Modern combinatorics has started from some mathematical works, for example, Pascal's triangle and the binomial coefficients, and Euler's problems on the partitions of integers and Konigsberg's bridge problem, and so on. In this paper, we investigate the origin of combinatorics by looking over some interesting ancient combinatorial problems and some important problems which have started various subfields of combinatorics. We also discuss a little on the role of combinatorics in mathematics and mathematics education.

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The Mediation of Embodied Symbol on Combinatorial Thinking

  • Cho, Han-Hyuk;Lee, Ji-Yoon;Lee, Hyo-Myung
    • Research in Mathematical Education
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    • v.16 no.1
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    • pp.79-90
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    • 2012
  • This research investigated if the embodied symbol using a turtle metaphor in a microworld environment works as a cognitive tool to mediate the learning of combinatorics. It was found that students were able to not only count the number of cases systematically by using the embodied symbols in a situated problem regarding Permutation and Combination, but also find the rules and infer a concept of Combination through the activities manipulating the symbols. Therefore, we concluded that the embodied symbol, as a bridge that connects learners' concrete experiences with abstract mathematical concepts, can be applied to introduction of various mathematical concepts as well as a combinatorics concept.

Epistemological Obstacles on Learning the Product Rule and the Sum Rule of Combinatorics (조합문제에서의 인식론적 장애 -곱의 법칙과 합의 법칙 중심으로-)

  • Kim, Suh-Ryung;Park, Hye-Sook;Kim, Wan-Soon
    • The Mathematical Education
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    • v.46 no.2
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    • pp.193-205
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    • 2007
  • In this paper, we focus on the product rule and sum rule which are considered as the most fundamental counting tools of Combinatorics. Despite of the importance of these rules in both educational and social aspects, they are taught superficially in class. We take the survey through both internet and questionaire to investigate how thoroughly students understand the rules. Then we discuss about the results of the survey and suggest effective teaching methods to improve students' understanding of these rules.

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ON THE MINIMUM WEIGHT OF A 3-CONNECTED 1-PLANAR GRAPH

  • Lu, Zai Ping;Song, Ning
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.763-787
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    • 2017
  • A graph is called 1-planar if it can be drawn in the Euclidean plane ${\mathbb{R}}^2$ such that each edge is crossed by at most one other edge. The weight of an edge is the sum of degrees of two ends. It is known that every planar graph of minimum degree ${\delta}{\geq}3$ has an edge with weight at most 13. In the present paper, we show the existence of edges with weight at most 25 in 3-connected 1-planar graphs.

Counting is an important ingredient of mathematics education (조합수학의 수학교육 내용요소로서의 적합성과 필요성)

  • Koh, Youngmee;Ree, Sangwook
    • Journal for History of Mathematics
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    • v.29 no.5
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    • pp.267-278
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    • 2016
  • Mathematics is a kind of language, and even a tool of cognition for human beings. Mathematics has been used to communicate and to develop the civilizations through the history. So mathematics is one of the most important subjects for human to teach and learn. Especially, developed countries believe that mathematics will play very important roles in the developments of future industries and so future society. In this article, we clarify that combinatorics which is mainly represented by counting is an important ingredient of future mathematics education. To do so, we investigate the characteristics of combinatorics from the educational and cognitive perspectives.

Eye Movements in Understanding Combinatorial Problems (순열 조합 이해 과제에서의 안구 운동 추적 연구)

  • Choi, In Yong;Cho, Han Hyuk
    • Journal of Educational Research in Mathematics
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    • v.26 no.4
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    • pp.635-662
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    • 2016
  • Combinatorics, the basis of probabilistic thinking, is an important area of mathematics and closely linked with other subjects such as informatics and STEAM areas. But combinatorics is one of the most difficult units in school mathematics for leaning and teaching. This study, using the designed combinatorial models and executable expression, aims to analyzes the eye movement of graduate students when they translate the written combinatorial problems to the corresponding executable expression, and examines not only the understanding process of the written combinatorial sentences but also the degree of difficulties depending on the combinatorial semantic structures. The result of the study shows that there are two types of solving process the participants take when they solve the problems : one is to choose the right executable expression by comparing the sentence and the executable expression frequently. The other approach is to find the corresponding executable expression after they derive the suitable mental model by translating the combinatorial sentence. We found the cognitive processing patterns of the participants how they pay attention to words and numbers related to the essential informations hidden in the sentence. Also we found that the student's eyes rest upon the essential combinatorial sentences and executable expressions longer and they perform the complicated cognitive handling process such as comparing the written sentence with executable expressions when they try the problems whose meaning structure is rarely used in the school mathematics. The data of eye movement provide meaningful information for analyzing the cognitive process related to the solving process of the participants.

PRODUCTS ON THE CHOW RINGS FOR TORIC VARIETIES

  • Park, Hye-Sook
    • Journal of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.469-479
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    • 1996
  • Toric variety is a normal algebraic variety containing algebraic torus $T_N$ as an open dense subset with an algebraic action of $T_N$ which is an extension of the group law of $T_N$. A toric variety can be described in terms of a certain collection, which is called a fan, of cones. From this fact, the properties of a toric variety have strong connection with the combinatorial structure of the corresponding fan and the relations among the generators. That is, we can translate the diffcult algebrogeometric properties of toric varieties into very simple properties about the combinatorics of cones in affine spaces over the reals.

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ON COMBINATORICS OF KONHAUSER POLYNOMIALS

  • Kim, Dong-Su
    • Journal of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.423-438
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    • 1996
  • Let L be a linear functional on the vector space of polynomials in x. Let $\omega(x)$ be a polynomial in x of degree d, for some positive integer d. We consider two sets of polynomials, ${R_n (x)}_{n \geq 0}, {S_n(x)}_{n \geq 0}$, such that $R_n(x)$ is a polynomial in x of degree n and $S_n(x)$ is a polynomial in $\omega(x)$ of degree n. (So $S_n(x)$ is a polynomial in x of degree dn.)

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FOCK SPACE REPRESENTATIONS OF QUANTUM AFFINE ALGEBRAS AND GENERALIZED LASCOUX-LECLERC-THIBON ALGORITHM

  • Kang, Seok-Jin;Kwon, Jae-Hoon
    • Journal of the Korean Mathematical Society
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    • v.45 no.4
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    • pp.1135-1202
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    • 2008
  • We construct the Fock space representations of classical quantum affine algebras using combinatorics of Young walls. We also show that the crystal graphs of the Fock space representations can be realized as the crystal consisting of proper Young walls. Finally, we give a generalized version of Lascoux-Leclerc-Thibon algorithm for computing the global bases of the basic representations of classical quantum affine algebras.

A CONSTRUCTION OF ONE-FACTORIZATION

  • Choi, Yoon-Young;Kim, Sang-Mok;Lim, Seon-Ju;Park, Bong-Joo
    • Journal of the Korean Mathematical Society
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    • v.45 no.5
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    • pp.1243-1253
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    • 2008
  • In this paper, we construct one-factorizations of given complete graphs of even order. These constructions partition the edges of the complete graph into one-factors and triples. Our new constructions of one-factors and triples can be applied to a recursive construction of Steiner triple systems for all possible orders ${\geq}$15.