ERIC ED472095: Manipulating Combinatorial Structures.

20 Pages·2000·0.49 MB·English

by  ERIC

DOCUMENT RESUME
ED 472 095
SE 066 382
Labelle, Gilbert
AUTHOR
Manipulating Combinatorial Structures.
TITLE
2000-00-00
PUB DATE
18p.; In: Canadian Mathematics Education Study Group = Groupe
NOTE
Canadien d'Etude en Didactique des Mathematiques. Proceedings
of the Annual Meeting (24th, Montreal, Quebec, Canada, May
26-30, 2000); see SE 066 379.
Non-Classroom (055) -- Speeches/Meeting Papers (150)
Guides
PUB TYPE
EDRS Price MF01/PC01 Plus Postage.
EDRS PRICE
Curriculum Development; Higher Education; Interdisciplinary
DESCRIPTORS
Approach; Mathematical Applications; *Mathematical Formulas;
Mathematical Models; *Mathematics Instruction; Modern
Mathematics; Teaching Methods
Combinatorics
IDENTIFIERS
ABSTRACT
This set of transparencies shows how the manipulation of
combinatorial structures in the context of modern combinatorics can easily
lead to interesting teaching and learning activities at every level of
education from elementary school to university. The transparencies describe:
(1) the importance and relations of combinatorics to science and social
(2) how analytic/algebraic combinatorics is similar to
activities;
(3) basic combinatorial structures with
analytic/algebraic geometry;
(5) power series associated with any
terminology; (4) species of structures;
(6) similarity of operations between power series and
species of structures;
(7) examples illustrating where the
the corresponding species of structures;
structures are manipulated; and (8) general teaching/learning activities.
(KHR)
Reproductions supplied by EDRS are the best that can be made
from the original document.
PERMISSION TO REPRODUCE AND
DISSEMINATE THIS MATERIAL HAS
O
BEEN GRANTED BY
Manipulating Combinatorial Structures
Gilbert Labelle
THE EDUCATIONAL RESOURCES
TO THE
Universite du Quebec a Montreal (UQAM)
INFORMATION CENTER (ERIC)
1
A combinatorial structure is a finite construction made on a finite set of elements. In real-life
situations, combinatorial structures often arise as "skeletons" or "schematic descriptions"
U.S. DEPARTMENT OF EDUCATION
Office of Educational Research and Improvement
of concrete objects. For example, on a road map, the elements can be cities and the finite
EDUCATIONAL RESOURCES INFORMATION
construction can be the various roads joining these cities. Similarly, a diamond can be con-
CENTER (ERIC)
This document has been reproduced as
sidered as a combinatorial structure: the plane facets of the diamond are connected together
received from the person or organization
originating it.
according to certain rules.
Minor changes have been made to
Combinatorics can be defined as the mathematical analysis, classification and enu-
improve reproduction quality.
meration of combinatorial structures. The main purpose of my presentation is to show how
Points of view or opinions stated in this
the manipulation of combinatorial structures, in the context of modern combinatorics, can
document do not necessarily represent
official OERI position cr policy.
easily lead to interesting teaching/learning activities at every level of education: from el-
ementary school to university.
The following pages of these proceedings contain a grayscale version of my (color)
transparencies (two transparencies on each page). I decided to publish directly my trans-
parencies (instead of a standard typed text) for two main reasons: my talk contained a great
amount of figures, which had to be reproduced anyway, and the short sentences in the
transparencies are easy to read and put emphasis on the main points I wanted to stress.
The first two transparencies are preliminary ones and schematically describe: a) the
importance and relations of combinatorics with science/ social activities, b) how analytic/
algebraic combinatorics is similar to analytic/ algebraic geometry. The next 7 transparencies
(numbered 1 to 7) contain drawings showing basic combinatorial structures together with
some terminology. Collecting together similar combinatorial structures give rise to the con-
cept of species of structures (transparencies 8 to 12). A power series is then associated to any
species of structures enabling one to count its structures (transparencies 13 to 15). Each op-
eration on power series (sum, product, substitution, derivation) is reflected by similar opera-
tions on the corresponding species of structures (transparencies 16 to 18). The power of this
correspondence is illustrated on explicit examples (transparencies 18 to 26) where the struc-
tures are manipulated (and counted) using various combinatorial operations. Transparen-
cies 27 to 30 suggest some general teaching/ learning activities (from elementary to advanced
levels) that may arise from these ideas. I hope that the readers will agree that manipulating
combinatorial structures constitutes a good activity to develop the mathematical mind.
References
The main references for the theory of combinatorial species are:
Joyal, A. (1981). Une theorie combinatoire des series formelles, Advances in Mathematics, 42, 1-82.
Bergeron, F., Labelle, G., & Leroux, P. (1998). Combinatorial Species and Tree-like Structures.
Encyclopedia of Math. and its Applications, Vol. 67, Cambridge University Press.
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