ECOS: The Educational

Combinatorial Object Server

THIS SITE IS NO LONGER MAINTAINED!
Use AMOF instead.

The official version of ECOS is on Canada's Schoolnet site. This site will be updated from time to time. Here is a summary of changes.

Object Type	Information	Generate
Subsets
Combinations
Permutations
8-Queens Problem
Pentominoes

Try It!

Click on a question mark icon to learn about that object or on the star icon to generate the object. To get back to this page click on the ECOS icon , which appears both at the top and at the bottom of all succeeding pages. In addition, the SchoolNet icon appears on every page as well; clicking on it will take you back to the SchoolNet home page.

What is a Combinatorial Object?

Combinatorial objects are everywhere! How many ways are there to make change for $1 using unlimited numbers of coins of all denominations? Each way is a combinatorial object known as a numerical partition. How many ways are there for Alice, Bob, and Carol to line up at the box office at a theatre? Each way is a combinatorial object known as a permutation. How many 5-card poker hands are there with a pair of aces; what is the probability of getting such a hand? To answer this question you need to know about combinatorial objects known as combinations.

But what is a combinatorial object? It's actually very difficult to define. Kind of like love: you know it when you see it, but it's hard to explain. The main feature of such objects is that there is only a finite number of any particular type. There is only a finite number of ways for persons to order themselves in line at a theatre, only a finite number of ways to make change for a dollar. On the other hand temperature does not take on a finite number of values (it could be 25.3315411 degrees), nor does the position of a ball on a pool table. The best way to learn about combinatorial objects is to study lots of examples of them and ECOS should help you in that study.

There are many, many types of combinatorial objects and we work with just a few of them in ECOS. Some are very simple, like permutations and subsets, and some are quite complicated, like the solutions to pentomino problems.

Combinatorial Objects and Discrete Mathematics

Discrete mathematics is that branch of mathematics that studies combinatorial objects. This is an area of mathematics of increasing importance in todays world. It is the mathematics that underlies computers and tele-communication. There is a recent but growing trend towards more discrete mathematics in the K-12 math curriculum. Discrete mathematics became a required part of study for mathematics majors in universities by 1980, and it was soon recommended that ideas of discrete mathematics should be developed earlier in the mathematics curriculum. Inclusion of discrete mathematics in all school mathematics program received the necessary impetus for serious consideration in 1989 with the inclusion of the topic as one of the secondary-level standards in the NCTM's Curriculum and Evaluation Standards for School Mathematics. The NCTM (National Council of Teachers of Mathematics) has since published a book, Discrete Mathematics Across the Curriculum K-12, that contains much material about discrete mathematics and its uses in the curriculum. We'll refer to this book in some of the information pages. Whenever you see [NCTM], we're referring to this book. Another book published by NCTM, Readings for Enrichment in Secondary School Mathematics, contains material on permutations, sets, Pascal's triangle, and other topics used by ECOS. The book Mathematical Activities: A Resource Book for Teachers by the popular author Brian Bolt contains useful material on Pentominoes, Pascal's triangle and many other topics of discrete mathematics.

Here is a list of some sites relevant to discrete mathematics education K-12.

Colorful Mathematics; PC games involving mathematical structures known as graphs.
The Discrete Mathematics chapter of the New Jersey Mathematics Standards and Curriculum Framework document.
DIMACS Leadership Program in Discrete Mathematics K-8.

What is the Combinatorial Object Server?

The (Combinatorial) Object Server (COS) is a WWW site which has the ability to generate (i.e., list) many different types of combinatorial objects. This is a highly interactive site that can be used to learn more about many types of discrete mathematical structures. ECOS is derived from COS.

Funding made possible through Industry Canada's
SchoolNet initiative

and through the cooperation
of the Canadian Mathematical Society.

Future Plans

Currently ECOS just generates a few of the combinatorial objects that are suitable for study at the elementary and secondary school levels. We would like to add the following objects to ECOS:

Numerical partitions. For example, ECOS could generate the different ways to make change from coins; an example mentioned in the first paragraph above.
Fibonacci objects. There are many combinatorial objects counted by the Fibonacci numbers; for example, the number of ways to climb n stairs taking one or two steps at a time.
Magic squares. These are often studied in schools, even by children learning to add. ECOS could generate them exhaustively.
Permutations of a Multiset. A natural extension of both the concept of a permutation and a combination is that of a permutation of a multiset. ECOS could be enhanced to generate permutations of a multiset.

We would like to enhance the existing pages as follows:

For each of the objects listed, Java programs could be written that would generate and and allow the user to manipulate the objects. Particularly useful in this regard would be programs for pentominoes and the Towers of Hanoi puzzle.
Instead of listing at most 200 objects, we would like ECOS to list some user-selectable number of objects, call it N, and then give the option of continuing the generation after the first N objects have be generated to list the next N objects, and then N more and so on, until all objects have been generated or the user has seen enough.
Expand the number of problems for classroom consideration. Provide solutions.