AMOF: The Amazing Mathematical Object Factory
(formerly known as ECOS)
|Permutations of a Multiset|
AMOF, the Amazing Mathematical Object Factory, shown on the left, produces lists of mathematical objects in response to customer orders. Today you are a customer and you must tell AMOF what you want produced. This factory is totally non-polluting and the objects produced are absolutely free! (There is a vicious rumour however, that the workers are underpaid and overworked.)
Click on a icon with an Indigo background, such as , to learn about that object, or on an icon with a Green background, such as to generate the object. Get it? Indigo = Information, Green = Generate; the first letters are the same! To get back to this page click on the AMOF 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.
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; so these are not combinatorial objects. The best way to learn about combinatorial objects is to study lots of examples of them and AMOF should help you in that study.
There are many, many types of combinatorial objects and we work with just a few of them in AMOF. Some are very simple, like permutations and subsets, and some are quite complicated, like the solutions to pentomino problems.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 and 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 programs 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 AMOF. 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.
- Colorful Mathematics; PC games involving mathematical structures known as graphs; from the University of Calgary.
- Mega-Mathematics; a collection of discrete mathematics activities geared towards a young audience.
- Math Forum's Discrete Math Page
- University of Colorado's Discrete Math Project (DMP)
- Discrete Math in the Classroom; a resource for teachers about the definition of and goals of discrete mathematics in education.
- DIMACS Leadership Program in Discrete Mathematics K-8.
- MacTutor's Mathematical games and recreations, most of which are discrete mathematical in nature.
The (Combinatorial) Object Server (COS) is a WWW site which has the ability to generate (i.e., list) many additional types of combinatorial objects besides those found on AMOF. COS is oriented more towards University students and researchers. However, the ideas behind AMOF (and originally, ECOS) are derived from COS.
Funding made possible through Industry Canada's
and through the cooperation
and the Computer Science Department of the University of Victoria.
The original ECOS site was created by Frank Ruskey and Susan Ruskey. ECOS evolved into "The Amazing Mathematical Object Factory", created by Frank Ruskey and Scott Lausch, 1998. ©Frank Ruskey, 1998.