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Patterns in Pascal's Triangle - with a Twist - More Patterns: PascGalois Triangles

Kathleen M. Shannon and Michael J. Bardzell

We can construct a  PascGalois Triangle -- the next twist on Pascal's Triangle -- by the following steps:

  1. Associate with every element in a finite group a different color.
  2. Place one element of the group down one side of the triangle.
  3. Place another (possibly different) group element down the the other side of the triangle.
  4. Using the group operation (generally referred to as "group multiplication"), generate the interior.

Pascal's triangle mod n for any integer n is one example of PascGalois Triangles, since the integers {0, 1, 2, ...,  - 1} under addition mod n are a finite group. We generally give names to groups so that we can easily refer to them. The set {0, 1, 2, ..., n-1} using addition mod n is called ZnThis group is also called the cyclic group of order n. (Why cyclic?) Remember that we have seen the "multiplication" tables for some of these groups.

Here is another example: Consider the set of ordered pairs {(0,0), (1,0), (0,1), (1,1)} and the operation of combining two members of the set by adding the first components mod 2 to get the first component of the result and adding the second components mod 2 to get the second component of the result. Here is the multiplication table for this operation:

* (0,0) (0,1) (1,0) (1,1)
(0,0) (0,0) (0,1) (1,0) (1,1)
(0,1) (0,1) (0,0) (1,1) (1,0)
(1,0) (1,0) (1,1) (0,0) (0,1)
(1,1) (1,1) (1,0) (0,1) (0,0)

Notice that this set is really like two copies of Z2 pasted next to each other. When two sets are put together to form a new set of ordered pairs in this manner, we call it a cross product. So this example is just the cross product of Z2 with itself, denoted Z2  x Z2. Notice that the operation involved is just the same operation from Z2, performed on each of the elements in the pair separately. In the next section we will see the PascGalois Triangle for this set.

Kathleen M. Shannon and Michael J. Bardzell, "Patterns in Pascal's Triangle - with a Twist - More Patterns: PascGalois Triangles," Convergence (December 2004)