Multi-dimensional arrays - Multimedia Introduction to Programming Using Java - page 302

Java syntax: Partial-array creation
new type [ int-expression ][]
Example : new int [n][]
Purpose : Create only the first dimension of a two-dimensional
array, with int-expression elements, all null .
Extension : For an n-dimensional array, any of the first dimen-
sions can be created, e.g. new int [5][3][][][] .

a6 a7 a8

2
4
7

1
3

c a6 a7

a8

Now, row 0 of c has 2 elements and row 1 has 3 elements. We call it a ragged
array : a two-dimensional array whose columns have different sizes.
In many cases, each row should have the same number of columns, and the
old method of creating both at the same time is the method to use. However, if
you want to save space by having each column have a different number of ele-
ments, use the new method, as just illustrated.

9.3.2 Pascal's triangle

In order to show one use of ragged arrays, we introduce Pascal's triangle, attrib-
uted to Blaise Pascal, a Swiss mathematician and philosopher, who first discov-
ered this little triangle and its properties. Below, we show Pascal's triangle in the
middle. On the right, we show it as we usually draw a two-dimensional array:

row 0 1 1

row 1 1 1 1 1

row 2 1 2 1 1 2 1

row 3 1 3 3 1 1 3 3 1

row 4 1 4 6 4 1 1 4 6 4 1

row 5 1 5 10 10 5 1 1 5 10 10 5 1

... ... ...

Each row r of this triangle contains r+1 integers. The first and last elements of
each row are 1 . Each other element t[r][c] is the sum of the two elements
above it:

t[r][c] = t[r - 1][c - 1] + t[r - 1][c] ( for 0<r , 0<c<r)

Pascal's triangle and combinatorics
Pascal's triangle is important in the field called combinatorics . The integer