Exercise 7.2.9

Answers

1.

This is because we drawn the dot diagram in the order such that the length of cycles are decreasing.

2.

We know that

p_{j}

and

r_{i}

are decreasing as

i

and

j

become greater. So

p_{j}

is number of rows who contains more than or equal to

j

dots. Hence

p_{j} = \max {i : r_{i} \geq j} .

Similarly, $r_{i}$ is the number of columns that contains more than or equal to $i$ dots. Hence

r_{i} = \max {j : p + j \geq i} .

3.

It comes from the fact that

p_{j}

decreases.

4.

There is only one way to draw a diagram such that it’s

i

-th row contains exactly

r_{i}

dots. Once the diagram has been determined, those

p_{j}

’s are determined.

jephianlin

2011-06-27 00:00