Combinatorics Through Guided Discovery, 2004a

14 1. What is Combinatorics?
k =0 1 2 3 4 5 6 7
n =0 1 0 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0
2 1 2 1 0 0 0 0 0
3 1 3 3 1 0 0 0 0
4 1 4 6 4 1 0 0 0
5 1 5 10 10 5 1 0 0
6 1 6 15 20 15 6 1 0
7 1 7 21 35 35 21 7 1
Table 1.2.6: Pascal’s Rectangle
⇒
Problem 32. Because our definition told us that ( n )
k
is 0 when k > n,wegot
a rectangular table of numbers that satisfies the Pascal Equation.
(a) Is there any other way to define ( n )
k
when k > n in order to get a
rectangular table that agrees with Pascal’s Right Triangle for k ≤ n
and satisfies the Pascal Equation? (h)
(b) Suppose we want to extend Pascal’s Rectangle to the left and define
( n )
−k
for n ≥ 0 and k > 0 so that −k < 0. What should we put into row
n and column −k of Pascal’s Rectangle in order for the Pascal Equation
to hold true? (h)
(c) What should we put into row −n and column k or column −k in order
for the Pascal Equation to continue to hold? Do we have any freedom
of choice? (h)
Problem 33. There is yet another bijection that lets us prove that a set of
size n has 2 n subsets. Namely, for each subset S of [n] ={1, 2,...,n}, define
a function (traditionally denoted by χ S ) as follows. a
χ S (i) =
{
1 if i ∈ S
0 if i S
The function χ S is called the characteristic function of S. Notice that the
characteristic function is a function from [n] to {0, 1}.
(a) For practice, consider the function χ {1,3} for the subset {1, 3} of the set
{1, 2, 3, 4}. What are
(i) χ {1,3} (1)?
(ii) χ {1,3} (2)?
(iii) χ {1,3} (3)?
(iv) χ {1,3} (4)?