math-combo-01

# Counting Idempotent Self-Maps

Let $A = \{1, 2, 3, 4, 5\}$. Count the number of functions $f : A \to A$ that satisfy
$$f(f(x)) = f(x) \qquad \text{for every } x \in A.$$

(Such a function acts as the identity on its own range.)

Give your answer as a single integer.

Alright, I have a problem to solve:

**Problem Statement:**
Let \( A = \{1, 2, 3, 4, 5\} \). Count the number of functions \( f : A \to A \) that satisfy \( f(f(x)) = f(x) \) for every \( x \in A \).

### Understanding the Problem
First, I need to understand what's being asked. We need to find all functions from the set \( A \) to itself such that when you apply the function twice, it's the same as applying it once. 

### Rewriting the Condition
The condition \( f(f(x)) = f(x) \) is known as the idempotent condition. It's similar to the property of projection operators in linear algebra, but here we're dealing with functions.

### Understanding Idempotent Functions
An idempotent function \( f \) has the property that for any \( x \) in its domain, \( f \) applied to \( f(x) \) gives back \( f(x) \). 

This means that \( f \) acts as the identity on its image. In other words, the image of \( f \) is a subset of \( A \) where \( f \) is the identity function.

### Approach to the Solution
I recall that for any function \( f \) to be idempotent, it must satisfy \( f \circ f = f \). 

But how can I count all such functions?

### Step 1: Understand the Structure of Idempotent Functions
An idempotent function \( f \) must satisfy \( f \) is the identity on its image. 

- **Image of \( f \)**: Let's say the image of \( f \) is a subset \( B \) of \( A \). Since \( f \) is idempotent, \( f \) must be the identity on \( B \). 

But what about the elements not in \( B \)? For any \( x \) not in \( B \), \( f(x) \) must be in \( B \) because \( B \) is the image of \( f \).

### Step 2: Choose the Image \( B \)
The image \( B \) can be any subset of \( A \). The size of \( B \) can range from 1 to 5.

For each possible size \( k \) of \( B \), we need to:
1. Choose \( k \) elements from \( A \) to be the image \( B \).
2. Define \( f \) to be the identity on \( B \).
3. For the remaining \( 5 - k \) elements, \( f \) must map them to any element in \( B \).

### Step 3: Counting for Each \( k \)
#### Case \( k = 1 \)
- **Choose \( B \)**: There are \( \binom{5}{1} = 5 \) ways to choose \( B \).
- **Define \( f \)**: For the remaining 4 elements, \( f \) must map them to the single element in \( B \).
  - Number of functions: \( 1^4 = 1 \) (since there's only one choice for each of the 4 elements).

Total for \( k=1 \): \( 5 \times 1 = 5 \).

#### Case \( k = 2 \)
- **Choose \( B \)**: \( \binom{5}{2} = 10 \) ways.
- **Define \( f \)**: 
  - For the 2 elements in \( B \), \( f \) must be the identity.
  - For the remaining 3 elements, each must map to any of the 2 elements in \( B \).
    - Number of functions: \( 2^3 = 8 \).

Total for \( k=2 \): \( 10 \times 8 = 80 \).

#### Case \( k = 3 \)
- **Choose \( B \)**: \( \binom{5}{3} = 10 \) ways.
- **Define \( f \)**:
  - For the 3 elements in \( B \), \( f \) is the identity.
  - For the remaining 2 elements, each must map to any of the 3 elements in \( B \).
    - Number of functions: \( 3^2 = 9 \).

Total for \( k=3 \): \( 10 \times 9 = 90 \).

#### Case \( k = 4 \)
- **Choose \( B \)**: \( \binom{5}{4} = 5 \) ways.
- **Define \( f \)**:
  - For the 4 elements in \( B \), \( f \) is the identity.
  - For the remaining 1 element, it must map to any of the 4 elements in \( B \).
    - Number of functions: \( 4^1 = 4 \).

Total for \( k=4 \): \( 5 \times 4 = 20 \).

#### Case \( k = 5 \)
- **Choose \( B \)**: \( \binom{5}{5} = 1 \) way.
- **Define \( f \)**:
  - For all 5 elements, \( f \) must be the identity.
    - Number of functions: \( 5^0 = 1 \) (since there are no elements outside \( B \)).

Total for \( k=5 \): \( 1 \times 1 = 1 \).

### Summing Up All Cases
Now, add up the totals from all cases:
- \( k=1 \): 5
- \( k=2 \): 80
- \( k=3 \): 90
- \( k=4 \): 20
- \( k=5 \): 1

Total number of idempotent functions = \( 5 + 80 + 90 + 20 + 1 = 196 \).

### Verification
I recall that for a finite set \( A \) with \( n \) elements, the number of idempotent functions is given by the sum of t