We are tasked with determining the number of walk functions $f$ from $\{0, \dots, 15\}$ to $\mathbb{Z}$, with the condition that consecutive values of $f(i)$ differ by exactly 1, i.e., $|f(i) - f(i-1)| = 1$ for all $i = 1, \dots, 15$, and $f(0) = 0$.

Problem Restatement:

We need to find how many walk functions exist from $\{0, \dots, 15\}$ to $\mathbb{Z}$, such that:

1. $f(0) = 0$,
2. For each $i = 1, 2, \dots, 15$, we have $|f(i) - f(i-1)| = 1$.

This type of problem is related to counting restricted paths, often referred to as "Dyck paths" or "simple random walks," constrained to the property that each step is either an increment or a decrement by 1.

Solution Outline:

At each step $i$ (for $i = 1, 2, \dots, 15$), the function $f(i)$ can either increase by 1 or decrease by 1 from $f(i-1)$. The function $f(i)$ represents the current position of a walk, and this walk cannot go arbitrarily far from 0; it must stay within a constrained region to ensure it eventually returns to 0.

Step-by-step approach:

1. Representation as a Path Problem:

   • We are essentially dealing with a walk on the integers that starts at 0, and each step is either up (+1) or down (-1), with $f(0) = 0$.
   • A valid walk of this kind must satisfy the condition that the walk does not "drift off to infinity," meaning it remains relatively close to the origin.

2. Interpretation as Dyck Paths:

   • These walks can be interpreted as a form of Dyck paths, which are paths from $(0, 0)$ to $(15, 0)$ on the integer grid, where each step is either an upward step (+1) or a downward step (-1), and the path never goes below the x-axis (i.e., the walk never dips below $f(i) = 0$).

3. Catalan Number:

   • The number of such paths from $(0, 0)$ to $(15, 0)$, staying above the x-axis, is given by the Catalan number.
   • The general formula for the Catalan number $C_n$ is:

   $C_n = \frac{1}{n+1} \binom{2n}{n}$

   In our case, the number of steps is 15, so we need to find $C_7$ (since for 15 steps, we consider 7 pairs of up and down steps):

   $C_7 = \frac{1}{7+1} \binom{14}{7} = \frac{1}{8} \binom{14}{7}$

4. Calculation:

   We calculate $\binom{14}{7}$:

   $\binom{14}{7} = \frac{14!}{7!7!} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 3432$

   Thus:

   $C_7 = \frac{1}{8} \times 3432 = 429$

Final Answer:

The number of walk functions from $\{0, \dots, 15\}$ to $\mathbb{Z}$ with $f(0) = 0$ is $\boxed{429}$.

Would you like further details or clarification on any part of the solution?

Here are 5 related questions to explore further:

1. How does the number of walks change for different values of $n$ in $\{0, \dots, n\}$?
2. What is the relation between this problem and the Fibonacci sequence?
3. Can you generalize the walk function problem to 3D space?
4. How do Catalan numbers appear in other combinatorial problems?
5. What is the asymptotic growth rate of Catalan numbers?

Tip: Catalan numbers frequently appear in problems involving recursive structures, like binary trees and balanced parentheses.