16.3.2. Categories of Strings

Let $s$ denote my function which is the number of valid sequences ending with $n$. A trivial recurrence condition for $s$ is $s = s + s + \ldots + s$.

This section introduces a function called $s$, which counts the number of valid sequences ending with a specific number (denoted as $n$). It presents a foundational recurrence relation, suggesting that the total number of such sequences can be calculated by summing up sequences that end with various previous numbers. The idea is that if you know how many sequences end with each number, you can combine them to find sequences that end with the current number.

Imagine a line of dominoes, where each domino represents a number in the sequence. If a domino can only fall or connect to its neighbor, the total count of “fallen” dominoes (valid sequences) can be determined by adding all the ways the last domino can connect to earlier ones.

The second last value in the sequence, namely $m$, can be 1. If that is the case, then the valid sequences starting and ending with 1 give one category of sequences. Similarly, if $m$ can be 2, you find all valid sequences starting with 1 and ending with 2.

This chunk discusses how the second last number in a valid sequence can influence what types of sequences are possible. For instance, if the second last number is 1, we create valid sequences that begin and end with 1. If it changes to 2, we look for sequences that start with 1 and end with 2. This illustrates how changing just one number can create different categories of valid sequences, contributing to calculating the total number of valid sequences.

Think of these valid sequences as routes connecting different cities where the second last city must lead to a destination. If the penultimate stop is city A, you can only reach city B by going directly from A to B, ensuring only specific routes are valid based on your choices.

All categories of sequences discussed are disjoint. The goal is to derive a more compact recurrence equation. The degree of the prior equation depends on multiple previous values.

Category 1 sequences occupy the second last position with $n - 1$, while Category 2 allows for other values.

This part emphasizes that the categories of sequences are distinct from each other; no valid sequence can fit more than one category at once. This lays the groundwork for establishing a more efficient (compact) mathematical model that counts these sequences. It also points out how these categories can be organized into a clearer structure based on which numbers are allowed in specific positions.

Consider different genres of movies. You can categorize them into action, comedy, and drama; each film can only fit into one genre. This is similar to how valid sequences work, as each one belongs to a specific category based on its characteristics.

The alternate recurrence equation derived states: $s = 2 \times s$. This is an equation of degree 1, meaning it simplifies the computation of the sequence.

Here, we arrive at a new, more efficient recurrence relation stating that the function $s$ can be derived from just the previous value of $s$. By refining the calculation to rely solely on this preceding term, it simplifies the problem significantly, making it easier to compute future values.

Think of a recipe for a dish where the amount of a key ingredient (like sugar) doubles with each batch. Rather than trying to remember the total amount used in several previous batches, you only need to know the last batch to calculate the sugar needed for the next, simplifying the whole cooking process.

The recurrence relation requires two initial conditions: $s(1) = 1$ and $s(2) = 1$. These conditions are crucial for the function to start correctly. With larger values of $n$, we get more varying sequences.

The necessity of initial conditions is highlighted. For instance, when $n = 1$, there’s only one valid sequence, which is trivially straight forward. By establishing these initial conditions, it allows the recurrence relation to generate further values accurately, ensuring a solid foundation for all computations.

Think about learning to ride a bicycle. The first thing to learn is how to balance (initial condition), and once you get that, you can easily learn to pedal, steer, and stop (subsequent values of the sequence). The initial skill sets the stage for further development.

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