These objects have naturally led us to identify and study integer sequences and triangles, such as powers of two, binomial coefficients, and Fibonacci numbers.
One aspect of combinatorics and graph theory that makes it have a different feel than courses such as advanced calculus, modern algebra, topology, or geometry is that combinatorics and graph theory tend to initially feel "broad" instead of "deep". So, rather than picking one topic, such as binomial coefficients, and spending half of the course studying these, we will instead survey a broad range of combinatorial objects. Each family of combinatorial objects that we study will lead us to different integer sequences and triangles, and to different strategies for enumeration. While every subject in Mathematics eventually becomes both broad and deep, there are differences between subjects in how they are perceived by students upon a first encounter.
Recall the Table from MacLane’s book given in Figure 1.2.1. Which of the items in the table have we been engaged with, either consciously or subconsciously, so far?
Recall the picture representing Tall’s Three Worlds framework given in Figure 1.2.2. Where in these three worlds of mathematics have we been situated so far with the math we have done?
