§ F1 or Fun : The field with one element
- Many combinatorial phenomena can be recovered as the "limit" of geometric phenomena over the "field with one element", a mathematical mirage.
§ Cardinality ~ Lines
- Consider projective space of dimension over . How many lines are there?
- Note that for each non-zero vector, we get a 'direction'. So there are potential directions.
- See that for any choice of direction , there are "linearly equivalent" directions, given by , , \dots, which are all distinct since field multiplication is a group.
- Thus, we have lines. This is equal to , which is
- If we plug in (study the "field with one element", we recover .
- Thus, "cardinality of a set of size " is the "number of lines of -dimensional projective space over !
- Since is the set of size , it is only natural that is defined to be the lines in . We will abuse notation and conflate with the cardinality, .
§ Permutation ~ Maximal flags
- Recall that a maximal flag is a sequence of subspaces . At each step, the dimension increases by , and we start with dimension . So we pick a line through the origin for . Then we pick a plane through the origin that contains the line through the origin. Said differently, we pick a plane spanned by . And so on.
- How many ways can we pick a line? That's . Now we need to pick another line orthogonal to the first line. So we build the quotient space , which is . Thus picking another line here is . On multiplying all of these, we get .
- In the case of finite sets, this gives us .
§ Combinations ~ Grassmanian
- Recall that a grassmanian consists of dimensional subspaces of an dimensional space.