Math History/A

29. Combinatorics

 * Key people : Paul Erdös, G.C. Rota, R. Stanley
 * Origins : counting/arrangment, graph theory, algebra/analysis' generating functions

Choosing k objects out of a collection of size n: $$ \frac{n!}{k!(n-k)!} = \binom{n}{k}$$

This is the dual of the Binomial Theorem.

The Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, ... But what is the nth term?

Answer:

$$ \begin{align*} f(x) & = & 1 & + x & +2x^2 & +3x^3 & +5x^4 & + 8x^5 & + \dots \\ x\cdot f(x) & = & & x & +x^2 & +2x^3 & +3x^4 & + 5x^5 & + \dots\\ x^2 \cdot f(x) & = & & & x^2 & +x^3 & + 2x^4 & + 3x^5 & + \dots \end{align*} $$

Therefore, $$f(x) - x\cdot{}f(x) -x^2\cdot{}f(x) = 1$$ and so $$ f(x) = \frac{1}{1-x-x^2}$$.