I will keep here miscellaneous notes related to material in the book.
As with many things in mathematics, Faà di Bruno was not the first person to state "Faà di Bruno's formula" (page 49 in the book). It seems that the credit should go to L. F. A. Arbogast, in his 1800 book Traité du Calcul des Dérivations. See Alex D. D. Craik, Prehistory of Faà di Bruno's Formula, American Mathematical Monthly 112 (2005), 119–130.
Craik suggests that Charles Babbage and Ada Lovelace knew about Arbogast's work, and were intending to use it in programming the Analytical Engine (based on Lovelace's notes on Menebrea's article on the Engine).
Here is a problem (the answer is known as Honsberger's Theorem). I was introduced to this problem by James East, of the University of Western Sydney.
Given a positive integer n, how many incongruent triangles are there with integer sides and perimeter n?In other words, how many partitions of the integer n into three parts such that the sum of any two parts is greater than the third?