Their endpoints cannot touch, so they pile up. In the case where the width of the space they are dropped on is fixed and finite, they have unit length and are dropped in a random column randomly with uniform probability is related to algebraic problems of random walks on monoids and groups. The height of the heaps then correpond to the length of reduced random words on the generator set of the monoid or group. Write a program that constructs random heaps and computes their heap. Compile data and try to conjecture a formula for the average height of the heaps as a function of the width and the number of segments of the heaps.
Braids are a generalization of permutations in combinatorics.
Write a program that counts, for a fixed number of strings, how many braids there are for each given pair of crossing and torsion numbers. Such grammars consist of an alphabet with terminal and non-terminal letters, and a set of rules. There are many types of grammars, depending on what we allow the rules to be. The combinatorial objects are built much in the same way as a sentence in a human language would be. Learn about Chomsky's theorem.