Playing with Paths: A Brief Tour of Enumerative Combinatorics
by Paul Fijn
Abstract: An introduction to the key ideas of enumerative combinatorics, and the presentation of some novel results relating to sets of paths on the integer lattice. Three main results of particular interest will be discussed:
(i) an involution for enumerating osculating lattice paths;
(ii) a combinatorial proof of a lemma which enables a purely combinatorial method for enumerating non-intersecting paths; and
(iii) a combinatorial proof of another recurrence which, in addition to the previous lemma, enables combinatorial interpretations of product forms.
The result in (i) is the first result for osculating paths which allows an arbitrary number of paths. Previous methods have been found for two or three paths, but none previously known were able to be generalised to the full problem. The results in (ii) and (iii) are the first known wholly combinatorial proofs for so-called `product forms'. These represent the first step in a combinatorial approach to solving the remainder of this very large class of problems.