Towards a Human Proof of Gessel’s Conjecture

We interpret walks in the first quadrant with steps {(1, 1), (1, 0), (−1, 0), (−1,−1)} as a generalization of Dyck words with two sets of letters. Using this language, we give a formal expression for the number of walks using the steps above, beginning and ending at the origin. We give an explicit formula for a restricted class of such words using a correspondence between such words and Dyck paths. This explicit formula is exactly the same as that for the degree of the polynomial satisfied by the square of the area of cyclic n-gons conjectured by Robbins, although the connection is a mystery. Finally we remark on another combinatorial problem in which the same formula appears and argue for the existence of a bijection.