Towards Reducing the HMZ Shuffle Conjecture

In her dissertation, Angela Hicks showed that, to reduce the HMZ shuffle conjecture, it is sufficient to find a certain bijection between two sets of two-part parking functions which, roughly speaking, either swaps the two parts of the parking function or transfers a car from the second part to the first part and which satisfies certain nice properties. I will consider a special case where this bijection, if it exists, must swap the two parts of the parking function, and I will give a simple operation which results in a bijection achieving this swap and satisfying the desired properties. 1 Background and Notations First we define Dyck paths. Definition 1 (Dyck path). Start with an n × n grid with a main diagonal running from its southwest corner to is northeast corner; we will label the main diagonal 0, the diagonal above it 1, the diagonal above that 2, and so on. An (n, n)-Dyck path is a series of north and east steps from the southwest corner to the northeast corner, such that the path never crosses the main diagonal. Example 2. Here is a (5, 5)-Dyck path that occupies diagonal 0 and diagonal 1: For parking functions, we use the following definition for [1]: Definition 3 (Parking function). An (n, n)-parking function is a labeled (n, n)-Dyck path, whose north steps are labeled with the integers 1, . . . , n so that the labels of consecutive north steps are increasing from bottom to top. We require each such integer to lie in the cell to the right of the north step it labels, so that we can treat these integers as n “cars” parked along the north steps. If label i lies on diagonal di, we say that di is the diagonal of car i. Definition 4 (Composition). The composition of a parking function is the composition c = (c1, c2, . . . , ck) of n where ci is the number of north steps after the ith intersection of the Dyck