Naiomi T. Cameron

Counting on Determinants

1. THE PROBLEM OF THE DETERMINED ANTS. Imagine four determined ants who simultaneously walk along the edges of the picnic table graph of Figure 1. The ants can move only to the right (northeast, southeast, and sometimes due east) with the goal of reaching four different morsels.

k-ribbon Fibonacci tableaux

We extend the notion of k-ribbon tableaux to the Fibonacci lattice, a differential poset defined by R. Stanley in 1975. Using this notion, we describe an insertion algorithm that takes k-colored permutations to pairs of k-ribbon Fibonacci tableaux of the same shape, and we demonstrate a color-to-spin property, similar to that described by Shimozono and White for ribbon tableaux. We describe a geometric interpretation of k-ribbon Fibonacci tableaux and use this interpretation to describe a notion of P equivalence for k-ribbon Fibonacci tableaux. In addition, we give an evacuation algorithm which relates the pair of k-ribbon Fibonacci tableaux obtained through the insertion algorithm to the pair of k-ribbon Fibonacci tableaux obtained using Fomins growth diagrams.