kk-ribbon Fibonacci tableaux

We extend the notion of kk-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 kk-colored permutations to pairs of kk-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 kk-ribbon Fibonacci tableaux and use this interpretation to describe a notion of PP equivalence for kk-ribbon Fibonacci tableaux. In addition, we give an evacuation algorithm which relates the pair of kk-ribbon Fibonacci tableaux obtained through the insertion algorithm to the pair of kk-ribbon Fibonacci tableaux obtained using Fomin’s growth diagrams.