Tableaux problem 7

7.[Wor] Develop a similar theory of limit shapes for Young–Fibonacci diagrams. The initial challenge is defining a concept of scaling between smaller and larger diagrams. E.g., if the scaling is a doubling, we need to define an embedding $D$ of $\mathbb{YF}_{r}$ into $\mathbb{YF}_{r}$ that maps words (diagrams) of size $k$ into words of size $2k$, $\mathbb{YF}_{r}^{(k)}\rightarrow\mathbb{YF}_{r}^{(2k)}$. The embedding should be “nearly isometric” in that it maps words that are “near” each other in $\mathbb{YF}_{r}$ (in the distance metric of the Hasse diagram) into words that are near each other $\mathbb{YF}_{r}$, and vice-versa. And every word in $\mathbb{YF}_{r}$ must be near an element of the image of $D$.

Given such a doubling map, it defines a limit space within which the question can be asked whether randomly-chosen $r$-colored permutations have a limit shape. If there is a limit shape, determine it. Since $\mathbb{YF}_{r}$ as dual graded graphs has a unique weighting 1, there are no colorings of the tableaux, so there are only three distributions of diagrams to be considered.