Tableaux problem 43

43.A central point of [LiangSag2024a]*Th. 3.1 is that for any partition $\lambda=(\lambda_{1},\lambda_{2},\ldots\lambda_{\ell(\lambda)})$ with $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{\ell(\lambda)}>0$, if we define $\lambda+i=(\lambda_{1}+i,\lambda_{2}+i,\ldots,\lambda_{\ell(\lambda)}+i)$, then the set of partitions $(\lambda+i)_{i\geq 0}$ has the property that for every $n$, the set of intervals in the partition lattice $\{[\lambda+n,\lambda+(n+k)]\,|\,k\geq 0\}$ are all isomorphic. This series of elements allows their Order Ideal Lemma (Lem. 1.1) to be applied.

Similar series of elements can be found in other distributive Fomin lattices, the lattice of strict partitions, the lattice of $k$-row partitions, and the cylindric partitions. This is surprising since this property does not seem to be forced by the hypotheses of distributive Fomin lattices.

Can this observation be extended to the Young–Fibonacci lattices? That is, are there sequences of elements in $\mathbb{YF}_{r}$, $(p_{i})_{i\geq 0}$ for which for every $n$, the set of intervals $\{[p_{n},p_{n+k}]\,|\,k\geq 0\}$ are all isomorphic? If there are, can they be used to prove an analog of their Th. 3.1? If so, does this provide a clue to a “semi-distributive” property of $\mathbb{YF}_{r}$?