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on Tableaux problem 17

The statement

17.[Stan1988a]*sec. 6 Prob. 6 Fix $r\in\mathbb{P}$ . What is the least number of elements of rank $n$ that an $r$ -differential poset can have? It seems plausible that the minimum value is achieved by $\mathbb{Y}^{\times r}$ , the $r$ -th cartesian power of the lattice of partitions. (The maximum value is attained by $\mathbb{YF}_{r}$ , the $r$ -differential Young–Fibonacci lattice.[Byrn2012a]*Th. 1.2)

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.