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: <math>p(n) \le r_n \le F_n,</math>
where ''p''(''n'') is the [[partition function (number theory)|number of integer partitions]] of ''n'' and {{math|''F''<sub>''n''</sub>}} is the ''n''th [[Fibonacci number]]. In other words, the conjecture states that at every rank, every differential poset has a number of vertices lying between the numbers for Young's lattice and the Young-Fibonacci lattice. The upper bound was [[mathematical proof|proved]] in {{harvtxt|Byrnes|2012}}, while the lower bound remains open. {{harvtxt|Stanley|Zanello|2012}} proved an [[asymptotic analysis|asymptotic]] version of the lower bound, showing that
: <math> r_n \gg n^a \exp(2\sqrt{n}) </math>
for every differential poset and some constant ''a''. By comparison, the partition function has asymptotics
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