The lattice dimension of the partial cube of partitions of $$n$$ is $$\Theta(\sqrt n)$$.

For the lower bound, consider how many neighbors the triangular partition $$1+2+3+\cdots$$ has.

For the upper bound, assign to each partition $\sum_{i\ge 0} x_i ,\quad x_i\ge x_{i+1},$ the coordinates $(x_1,x_2,\dots,x_{\sqrt n}, y_1,y_2,\dots, y_{\sqrt n}),$ where $$y_j$$ denotes the number of values greater than or equal to $$j$$ among the $$x_i$$ with $$i\gt\sqrt n$$.