What is the dimension of the set of partitions?

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 \( i\gt\sqrt n \).