There is a determinantal formula for the number of elements of the interval $[\mu,\lambda]$ of Young's lattice between two partitions due to Kreweras and MacMahon in the case of $\mu=\varnothing$ (see section 2.3.7 of http://www.numdam.org/item/BURO_1965__6__9_0/ or Stanley, EC1, exercise 3.149).

Is there a similar formula for the number of edges of the Hasse diagram of this interval $[\mu,\lambda]$? Or, is there any kind of reasonable formula at all for this number?