Formula for number of edges in Hasse diagram of Young's lattice interval

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?

• (In fact the Kreweras/MacMahon formula is for the number of $m$-multichains of this interval, but we can take $m=1$.) – Sam Hopkins Mar 3 '19 at 18:50
• You are asking for the number of skew plane partitions of shape $\lambda/\mu$ with parts $1,2,3$ and with exactly one part equal to 2. This doesn't fit naturally into what is known about plane partitions, so I wouldn't be surprised if the problem does not have a nice solution. – Richard Stanley Mar 4 '19 at 19:47
• @RichardStanley: thanks, good to know there is likely not a nice answer. (A kind of answer is given by Theorem 3.4 of cambridge.org/core/journals/forum-of-mathematics-sigma/article/… but it involves summing over all corners of the skew shape and so is pretty unwieldy.) – Sam Hopkins Mar 4 '19 at 19:53