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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?

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  • $\begingroup$ (In fact the Kreweras/MacMahon formula is for the number of $m$-multichains of this interval, but we can take $m=1$.) $\endgroup$ – Sam Hopkins Mar 3 '19 at 18:50
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    $\begingroup$ 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. $\endgroup$ – Richard Stanley Mar 4 '19 at 19:47
  • $\begingroup$ @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.) $\endgroup$ – Sam Hopkins Mar 4 '19 at 19:53

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