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A well-known hook length formula states that the number of ways to arrange the elements of $[n]$ in a Young tableau with $n$ cells so that all columns and rows are increasing is $\frac{n!}{\prod_c h(c)}$, where $h(c)$ is the "hook length" of a cell $c$, that is, the number of cells immediately above or to the right of the cell (including the cell $c$).

Consider a "multidimensional" generalization of Young tableaux. Formally, a $k$-dimensional Young tableau is a sequence of $(k - 1)$-dimensional tableaux so that each tableau contains the next one, in the sense that $A$ contains $B$ if it has at least as many layers as $B$, and $i$-th of these layers contains the $i$-th layer of $B$ for each $i$. $0$-dimensional tableaux is just a single cell, and is considered to contain itself.

Is there a generalization of the hook length formula for the number of monotonous arrangements of numbers in a $k$-dimensional tableau? Can it be computed within reasonable (say, in $O(n^{f(k)})$) time?

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    $\begingroup$ Well the word "contains" which you put in scare quotes is not well-defined and so this question cannot have a precise answer, but I think the general belief is that hook-length formulas are very special and somehow related to the 2-dimensional (or 3-dimensional, depending on your perspective) nature of SYTs; see for example MacMahon's formula for the number of plane partitions (en.wikipedia.org/wiki/Plane_partition); nothing like this holds in higher dimensions (despite the fact that MacMahon conjectured it would!) $\endgroup$
    – Sam Hopkins
    Commented Sep 19, 2017 at 21:21
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    $\begingroup$ OTOH, if you are looking for somewhere else where we have hook length formulas, then you could search for the "$d$-complete posets" that have been much studied by R. Proctor. $\endgroup$
    – Sam Hopkins
    Commented Sep 19, 2017 at 21:24
  • $\begingroup$ Note that there is a hook-length formula for trees (Knuth proved this if I recall). $\endgroup$
    – Per Alexandersson
    Commented Sep 20, 2017 at 9:59
  • $\begingroup$ You can recursively "define" the hooks if you want them to only depend on the 'upset' shape in the poset, similar to how hook values in Young diagrams and subtrees are defined. The hook value of a box should only depend on the sub-structure depending on the shape defined by the 'upset'. You want that the number of linear extensions then is given by $n!$ divided by product of hook values. If these turn out to be integers, you have something interesting - however, I suspect this is not the case here. $\endgroup$
    – Per Alexandersson
    Commented Sep 20, 2017 at 10:03

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