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 below or to the left 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?

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?

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 below or to the left 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, with $0$-dimensional tableaux being a single cell. 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?

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 below or to the left 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?