How many arrangements? $(a, b, c, d, x, y, z, w)$ I would appreciate if somebody could help me with the following problem:
Q: How many arrangements?

My Study
I solved by mathematica11.0
f[n_] := {a, b, c, d, x, y, z, w} /. 
  Solve[0 < a < b < c < d < n && 0 < x < y < z < w < n && a < x && 
    b < y && c < z && d < w, {a, b, c, d, x, y, z, w}, Integers]
f[9]=490

 A: This is only a partial solution.
The problem is somehow related to Young tableaux.  If we were required to use all the numbers from $1$ to $8$, we could compute the number of tableau of this shape with the hook-length formula which in this instance gives $$\frac{8!}{5\cdot4\cdot3\cdot2\cdot4\cdot3\cdot2\cdot1}=14$$
Here are the hook lengths:
$$5\ 4\ 3\ 2\\
  4\ 3\ 2\ 1$$
It is apparent that we can have any or all of the equalities $b=x,\ c=y, d=z$. For example, if
$$abcd\\
xyzw$$  is a standard tableau, we could replace it by either $$abcd\\byzw$$ or $$axcd\\xyzw$$  For each of the $3$ diagonals, we have $3$ choices: leave it unchanged, or make it constant in one of two ways, which brings us up to $27\cdot14=378$ solutions.
So far all our solutions have $a=1,\ w=8$, but there are others, for example $$2345\\3456$$  I haven't been able to think of a simple way to count these.
I've Googled "Young tableaux with repeated elements," but I didn't find anything bearing directly on this problem.  Admittedly, I've just spent a few minutes on this.  This paper discusses a number of variants, but a brief perusal didn't turn up anything directly relevant.
