The entries in an array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? (2010 AMC12 B)
The answer is 42. Isn't there a specific process that answers this, such as Involution numbers? In other words, is there a specific list of numbers like Catalan numbers that can solve this or a formula, or is there a solution that involves Involution/telephone numbers?
What if the question was generalized to digits to $n^2$ in an $n \times n$ standard Young Tableaux? How can that be easily solved?
The number of standard Young tableaux of a given shape is given by the hook-length formula, see the formula for $\dim \pi_\lambda$ and the image in the link below:
http://en.wikipedia.org/wiki/Hook-length_formula#Dimension_of_a_representation
In particular, the number of standard Young tableaux of a $n\times n$ square is given by $$\frac{(n^2)!}{\frac{n!(n+1)!\dots (2n-1)!}{0!1!\dots(n-1)!}}=\frac{(n^2)!}{\prod_{k=1}^{2n-1}k^{\min(k,2n-k)}}.$$
Compare your $n=3$ case: $$\frac{9\cdot 8\cdot 7 \cdot 6 \cdot 5 \cdot 4\cdot 3\cdot 2 \cdot 1}{1\cdot2^2\cdot 3^3\cdot 4^2\cdot 5}=7 \cdot 6 =42.$$