What is the number of functions $f$ from the set $\{1, 2, . . . , 2n\}$ to $\{1, 2, . . . , n\}$ so that $f(x) \leq \lceil x/2 \rceil$ for all $x$?

Question

What is the number of functions $f$ from the set $\{1, 2, . . . , 2n\}$ to $\{1, 2, . . . , n\}$ so that $f(x) \leq \lceil x/2 \rceil$ for all $x$?

I'm a complete beginner at solving something like this, and I tried to use the product rule. I'm 99% sure I've done it wrong.

So for elements $1, 2, 3, 4, 5, 6, 7, 8$, there are $1,1,2,2,3,3,4,4$ possibilities respectively (if I'm understanding this right)

If this continues for element $2n$ there will be $n$ possibilities. I'm not sure how to continue to how many functions there are from here. If I had to guess since there are $1/2$ as many possibilities as the element number.

There are $n^{m}$ functions from $m$ elements to $n$ elements so would the answer be $\frac{n^{m}}{2}$ ? Is my logic correct?

Answer 1

As you state in your question, there is $1$ place the value $1$ can map to. There is $1$ place that $2$ can map to.

There is $i$ place that $2i-1$ can map to & $i$ places that $2i$ can map to.

So there will be $\color{red}{(n!)^2}$ such mappings.