Let $n \in \mathbb{N}, n \geq 2$ and $M = \{1, 2, \ldots, n\}$. Show that there are more than $2^n$ surjective functions $f : \mathcal{P}(M) \rightarrow \{0, 1, \ldots, n\}$ such that $f(A) \leq f(B)$ for all $A \subseteq B \subseteq M$.
($\mathcal{P}(M)$ is the power set of $M$)
My first thought was representing each subset as a bitset of length $n$, then defining a function for each bitset $A$, for example $f_A(B) = \text{popcount(A XOR B)}$, and playing around with other operations such as sum/difference $\pmod{2^n}$. Then, I would find another unrelated function which satisfies the property (for example, the maximum function or the cardinality function) and state that there would be at least $2^n + 1$ functions. The problem is that the bitwise functions have some edge cases that fail to satisfy the property.
Then, I tried to actually compute the number of functions but I don't know how to make sure the property holds while calculating. I suppose the final formula will probably involve ordered Bell numbers due to inclusion being a partial ordering.
The following stronger result holds:
Proof: For a fixed permutation $\sigma:M\to M$ define a function $f_{\sigma}:\mathcal{P}(M)\to \{0,\dots, n\}$ as follows:
Set $f_\sigma(\emptyset)=0$
For singletons $\{i\}$ set $f_\sigma(\{i\})=\sigma(i)$
For any $S\subseteq M$ with $|S|\geq 2$ set $$f_\sigma(S) = \min\left(\sum_{s\in S}\sigma(s), n\right)$$
It's easy to verify that $f_\sigma$ is surjective, $f_\sigma(A)\leq f_\sigma(B)$ when $A\subseteq B$ and that two different permutations $\sigma$, $\sigma'$ give rise to two different functions $f_\sigma$ and $f_{\sigma'}$. This procedure generates at least $n!$ such surjective functions.
For $n\geq 4$ this result is stronger than what you want to prove. One could work out the cases $n=2$ and $n=3$ by hand. For the case $n=3$ notice that the above already gives you $3!=6$ functions so it's easier if you try to expand on that set by trying to find $3$ more different functions (notice that functions assigning the same value to two or more singletons are guaranteed to be different from the already constructed ones).