Subsets and Sets Question in New Zealand MO

Question

For any positive integer $n$, let $f(n)$ be the number of subsets of ${1, 2, . . . , n}$ whose sum is equal to $n$. Does there exist infinitely many positive integers $m$ such that $f(m) = f(m + 1)$? (Note that each element in a subset must be distinct.)

This is a New Zealand MO question and in their solution, they state that, by Michael Albert, that if we make $S(n)$ the set that fulfills what the question asks for, then $S(n)→S(n+1)$ is considered as a 'map', then this map would be an injection (needs proof but easy) and not a surjection provided that $S(n + 1)$ contains a set whose largest and second largest elements differ by one.

They state that a proof is needed and I would appreciate it if anybody could explain this to me, with the proof, as this will help me to understand the question better.

FYI: I have added my solution below.

Thank you very much!

Answer 1

Let $S(n)$ be the set of such subsets. Consider the $S(n)$ to $S(n + 1)$ that adds one to the largest element of each $A ∈ S(n)$.

For even $n = 2k ⩾ 2$ this is true since we can take ${k, k + 1} ∈ S(n + 1)$ and for odd $n = 2k + 1 ⩾ 5$ this is true since we can take ${1, k, k + 1}$. So for $n ⩾ 5$, we must have $f(n) < f(n + 1)$ and there do not exist infinitely many such pairs.