Question 118
Let $k$ be a positive integer, and let $n = 2^k$, $N = \{1, 2, \dots, n\}$. For a bijective function $f$ from $N$ to $N$, if a set $A \subseteq N$ contains an element $a$ such that $\{a, f(a), f(f(a)), \dots\} = A$, then we call $A$ a **cycle** of $f$. Prove that among all bijective functions $f$ from $N$ to $N$, at least $\frac{n!}{2}$ of them have at most $2k-1$ cycles.