Question 136
Let $n \ge 2$ be a natural number, and $\mathcal{F}$ be the set of functions $f : \{1, 2, \dots, n\} \to \{1, 2, \dots, n\}$ such that $f(k) \le f(k+1) \le f(k)+1$, for every $k \in \{1, 2, \dots, n-1\}$.
a) Determine the cardinality of the set $\mathcal{F}$.
b) Determine the total number of fixed points of the functions in $\mathcal{F}$.
(A *fixed point* of the function $f$ is a number $x \in \{1, 2, \dots, n\}$ such that $f(x) = x$.)
a) Determine the cardinality of the set $\mathcal{F}$.
b) Determine the total number of fixed points of the functions in $\mathcal{F}$.
(A *fixed point* of the function $f$ is a number $x \in \{1, 2, \dots, n\}$ such that $f(x) = x$.)