Putnam 2024 B5

Let $k$ and $m$ be positive integers. For a positive integer $n$, let $f(n)$ be the number of integer sequences $x_1,\dots,x_k,y_1,\dots,y_m,z$ satisfying $1 \leq x_1 \leq \cdots \leq x_k \leq z \leq n$ and $1 \leq y_1 \leq \cdots \leq y_m \leq z \leq n$. Show that $f(n)$ can be expressed as a polynomial in $n$ with nonnegative coefficients.