= Solution
Let $B$ be the set of bins that are occupied in configuration $x$ but empty in configuration $y$. For each $b\in B$, choose the lowest-numbered ball $j_b$ lying in $b$ under $x$. Then $\alpha_{j_b}(x)=1$ and $x_{j_b}\ne y_{j_b}$. Distinct bins choose distinct balls, so
$$
|B|\leq\sum_{j=1}^m\alpha_j(x)\mathbf1_{\{x_j\ne y_j\}}.
$$
Every increase in the number of empty bins is accounted for by a newly empty bin, while newly occupied bins only decrease that number. Hence $f(y)-f(x)\leq|B|$, proving the stated inequality.
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