I am studying "A Walk Through Combinatorics" and I encountered this equation $$$$$S(n,k) \le c(n,k)$ $$$$ The author wants us to prove that. I can verify this equation holds via induction on both k and n. But it feels not right to prove via induction so I wonder if there exists a more elegant way to show that. I wonder if is it possible to show that the inequality holds by using the duality between Stirling numbers of the first kind and Stirling numbers of the second kind and the definition of $c(n,k) = (-1)^{n-k}s(n,k).$ $$$$ I would be very grateful for your patience.
2026-03-28 00:00:12.1774656012
$S(n,k) \le c(n,k)$
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