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I'm trying to prove that the number of integer sequences of length $n$: $(a_1, a_2, \ldots, a_n)$ such that $0 \leq a_i \leq (n-i)$, $1 \leq i \leq n$ and exactly $k$ entries are equal to $0$ is given by the Stirling number of the first kind i. e. the number of permutations which break into k disjoint cycles.

Have no results at the moment. It's difficult to notice any connection between those things.

user492754
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    Check out Lehmer code. – Phicar Apr 25 '23 at 12:30
  • The (unsigned) Stirling number of the first kind, $c(n,k)$, is also the number of $n$-permutations with $k$ left-to-right-maxima. Think about inversion bottoms, left-to-right maxima, Lehmer code, and try to put it all together. – Alexander Burstein Apr 25 '23 at 18:50

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