# 2021-06 A nondecreasing subsequence

Let $$\mathcal{A}_n$$ be the collection of all sequences $$\mathbf{a}= (a_1,\dots, a_n)$$ with $$a_i \in [i]$$ for all $$i\in [n]=\{1,2,\dots, n\}$$. A nondecreasing $$k$$-subsequence of $$\mathbf{a}$$ is a subsequence $$(a_{i_1}, a_{i_2},\dots, a_{i_k})$$ such that $$i_1< i_2< \dots < i_k$$ and $$a_{i_1}\leq a_{i_2}\leq \dots \leq a_{i_k}$$. For given $$k$$, determine the smallest $$n$$ such that any sequence $$\mathbf{a}\in \mathcal{A}_n$$ has a nondecreasing $$k$$-subsequence.

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