Fluffy and Alternating Subsequence

Today, Fluffy decided to play with a sequence of integers, $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ which is coincidentally a permutation of the numbers $1$, $2$, $\ldots$, $n$. Fluffy thinks that a sequence $b_{1}$, $b_{2}$, $\ldots$, $b_{k}$ is alternating if and only if either of the following holds : - For all $i$, $b_{i} < b_{i + 1}$ if and only if $i$ is odd - For all $i$, $b_{i} < b_{i + 1}$