2023 Putnam B3

A sequence $y_1,y_2,\dots,y_k$ of real numbers is called *zigzag* if $k=1$, or if $y_2-y_1, y_3-y_2, \dots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1,X_2,\dots,X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1,X_2,\dots,X_n)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1,i_2,\dots,i_k$ such that $X_{i_1},X_{i_2},\dots,X_{i_k}$ is zigzag. Find the expected value of $a(X_1,X_2,\dots,X_n)$ for $n \geq 2$.

如果 $k=1$，或者 $y_2-y_1、y_3-y_2、\dots、y_k-y_{k-1}$ 非零且符号交替，则实数序列 $y_1,y_2,\dots,y_k$ 称为 *zigzag*。让 $X_1,X_2,\dots,X_n$ 独立于 $[0,1]$ 上的均匀分布进行选择。设 $a(X_1,X_2,\dots,X_n)$ 为 $k$ 的最大值，其中存在递增的整数序列 $i_1,i_2,\dots,i_k$，使得 $X_{i_1},X_{i_2},\dots,X_{i_k}$ 呈锯齿形。求 $n \geq 2$ 的 $a(X_1,X_2,\dots,X_n)$ 的期望值。