ELMO 2024 Problem 2

For positive integers $a$ and $b$ , an $(a,b)$ -shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$ , $k$ , $a_1$ , $a_2$ , $\dots$ , $a_k$ , $b_1$ , $b_2$ , $\dots$ , $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$ . Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$ -shuffle for each $1\leq i\leq k$ , in ascending order of $i$ . Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards.

Linus Tang