Problem

Source: 2023 Czech-Polish-Slovak Match Junior, Team p2 CPSJ

Tags: combinatorics



The numbers $1, 2,..., 2023$ are written on the board in this order. We can repeatedly perform the following operation with them: We select any odd number of consecutively written numbers and write these numbers in reverse order. How many different orders of these $2023$ numbers can we get? Example: If we start with only the numbers $1, 2, 3, 4, 5, 6$, we can perform the following steps $$1, 2, 3, 4, 5, 6 \to 3, 2, 1,4, 5, 6 \to 3, 6, 5, 4, 1, 2 \to 3, 6, 1, 4, 5, 2 \to ...$$