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permutations of numbers 1-2023

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

May 5, 2024
combinatorics

Problem Statement

The numbers 1,2,...,20231, 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 20232023 numbers can we get?
Example: If we start with only the numbers 1,2,3,4,5,61, 2, 3, 4, 5, 6, we can perform the following steps 1,2,3,4,5,6→3,2,1,4,5,6→3,6,5,4,1,2→3,6,1,4,5,2→...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 ...
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