Count Bitonic Permutations

What this bitonic permutation counting problem tests

This is a medium-difficulty combinatorial puzzle that requires you to count permutations with a specific structure: a sequence that rises to a single peak and then falls. It tests whether you can identify the combinatorial constraints that define the problem, set up a counting argument, and avoid over- or under-counting edge cases.

The key insight is recognizing that a bitonic permutation is fully determined once you decide which element sits at the peak and how the remaining elements are split between the ascending and descending portions. Candidates typically succeed by thinking about the problem in terms of choosing positions and elements, rather than trying to enumerate all valid sequences. The solution rewards clear logical reasoning and careful handling of the boundary cases where all elements ascend or all descend.

• Combinatorial structure and constraint satisfaction
• Counting via partitioning and selection
• Edge cases in structural definitions