A bitonic sequence is a special type of sequence that has a distinct pattern of elements. It can be divided into two parts: the first part is monotonically increasing, and the second part is monotonically decreasing.

The point at which the transition from increasing to decreasing occurs is known as the bitonic point. In the case of a cyclic bitonic sequence, it means that the sequence eventually loops back to the beginning. To determine which of the provided sequences is not bitonic, we will analyze each option in detail.

A. {8, 6, 4, 2, 3, 5, 7, 9}:

This sequence starts in a decreasing order from 8 down to 2 and then changes direction at the bitonic point (2), increasing from 2 to 9. So, this is a bitonic sequence.

B. {0, 4, 8, 9, 2, 1}:

This sequence starts in an increasing order from 0 to 9 without a distinct bitonic point. It then continues in a decreasing order from 9 down to 1. This sequence is not bitonic because there is no point where it transitions from increasing to decreasing monotonically.

C. {3, 5, 7, 9, 8, 6, 4, 2}:

This sequence starts in an increasing order from 3 to 9 and then changes direction at the bitonic point (9), decreasing from 9 to 2. So, this is a bitonic sequence.

D. {1, 2, 4, 7, 6, 0, 1}:

This sequence starts in an increasing order from 1 to 7 and then changes direction at the bitonic point (7), decreasing from 7 to 0. However, it then increases again from 0 to 1. This means that it does not strictly follow the definition of a bitonic sequence because it reverses direction more than once. Therefore, option D is not a bitonic sequence.

Now, let’s delve into the reasons why the other options are not correct and why option D is indeed the correct answer.

Option A: {8, 6, 4, 2, 3, 5, 7, 9}

This sequence starts with decreasing numbers from 8 down to 2, and at the bitonic point (2), it transitions to an increasing order from 2 to 9. This pattern of a clear transition from decreasing to increasing is characteristic of a bitonic sequence. Therefore, option A is indeed a bitonic sequence.

Option B: {0, 4, 8, 9, 2, 1}

Option B is not a bitonic sequence because it lacks a distinct bitonic point where the direction of the sequence changes from increasing to decreasing. It starts with an increasing order from 0 to 9 and continues in a decreasing order from 9 down to 1. There is no point at which the sequence switches from increasing to decreasing monotonically. Thus, it does not meet the criteria for a bitonic sequence.

Option C: {3, 5, 7, 9, 8, 6, 4, 2}

This sequence starts with an increasing order from 3 to 9 and then changes direction at the bitonic point (9), decreasing from 9 to 2. This clear transition from increasing to decreasing is a characteristic of a bitonic sequence. Therefore, option C is indeed a bitonic sequence.

Option D: {1, 2, 4, 7, 6, 0, 1}

Option D initially follows the pattern of a bitonic sequence by starting with an increasing order from 1 to 7 and then changing direction at the bitonic point (7) as it decreases from 7 to 0.

However, the problem with option D is that it reverses direction again, increasing from 0 to 1. This secondary increase violates the definition of a bitonic sequence, which requires that the sequence transitions from increasing to decreasing only once. Thus, option D is not a bitonic sequence.

In conclusion, a bitonic sequence is characterized by a distinct transition point where it changes from monotonically increasing to monotonically decreasing. Options A and C both exhibit this behavior and are indeed bitonic sequences.

Option B, on the other hand, lacks a clear transition point and is not a bitonic sequence. Option D initially appears to be a bitonic sequence but violates the criteria by reversing direction more than once, making it not a bitonic sequence. Therefore, the correct answer is D.