Given an integer N, calculate the number of permutations of A = [1, 2, …, N] which are first decreasing and then increasing.
Input: N = 5
Output : 14
Following are the sub-sequences which are first decreasing and then increasing:
[2, 1, 3, 4, 5], [3, 1, 2, 4, 5], [4, 1, 2, 3, 5], [5, 1, 2, 3, 4],
[3, 2, 1, 4, 5], [4, 2, 1, 3, 5], [5, 2, 1, 3, 4], [4, 3, 1, 2, 5],
[5, 3, 1, 2, 4], [5, 4, 1, 2, 3], [5, 4, 3, 1, 2], [5, 4, 2, 1, 3],
[5, 3, 2, 1, 4], [4, 3, 2, 1, 5]
Input : N = 1
Output : 0
Approach: It is clear that the point at which sequence becomes increasing from decreasing is occupied by the smallest element of the permutation which is 1. Also, decreasing sequence is always followed by an increasing sequence which means that the smallest element can have positions ranging [2, …, N-1]. Otherwise, it will result a fully increasing or fully decreasing sequence.
For example, consider N = 5 and position = 2, i.e smallest element at position 2 in the sequence. Count all possible sequences with Configuration = [_, 1, _, _, _].
Select any 1 element from the remaining 4 elements (2, 3, 4, 5) to fill position 1. For example, we select element = 3. Configuration looks like [3, 1, _, _, _]. Only 1 sequence is possible i.e [3, 1, 2, 4, 5]. Thus for every selected element to fill at position 1, a sequence is possible. With this configuration, total of 4C1 permutations are possible.
Now, consider the configuration = [_, _, 1, _, _].
A similar approach can be applied by selecting any 2 elements from the remaining 4 elements and for every pair of elements, a single valid permutation is possible. Hence, the number of permutations for this configuration = 4C2
The final configuration possible for N = 5 is [_, _, _, 1, _]
Select any 3 of the remaining 4 elements and for every triplet, a permutation is obtained. Number of permutations in this case = 4C3
Final count = 4C1 + 4C2 + 4C3 for N = 5
Generalized result for N:
Count = N-1C1 + N-1C2 + ... + N-1CN-2 which simplifies to, N-1Ci = 2N-1 - 2 (from binomial theorem)
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