Assignment #10 Solution

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Suppose p_1, p_2, …, p_n is a permutation of the numbers 1 to n. A sequence

a_1, …, a_n of distinct integers is said to be of type p_1,…,p_n

if and only if for all 1 <= i < j <= n, a_i < a_j iff p_i < p_j. Thus a strictly

increasing sequence is of type 1, 2, …, n while a strictly decreasing

sequence is of type n, n-1, …., 1. The type of a sequence of distinct numbers is

uniquely determined by the sorted order of the elements.

Given a permutation of the numbers 1, …, n, you have to count the number

of subsequences of the permutation of type 1, 3, 2 and 3, 1, 2. In other words,

count the number of triples (i,j,k) such that i < j < k and p_i < p_k < p_j

for the first part, and p_j < p_k < p_i for the second.

Input Format

The first line of input specifies n, 3 <= n <= 500000, the number of elements in the

permutation. The next line contains n numbers specifying the permutation.

Output Format

Output the two numbers on a single line separated by a single space. Note that

the numbers may be large so use long long for counting.

Sample Input1


5 2 3 4 6 1

Sample Output 1

0 3

Sample Input2


6 2 4 8 1 5 7 3

Sample Output 2

11 11

You will get 1/2 credit if you do one of the parts correctly. Print out 2 numbers

even if you have done only one part, otherwise it will not be evaluated correctly.

Submit a single file named RollNo_10.cpp .

Hint: Consider counting number of subsequences of type 2, 1. This is known as

the number of inversion pairs in the permutation, and there are many ways of

doing it in O(nlog n) time. One way is to use merge sort and do some more work

in the merging step. Another way is to use prefix sums. Keep an array A of 0,1

entries and at ith step increment value of A[p_i]. Compute the prefix sum of

the A array up to p_i. Then p_i – prefix_sum[p_i] is the number of numbers

< p_i that are still to be inserted. These form inversion pairs with p_i.

Use a similar idea for 3 term subsequences.

Homework: Suppose you are given a permutation of length k. Try to find an algorithm

for counting subsequences of length k in a permutation of n, of the given type,

in time which may be exponential in k, but is polynomial in n with degree a

constant independent of k. A brute force algorithm trying all subsequences of

length k, takes O(k^2n^k) time. For an increasing subsequence, it can be done in

O(knlog n) time.