<p>You are given an integer array <code>nums</code> of length <code>n</code> which represents a permutation of all the integers in the range <code>[0, n - 1]</code>.</p>

<p>The number of <strong>global inversions</strong> is the number of the different pairs <code>(i, j)</code> where:</p>

<ul>
	<li><code>0 &lt;= i &lt; j &lt; n</code></li>
	<li><code>nums[i] &gt; nums[j]</code></li>
</ul>

<p>The number of <strong>local inversions</strong> is the number of indices <code>i</code> where:</p>

<ul>
	<li><code>0 &lt;= i &lt; n - 1</code></li>
	<li><code>nums[i] &gt; nums[i + 1]</code></li>
</ul>

<p>Return <code>true</code> <em>if the number of <strong>global inversions</strong> is equal to the number of <strong>local inversions</strong></em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,0,2]
<strong>Output:</strong> true
<strong>Explanation:</strong> There is 1 global inversion and 1 local inversion.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,0]
<strong>Output:</strong> false
<strong>Explanation:</strong> There are 2 global inversions and 1 local inversion.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>n == nums.length</code></li>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= nums[i] &lt; n</code></li>
	<li>All the integers of <code>nums</code> are <strong>unique</strong>.</li>
	<li><code>nums</code> is a permutation of all the numbers in the range <code>[0, n - 1]</code>.</li>
</ul>

class Solution {
public:


    long int inversion(vector<int>&nums, int low, int high){
        if(low==high){return 0;}
        long int mid = (low+high)/2;
        long int x = inversion(nums, low, mid);
        long int y = inversion(nums, mid+1, high);
        long int z = merge(nums, low, mid+1, high);
        return x+y+z;

    }
    long int merge(vector<int>&nums, int x, int y, int z){
        int p1 = x;
        int p2 = y;
        int p3 = 0;
        long int count =0;
        vector<int>temp(z-x+1, 0);
        while(p1<y && p2<=z){
            if(nums[p1]<=nums[p2]){
                temp[p3++] = nums[p1++];
            }
            else{
                temp[p3++] = nums[p2++];
                count+=(y-p1);
            }
        }
        while(p1<y){
            temp[p3++] = nums[p1++]; 
        }
        while(p2<=z){
            temp[p3++] = nums[p2++];
        }
        // copy temp
        for(int i = 0; i<z-x+1; i++){
            nums[i+x] = temp[i];
        }
        return count;
    }
    bool isIdealPermutation(vector<int>& nums) {
        // local count 
        long int clocal = 0;
        for(int i =0; i<nums.size()-1; i++){
            if(nums[i]>nums[i+1]){
                clocal = clocal+1;
            }
        }
        // global count 
        long int cglobal = 0;
        int low = 0;
        int high = nums.size()-1;
        cglobal = inversion(nums, low, high);
        if(cglobal == clocal){return true;}
        return false;
    }
};
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