Number of n-element permutations with exactly k in

问题:

I am trying to efficiently solve SPOJ Problem 64: Permutations.

Let A = [a1,a2,...,an] be a permutation of integers 1,2,...,n. A pair of indices (i,j), 1<=i<=j<=n, is an inversion of the permutation A if ai>aj. We are given integers n>0 and k>=0. What is the number of n-element permutations containing exactly k inversions?

For instance, the number of 4-element permutations with exactly 1 inversion equals 3.

To make the given example easier to see, here are the three 4-element permutations with exactly 1 inversion:

(1, 2, 4, 3)
(1, 3, 2, 4)
(2, 1, 3, 4)

In the first permutation, 4 > 3 and the index of 4 is less than the index of 3. This is a single inversion. Since the permutation has exactly one inversion, it is one of the permutations that we are trying to count.

For any given sequence of n elements, the number of permutations is factorial(n). Thus if I use the brute force n² way of counting the number of inversions for each permutation and then checking to see if they are equal to k, the solution to this problem would have the time complexity O(n! * n²).

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Previous Research

A subproblem of this problem was previously asked here on StackOverflow. An O(n log n) solution using merge sort was given which counts the number of inversions in a single permutation. However, if I use that solution to count the number of inversions for each permutation, I would still get a time complexity of O(n! * n log n) which is still very high in my opinion.

This exact question was also asked previously on Stack Overflow but it received no answers.

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My goal is to avoid the factorial complexity that comes from iterating through all permutations. Ideally I would like a mathematical formula that yields the answer to this for any n and k but I am unsure if one even exists.

If there is no math formula to solve this (which I kind of doubt) then I have also seen people giving hints that an efficient dynamic programming solution is possible. Using DP or another approach, I would really like to formulate a solution which is more efficient than O(n! * n log n), but I am unsure of where to start.

Any hints, comments, or suggestions are welcome.

EDIT: I have answered the problem below with a DP approach to computing Mahonian numbers.

回答1:

The solution needs some explanations. Let's denote the number of permutations with n items having exactly k inversions by I(n, k)

Now I(n, 0) is always 1. For any n there exist one and only one permutation which has 0 inversions i.e., when the sequence is increasingly sorted

Now I(0, k) is always 0 since we don't have the sequence itself

Now to find the I(n, k) let's take an example of sequence containing 4 elements {1,2,3,4}

for n = 4 below are the permutations enumerated and grouped by number of inversions

|___k=0___|___k=1___|___k=2___|___k=3___|___k=4___|___k=5___|___k=6___|
| 1234    | 1243    | 1342    | 1432    | 2431    | 3421    | 4321    |
|         | 1324    | 1423    | 2341    | 3241    | 4231    |         |
|         | 2134    | 2143    | 2413    | 3412    | 4312    |         |
|         |         | 2314    | 3142    | 4132    |         |         |
|         |         | 3124    | 3214    | 4213    |         |         |
|         |         |         | 4123    |         |         |         |
|         |         |         |         |         |         |         |
|I(4,0)=1 |I(4,1)=3 |I(4,2)=5 |I(4,3)=6 |I(4,4)=5 |I(4,5)=3 |I(4,6)=1 |
|         |         |         |         |         |         |         |

Now to find the number of permutation with n = 5 and for every possible k we can derive recurrence I(5, k) from I(4, k) by inserting the nth (largest) element(5) somewhere in each permutation in the previous permutations, so that the resulting number of inversions is k

for example, I(5,4) is nothing but the number of permutations of the sequence {1,2,3,4,5} which has exactly 4 inversions each. Let's observe I(4, k) now above until column k = 4 the number of inversions is <= 4 Now lets place the element 5 as shown below

|___k=0___|___k=1___|___k=2___|___k=3___|___k=4___|___k=5___|___k=6___|
| |5|1234 | 1|5|243 | 13|5|42 | 143|5|2 | 2431|5| | 3421    | 4321    |
|         | 1|5|324 | 14|5|23 | 234|5|1 | 3241|5| | 4231    |         |
|         | 2|5|134 | 21|5|43 | 241|5|3 | 3412|5| | 4312    |         |
|         |         | 23|5|14 | 314|5|4 | 4132|5| |         |         |
|         |         | 31|5|24 | 321|5|4 | 4213|5| |         |         |
|         |         |         | 412|5|3 |         |         |         |
|         |         |         |         |         |         |         |
|    1    |    3    |    5    |    6    |    5    |         |         |
|         |         |         |         |         |         |         |

Each of the above permutation which contains 5 has exactly 4 inversions. So the total permutation with 4 inversions I(5,4) = I(4,4) + I(4,3) + I(4,2) + I(4,1) + I(4,0) = 1 + 3 + 5 + 6 + 5 = 20

Similarly for I(5,5) from I(4,k)

|___k=0___|___k=1___|___k=2___|___k=3___|___k=4___|___k=5___|___k=6___|
|   1234  | |5|1243 | 1|5|342 | 14|5|32 | 243|5|1 | 3421|5| | 4321    |
|         | |5|1324 | 1|5|423 | 23|5|41 | 324|5|1 | 4231|5| |         |
|         | |5|2134 | 2|5|143 | 24|5|13 | 341|5|2 | 4312|5| |         |
|         |         | 2|5|314 | 31|5|44 | 413|5|2 |         |         |
|         |         | 3|5|124 | 32|5|14 | 421|5|3 |         |         |
|         |         |         | 41|5|23 |         |         |         |
|         |         |         |         |         |         |         |
|         |    3    |    5    |    6    |    5    |    3    |         |
|         |         |         |         |         |         |         |

So the total permutation with 5 inversions I(5,5) = I(4,5) + I(4,4) + I(4,3) + I(4,2) + I(4,1) = 3 + 5 + 6 + 5 + 3 = 22

So I(n, k) = sum of I(n-1, k-i) such that i < n && k-i >= 0

Also, k can go up to n*(n-1)/2 this occurs when the sequence is sorted in decreasing order https://secweb.cs.odu.edu/~zeil/cs361/web/website/Lectures/insertion/pages/ar01s04s01.html http://www.algorithmist.com/index.php/SPOJ_PERMUT1

#include <stdio.h>

int dp[100][100];

int inversions(int n, int k)
{
    if (dp[n][k] != -1) return dp[n][k];
    if (k == 0) return dp[n][k] = 1;
    if (n == 0) return dp[n][k] = 0;
    int j = 0, val = 0;
    for (j = 0; j < n && k-j >= 0; j++)
        val += inversions(n-1, k-j);
    return dp[n][k] = val;
}

int main()
{
    int t;
    scanf("%d", &t);
    while (t--) {
        int n, k, i, j;
        scanf("%d%d", &n, &k);
        for (i = 1; i <= n; i++)
            for (j = 0; j <= k; j++)
                dp[i][j] = -1;
        printf("%d\n", inversions(n, k));
    }
    return 0;
}

回答2:

It's one day later and I have managed to solve the problem using dynamic programming. I submitted it and my code was was accepted by SPOJ so I figure I'll share my knowledge here for anyone who is interested in the future.

After looking in the Wikipedia page which discusses inversion in discrete mathematics, I found an interesting recommendation at the bottom of the page.

Numbers of permutations of n elements with k inversions; Mahonian numbers: A008302

I clicked on the link to OEIS and it showed me an infinite sequence of integers called the Triangle of Mahonian numbers.

1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1 . . .

I was curious about what these numbers were since they seemed familiar to me. Then I realized that I had seen the subsequence 1, 3, 5, 6, 5, 3, 1 before. In fact, this was the answer to the problem for several pairs of (n, k), namely (4, 0), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6). I looked at what was on both sides of this subsequence and was amazed to see that it was all valid (i.e. greater than 0 permutations) answers for n < 4 and n > 4.

The formula for the sequence was given as:

coefficients in expansion of Product_{i=0..n-1} (1+x+...+x^i)

This was easy enough for me to understand and verify. I could basically take any n and plug into the formula. Then the coefficient for the x^k term would be the answer for (n, k).

I will show an example for n = 3.

(x0)(x0 + 1)(x0 + x1 + x2) = (1)(1 + x)(1 + x + x2) = (1 + x)(1 + x + x2) = 1 + x + x + x2 + x2 + x3 = 1 + 2x + 2x2 + x3

The final expansion was 1 + 2x + 2x2 + x3 and the coefficients of the x^k terms were 1, 2, 2, and 1 for k = 0, 1, 2, 3 respectively. This just happens to be all valid numbers of inversions for 3-element permutations.

1, 2, 2, 1 is the 3rd row of the Mahonian numbers when they are laid out in a table as follows:

1
1 1
1 2 2 1
1 3 5 6 5 3 1
etc.

So basically computing my answer came down to simply calculating the nth Mahonian row and taking the kth element with k starting at 0 and printing 0 if the index was out of range. This was a simple case of bottom-up dynamic programming since each ith row could be used to easily compute the i+1st row.

Given below is the Python solution I used which ran in only 0.02 seconds. The maximum time limit for this problem was 3 seconds for their given test cases and I was getting a timeout error before so I think this optimization is rather good.

def mahonian_row(n):
    '''Generates coefficients in expansion of 
    Product_{i=0..n-1} (1+x+...+x^i)
    **Requires that n is a positive integer'''
    # Allocate space for resulting list of coefficients?
    # Initialize them all to zero?
    #max_zero_holder = [0] * int(1 + (n * 0.5) * (n - 1))

    # Current max power of x i.e. x^0, x^0 + x^1, x^0 + x^1 + x^2, etc.
    # i + 1 is current row number we are computing
    i = 1
    # Preallocate result
    # Initialize to answer for n = 1
    result = [1]
    while i < n:
        # Copy previous row of n into prev
        prev = result[:]
        # Get space to hold (i+1)st row
        result = [0] * int(1 + ((i + 1) * 0.5) * (i))
        # Initialize multiplier for this row
        m = [1] * (i + 1)
        # Multiply
        for j in range(len(m)):
            for k in range(len(prev)):
                result[k+j] += m[j] * prev[k]
        # Result now equals mahonian_row(i+1)
        # Possibly should be memoized?
        i = i + 1
    return result


def main():
    t = int(raw_input())
    for _ in xrange(t):
        n, k = (int(s) for s in raw_input().split())
        row = mahonian_row(n)
        if k < 0 or k > len(row) - 1:
            print 0
        else:
            print row[k]


if __name__ == '__main__':
    main()

I have no idea of the time complexity but I am absolutely certain this code can be improved through memoization since there are 10 given test cases and the computations for previous test cases can be used to "cheat" on future test cases. I will make that optimization in the future, but hopefully this answer in its current state will help anyone attempting this problem in the future since it avoids the naive factorial-complexity approach of generating and iterating through all permutations.

回答3:

If there is a dynamic programming solution, there is probably a way to do it step by step, using the results for permutations of length n to help with the results for permutations of length n+1.

Given a permutation of length n - values 1-n, you can get a permutation of length n+1 by adding value (n+1) at n+1 possible positions. (n+1) is larger than any of 1-n so the number of inversions you create when you do this depends on where you add it - add it at the last position and you create no inversions, add it at the last but one position and you create one inversion, and so on - look back at the n=4 cases with one inversion to check this.

So if you consider one of n+1 places where you can add (n+1) if you add it at place j counting from the right so the last position as position 0 the number of permutations with K inversions this creates is the number of permutations with K-j inversions on n places.

So if at each step you count the number of permutations with K inversions for all possible K you can update the number of permutations with K inversions for length n+1 using the number of permutations with K inversions for length n.

回答4:

A major problem in computing these coefficients is the size of the order of the resultant product. The polynomial Product i=1,2,..,n {(1+x).(1+x+x^2)....(1+x+x^2+..+x^i)+...(1+x+x^2+...+x^n) will have an order equivalent to n*(n+1). Consequently, this puts a restrictive computational limit on the process. If we use a process where the previous results for the Product for n-1 are used in the process for computation of the Product for n, we are looking at the storage of (n-1)*n integers. It is possible to use a recursive process, which will be much slower, and again it is limited to integers less than the square root of the common size of the integer. The following is some rough and ready recursive code for this problem. The function mahonian(r,c) returns the c th coefficient for the r th Product. But again it is extremely slow for large Products greater than 100 or so. Running this it can be seen that recursion is clearly not the answer.

unsigned int numbertheory::mahonian(unsigned int r, unsigned int c)
  {
      unsigned int result=0;
      unsigned int k;

     if(r==0 && c==0)
       return 1;
     if( r==0 && c!=0)
      return 0;

   for(k=0; k <= r; k++)
       if(r > 0 && c >=k)
           result = result + mahonian(r-1,c-k);

   return result;

}

As a matter of interest I have included the following which is a c++ version of Sashank which is lot more faster than my recursion example. Note I use the armadillo library.

uvec numbertheory::mahonian_row(uword n){
 uword i = 2;
 uvec current;
 current.ones(i);
 uword current_size;
 uvec prev;
 uword prev_size;

 if(n==0){
   current.ones(1);
   return current;
 }

 while (i <= n){                  // increment through the rows
   prev_size=current.size();     // reset prev size to current size
   prev.set_size(prev_size);     // set size of prev vector
   prev= current;                //copy contents of current to prev vector
   current_size =1+ (i*(i+1)/2);      // reset current_size
   current.zeros(current_size);    // reset current vector with zeros

   for(uword j=0;j<i+1; j++)       //increment through current vector
      for(uword k=0; k < prev_size;k++)
         current(k+j) += prev(k);
   i++;                                        //increment to next row
}
return current;                                //return current vector
 }

 uword numbertheory::mahonian_fast(uword n, uword c) {
**This function returns the coefficient of c order of row n of
**the Mahonian numbers
    // check for input errors
    if(c >= 1+ (n*(n+1)/2)) {
        cout << "Error. Invalid input parameters" << endl;
   }
   uvec mahonian;
   mahonian.zeros(1+ (n*(n+1)/2));
   mahonian = mahonian_row(n);
   return mahonian(c);
 }

回答5:

We can make use to dynamic programming to solve this problem. we have n place to fill with numbers to from 1 to n, _ _ _ _ _ _ _ take n=7, then at very first place we can achieve atmost n-1 inversion and at least 0 , similarly for second place we can achieve atmost n-2 inversion and at least 0, in general, we can achieve atmost n-i inversions at ith index, irrespective of the choice of number we place before. our recursive formula will look like :

f(n,k) = f(n-1,k) + f(n-1,k-1) + f(n-1,k-2) ............. f(n-1,max(0,k-(n-1)) no inversion one inversion two inversion n-1 inversion we can achieve 0 inversions by placing smallest of the remaining number from the set (1,n) 1 inversion by placing second smallest and so on,

base condition for our recursive formula will be.

if( i==0 && k==0 ) return 1(valid permutation)

if( i==0 && k!=0 ) return 0 (invalid permutation).

if we draw recursion tree we will see subproblems repeated multiple times, Hence use memoization to reduce complexity to O(n*k).