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3426. Manhattan Distances of All Arrangements of Pieces

Description

You are given three integers m, n, and k.

There is a rectangular grid of size m × n containing k identical pieces. Return the sum of Manhattan distances between every pair of pieces over all valid arrangements of pieces.

A valid arrangement is a placement of all k pieces on the grid with at most one piece per cell.

Since the answer may be very large, return it modulo 109 + 7.

The Manhattan Distance between two cells (xi, yi) and (xj, yj) is |xi - xj| + |yi - yj|.

 

Example 1:

Input: m = 2, n = 2, k = 2

Output: 8

Explanation:

The valid arrangements of pieces on the board are:

  • In the first 4 arrangements, the Manhattan distance between the two pieces is 1.
  • In the last 2 arrangements, the Manhattan distance between the two pieces is 2.

Thus, the total Manhattan distance across all valid arrangements is 1 + 1 + 1 + 1 + 2 + 2 = 8.

Example 2:

Input: m = 1, n = 4, k = 3

Output: 20

Explanation:

The valid arrangements of pieces on the board are:

  • The first and last arrangements have a total Manhattan distance of 1 + 1 + 2 = 4.
  • The middle two arrangements have a total Manhattan distance of 1 + 2 + 3 = 6.

The total Manhattan distance between all pairs of pieces across all arrangements is 4 + 6 + 6 + 4 = 20.

 

Constraints:

  • 1 <= m, n <= 105
  • 2 <= m * n <= 105
  • 2 <= k <= m * n

Solutions

Solution 1

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