handshakes-that-dont-cross

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1259. Handshakes That Don't Cross (Hard)

You are given an even number of people num_people that stand around a circle and each person shakes hands with someone else, so that there are num_people / 2 handshakes total.

Return the number of ways these handshakes could occur such that none of the handshakes cross.

Since this number could be very big, return the answer mod 10^9 + 7

 

Example 1:

Input: num_people = 2
Output: 1

Example 2:

Input: num_people = 4
Output: 2
Explanation: There are two ways to do it, the first way is [(1,2),(3,4)] and the second one is [(2,3),(4,1)].

Example 3:

Input: num_people = 6
Output: 5

Example 4:

Input: num_people = 8
Output: 14

 

Constraints:

• 2 <= num_people <= 1000
• num_people % 2 == 0

Related Topics

[Math] [Dynamic Programming]

Hints

Hint 1

Use dynamic programming.

Hint 2

Let dp[n] be the number of ways that n people can handshake.

Hint 3

Then fix a person as a pivot and turn for every other person who will have a handshake, the answer is the sum of the products of the new two subproblems.