Entringer Number

The Entringer Number E(n, k) are the number of permutations of {1, 2, ..., n + 1}, starting with k + 1, which, after initially falling, alternatively fall then rise. The Entringer are given by: 

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For example, for n = 4 and k = 2, E(4, 2) is 4. 
They are: 
3 2 4 1 5 
3 2 5 1 4 
3 1 4 2 5 
3 1 5 2 4

Examples :

Input : n = 4, k = 2
Output : 4

Input : n = 4, k = 3
Output : 5

Below is program to find Entringer Number E(n, k). The program is based on above simple recursive formula. 

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Below is the implementation of finding Entringer Number using Dynamic Programming: 

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Time Complexity: O(n * n)
Auxiliary Space: O(n * k)

Efficient approach : Space optimization

In previous approach the current value dp[i][j] is only depend upon the current and previous row values of DP. So to optimize the space complexity we use a single 1D array to store the computations.

Implementation steps:

• Create a 1D vector dp of size K+1.
• Set a base case by initializing the values of DP .
• Now iterate over subproblems by the help of nested loop and get the current value from previous computations.
• Now Create a temporary 1d vector curr used to store the current values from previous computations.
• After every iteration assign the value of curr to dp for further iteration.
• At last return and print the final answer stored in dp[k].

Implementation:

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Time complexity: O(N*K)
Auxiliary Space: O(K)