Consider a n x n grid with indexes of top left corner as (0, 0). Dyck path is a staircase walk from bottom left, i.e., (n-1, 0) to top right, i.e., (0, n-1) that lies above the diagonal cells (or cells on line from bottom left to top right).
The task is to count the number of Dyck Paths from (n-1, 0) to (0, n-1).
Input : n = 1 Output : 1 Input : n = 2 Output : 2 Input : n = 3 Output : 5 Input : n = 4 Output : 14
The number of Dyck paths from (n-1, 0) to (0, n-1) can be given by the Catalan numberC(n).
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Below are the implementations to find count of Dyck Paths (or n’th Catalan number).
Number of Dyck Paths is 14
- Find number of sequences of 1 and -1 such that every sequence follows below constraints :
a) The length of a sequence is 2n
b) There are equal number of 1’s and -1’s, i.e., n 1’s, n -1s
c) Sum of prefix of every sequence is greater than or equal to 0. For example, 1, -1, 1, -1 and 1, 1, -1, -1 are valid, but -1, -1, 1, 1 is not valid.
- Number of paths of length m + n from (m-1, 0) to (0, n-1) that are restricted to east and north steps.
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