How many {(1,1),(1,−1),(2,0)}-lattice paths are there from the point (0,0) to the line x=8?
Details and assumptions
A lattice path is a path in the Cartesian plane between points with integer coordinates.
A step in a lattice path is a single move from one point with integer coordinates to another.
The size of the step from (x1,y1) to (x2,y2) is (x2−x1,y2−y1).
For a set S={(xi,yi)}ki=1, an S-lattice path is a lattice path where every step has size which is a member of S.
To determine the number of {(1,1),(1,−1),(2,0)}-lattice paths from (0,0) to the line x=8, we can use dynamic programming.
We'll define a 2D array dp[i][j] to represent the number of paths from (0,0) to the point (i,j). The base case for this problem is dp[0][0] = 1, as there is only one path from (0,0) to itself.
We can then iterate through each point (i,j) on the grid, starting from (0,0) and moving right. For each point, we'll calculate the number of paths to reach that point by summing up the number of paths from the previous points that can lead to this current point.
To calculate dp[i][j], we can look at the previous points that can lead to (i,j). In this case, we can reach (i,j) from three possible previous points, which are (i-1,j-1) using the step (1,1), (i-1,j+1) using the step (1,-1), and (i-2,j) using the step (2,0). Therefore, the formula to calculate dp[i][j] is as follows:
dp[i][j] = dp[i-1][j-1] + dp[i-1][j+1] + dp[i-2][j]
Once we iterate through all the points on the grid, the value of dp[8][j] for each j will represent the number of paths from (0,0) to the line x=8 using the given steps {(1,1),(1,−1),(2,0)}.
Here's an example implementation in Python:
```python
# Define the grid size
n = 8
# Create the 2D array for dp
dp = [[0] * n for _ in range(n)]
# Base case: There is only one path from (0,0) to itself
dp[0][0] = 1
# Iterate through each point on the grid
for i in range(1, n):
for j in range(n):
# Calculate the number of paths to reach (i,j)
dp[i][j] = dp[i-1][j-1] + dp[i-1][j+1] + dp[i-2][j]
# Find the number of paths from (0,0) to the line x=8
total_paths = sum(dp[n-1])
print("The number of {(1,1),(1,−1),(2,0)}-lattice paths from (0,0) to x=8 is:", total_paths)
```
By running this code, you will get the number of {(1,1),(1,−1),(2,0)}-lattice paths from (0,0) to the line x=8.