Delannoy Numbers
Questions about Delannoy numbers, their properties, recurrence relations, and applications in combinatorics
Questions
Question 1 Multiple Choice (Single Answer)
What is the Delannoy number D(n, m)?
- The number of paths from (0, 0) to (n, m) in the grid, where each step is either right or up.
- The number of paths from (0, 0) to (n, m) in the grid, where each step is either right or down.
- The number of paths from (0, 0) to (n, m) in the grid, where each step is either right or diagonal.
- The number of paths from (0, 0) to (n, m) in the grid, where each step is either up or diagonal.
Question 2 Multiple Choice (Single Answer)
What is the recurrence relation for the Delannoy numbers?
- D(n, m) = D(n - 1, m) + D(n, m - 1)
- D(n, m) = D(n - 1, m) - D(n, m - 1)
- D(n, m) = D(n - 1, m) * D(n, m - 1)
- D(n, m) = D(n - 1, m) / D(n, m - 1)
Question 3 Multiple Choice (Single Answer)
What is the generating function for the Delannoy numbers?
- F(x, y) = (1 - x - y)^-1
- F(x, y) = (1 + x + y)^-1
- F(x, y) = (1 - x * y)^-1
- F(x, y) = (1 + x * y)^-1
Question 4 Multiple Choice (Single Answer)
What is the asymptotic formula for the Delannoy numbers?
- D(n, m) ~ (2 / sqrt(pi)) * n^(3/2) * m^(3/2)
- D(n, m) ~ (2 / sqrt(pi)) * n^(1/2) * m^(1/2)
- D(n, m) ~ (2 / sqrt(pi)) * n^(5/2) * m^(5/2)
- D(n, m) ~ (2 / sqrt(pi)) * n^(7/2) * m^(7/2)
Question 5 Multiple Choice (Single Answer)
What are some applications of the Delannoy numbers?
- Counting lattice paths
- Counting tilings
- Counting permutations
- Counting graphs