Schröder Numbers

Description: Schröder Numbers are a sequence of integers that arise in various combinatorial problems. They are closely related to the Catalan numbers and have applications in counting lattice paths, triangulations, and other combinatorial structures.
Number of Questions: 15
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Tags: combinatorics schröder numbers catalan numbers
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What is the general formula for the Schröder number (S_n)?

  1. (S_n = \frac{1}{n+1}\binom{2n}{n})

  2. (S_n = \frac{1}{n}\binom{2n+1}{n})

  3. (S_n = \frac{1}{n+2}\binom{2n+2}{n})

  4. (S_n = \frac{1}{n-1}\binom{2n-1}{n})


Correct Option: A
Explanation:

The Schröder number (S_n) is given by the formula (S_n = \frac{1}{n+1}\binom{2n}{n}).

What is the value of (S_5)?

  1. 42

  2. 56

  3. 70

  4. 84


Correct Option: D
Explanation:

Using the formula (S_n = \frac{1}{n+1}\binom{2n}{n}), we have (S_5 = \frac{1}{6}\binom{10}{5} = \frac{1}{6}\cdot 252 = 42).

What is the relationship between Schröder numbers and Catalan numbers (C_n)?

  1. (S_n = C_n)

  2. (S_n = 2C_n)

  3. (S_n = C_n + 1)

  4. (S_n = C_n - 1)


Correct Option: C
Explanation:

The Schröder number (S_n) is related to the Catalan number (C_n) by the formula (S_n = C_n + 1).

What is the generating function for the Schröder numbers?

  1. (F(x) = \frac{1}{1-x-x^2})

  2. (F(x) = \frac{1}{1+x+x^2})

  3. (F(x) = \frac{1}{1-x^2})

  4. (F(x) = \frac{1}{1+x^2})


Correct Option: A
Explanation:

The generating function for the Schröder numbers is (F(x) = \frac{1}{1-x-x^2}).

What is the asymptotic behavior of the Schröder numbers?

  1. (S_n \sim \frac{1}{\sqrt{n}}\left(\frac{4}{3}\right)^n)

  2. (S_n \sim \frac{1}{\sqrt{n}}\left(\frac{3}{4}\right)^n)

  3. (S_n \sim \frac{1}{n}\left(\frac{4}{3}\right)^n)

  4. (S_n \sim \frac{1}{n}\left(\frac{3}{4}\right)^n)


Correct Option: A
Explanation:

The asymptotic behavior of the Schröder numbers is given by (S_n \sim \frac{1}{\sqrt{n}}\left(\frac{4}{3}\right)^n).

What is the number of ways to triangulate a convex (n)-gon?

  1. (S_n)

  2. (C_n)

  3. (S_n + C_n)

  4. (S_n - C_n)


Correct Option: A
Explanation:

The number of ways to triangulate a convex (n)-gon is given by the Schröder number (S_n).

What is the number of ways to parenthesize a product of (n) factors?

  1. (S_n)

  2. (C_n)

  3. (S_n + C_n)

  4. (S_n - C_n)


Correct Option: B
Explanation:

The number of ways to parenthesize a product of (n) factors is given by the Catalan number (C_n).

What is the number of ways to arrange (n) distinct objects in a circle?

  1. (S_n)

  2. (C_n)

  3. (S_n + C_n)

  4. (S_n - C_n)


Correct Option: A
Explanation:

The number of ways to arrange (n) distinct objects in a circle is given by the Schröder number (S_n).

What is the number of ways to color the faces of a cube with (k) colors, such that no two adjacent faces have the same color?

  1. (S_k)

  2. (C_k)

  3. (S_k + C_k)

  4. (S_k - C_k)


Correct Option: A
Explanation:

The number of ways to color the faces of a cube with (k) colors, such that no two adjacent faces have the same color, is given by the Schröder number (S_k).

What is the number of ways to partition a set of (n) elements into (k) non-empty subsets?

  1. (S_k)

  2. (C_k)

  3. (S_k + C_k)

  4. (S_k - C_k)


Correct Option: A
Explanation:

The number of ways to partition a set of (n) elements into (k) non-empty subsets is given by the Schröder number (S_k).

What is the number of ways to place (n) non-attacking rooks on an (n \times n) chessboard?

  1. (S_n)

  2. (C_n)

  3. (S_n + C_n)

  4. (S_n - C_n)


Correct Option: A
Explanation:

The number of ways to place (n) non-attacking rooks on an (n \times n) chessboard is given by the Schröder number (S_n).

What is the number of ways to construct a binary tree with (n) internal nodes?

  1. (S_n)

  2. (C_n)

  3. (S_n + C_n)

  4. (S_n - C_n)


Correct Option: B
Explanation:

The number of ways to construct a binary tree with (n) internal nodes is given by the Catalan number (C_n).

What is the number of ways to triangulate a convex (n)-gon with (k) interior points?

  1. (S_{n+k})

  2. (C_{n+k})

  3. (S_{n+k} + C_{n+k})

  4. (S_{n+k} - C_{n+k})


Correct Option: A
Explanation:

The number of ways to triangulate a convex (n)-gon with (k) interior points is given by the Schröder number (S_{n+k}).

What is the number of ways to partition a set of (n) elements into (k) non-empty subsets, such that each subset contains at least (2) elements?

  1. (S_k)

  2. (C_k)

  3. (S_k + C_k)

  4. (S_k - C_k)


Correct Option: D
Explanation:

The number of ways to partition a set of (n) elements into (k) non-empty subsets, such that each subset contains at least (2) elements, is given by (S_k - C_k).

What is the number of ways to arrange (n) distinct objects in a row, such that no two adjacent objects are the same?

  1. (S_n)

  2. (C_n)

  3. (S_n + C_n)

  4. (S_n - C_n)


Correct Option: B
Explanation:

The number of ways to arrange (n) distinct objects in a row, such that no two adjacent objects are the same, is given by the Catalan number (C_n).

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