Eulerian Numbers

Description: Eulerian Numbers Quiz
Number of Questions: 15
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Tags: combinatorics eulerian numbers
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What is the definition of Eulerian numbers?

  1. The number of permutations of n elements that have exactly k inversions.

  2. The number of permutations of n elements that have no inversions.

  3. The number of permutations of n elements that have exactly k fixed points.

  4. The number of permutations of n elements that have no fixed points.


Correct Option: A
Explanation:

Eulerian numbers are defined as the number of permutations of n elements that have exactly k inversions.

What is the recurrence relation for Eulerian numbers?

  1. $A(n, k) = A(n-1, k-1) + (n-k)A(n-1, k)$

  2. $A(n, k) = A(n-1, k) + (n-k+1)A(n-1, k-1)$

  3. $A(n, k) = A(n-1, k) + (n-k)A(n-1, k+1)$

  4. $A(n, k) = A(n-1, k) + (n-k+1)A(n-1, k+1)$


Correct Option: A
Explanation:

The recurrence relation for Eulerian numbers is $A(n, k) = A(n-1, k-1) + (n-k)A(n-1, k)$.

What is the initial condition for Eulerian numbers?

  1. $A(0, 0) = 1$

  2. $A(1, 0) = 1$

  3. $A(0, 1) = 1$

  4. $A(1, 1) = 1$


Correct Option: A
Explanation:

The initial condition for Eulerian numbers is $A(0, 0) = 1$.

What is the generating function for Eulerian numbers?

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

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

  3. $F(x) = \frac{1-x}{1+xe^x}$

  4. $F(x) = \frac{1+x}{1-xe^x}$


Correct Option: A
Explanation:

The generating function for Eulerian numbers is $F(x) = \frac{1-x}{1-xe^x}$.

What is the asymptotic formula for Eulerian numbers?

  1. $A(n, k) \sim \frac{1}{k!}n^k$

  2. $A(n, k) \sim \frac{1}{k!}n^{k+1}$

  3. $A(n, k) \sim \frac{1}{k!}n^{k-1}$

  4. $A(n, k) \sim \frac{1}{k!}n^{k+2}$


Correct Option: A
Explanation:

The asymptotic formula for Eulerian numbers is $A(n, k) \sim \frac{1}{k!}n^k$.

What is the relationship between Eulerian numbers and Stirling numbers of the second kind?

  1. $A(n, k) = \sum_{i=0}^k S(n, i)$

  2. $A(n, k) = \sum_{i=0}^k S(n+1, i)$

  3. $A(n, k) = \sum_{i=0}^k S(n, i+1)$

  4. $A(n, k) = \sum_{i=0}^k S(n+1, i+1)$


Correct Option: A
Explanation:

The relationship between Eulerian numbers and Stirling numbers of the second kind is $A(n, k) = \sum_{i=0}^k S(n, i)$.

What is the relationship between Eulerian numbers and Lah numbers?

  1. $A(n, k) = L(n, k+1)$

  2. $A(n, k) = L(n+1, k)$

  3. $A(n, k) = L(n, k)$

  4. $A(n, k) = L(n+1, k+1)$


Correct Option: C
Explanation:

The relationship between Eulerian numbers and Lah numbers is $A(n, k) = L(n, k)$.

What is the relationship between Eulerian numbers and Catalan numbers?

  1. $A(2n, n) = C_n$

  2. $A(2n+1, n) = C_n$

  3. $A(2n, n+1) = C_n$

  4. $A(2n+1, n+1) = C_n$


Correct Option: A
Explanation:

The relationship between Eulerian numbers and Catalan numbers is $A(2n, n) = C_n$.

What is the relationship between Eulerian numbers and Motzkin numbers?

  1. $A(2n, n) = M_{2n}$

  2. $A(2n+1, n) = M_{2n+1}$

  3. $A(2n, n+1) = M_{2n}$

  4. $A(2n+1, n+1) = M_{2n+1}$


Correct Option: A
Explanation:

The relationship between Eulerian numbers and Motzkin numbers is $A(2n, n) = M_{2n}$.

What is the relationship between Eulerian numbers and Narayana numbers?

  1. $A(n, k) = N(n, k+1)$

  2. $A(n, k) = N(n+1, k)$

  3. $A(n, k) = N(n, k)$

  4. $A(n, k) = N(n+1, k+1)$


Correct Option: C
Explanation:

The relationship between Eulerian numbers and Narayana numbers is $A(n, k) = N(n, k)$.

What is the relationship between Eulerian numbers and Delannoy numbers?

  1. $A(n, k) = D(n, k+1)$

  2. $A(n, k) = D(n+1, k)$

  3. $A(n, k) = D(n, k)$

  4. $A(n, k) = D(n+1, k+1)$


Correct Option: C
Explanation:

The relationship between Eulerian numbers and Delannoy numbers is $A(n, k) = D(n, k)$.

What is the relationship between Eulerian numbers and Schröder numbers?

  1. $A(n, k) = S(n, k+1)$

  2. $A(n, k) = S(n+1, k)$

  3. $A(n, k) = S(n, k)$

  4. $A(n, k) = S(n+1, k+1)$


Correct Option: C
Explanation:

The relationship between Eulerian numbers and Schröder numbers is $A(n, k) = S(n, k)$.

What is the relationship between Eulerian numbers and Genocchi numbers?

  1. $A(n, k) = G(n, k+1)$

  2. $A(n, k) = G(n+1, k)$

  3. $A(n, k) = G(n, k)$

  4. $A(n, k) = G(n+1, k+1)$


Correct Option: C
Explanation:

The relationship between Eulerian numbers and Genocchi numbers is $A(n, k) = G(n, k)$.

What is the relationship between Eulerian numbers and Bell numbers?

  1. $A(n, k) = B(n, k+1)$

  2. $A(n, k) = B(n+1, k)$

  3. $A(n, k) = B(n, k)$

  4. $A(n, k) = B(n+1, k+1)$


Correct Option: C
Explanation:

The relationship between Eulerian numbers and Bell numbers is $A(n, k) = B(n, k)$.

What is the relationship between Eulerian numbers and Fibonacci numbers?

  1. $A(n, k) = F(n+k)$

  2. $A(n, k) = F(n-k)$

  3. $A(n, k) = F(n+k+1)$

  4. $A(n, k) = F(n-k+1)$


Correct Option: C
Explanation:

The relationship between Eulerian numbers and Fibonacci numbers is $A(n, k) = F(n+k+1)$.

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