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Eulerian Numbers

Eulerian Numbers Quiz

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Questions

Question 1 Multiple Choice (Single Answer)

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.
Question 2 Multiple Choice (Single Answer)

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. $A(n, k) = A(n-1, k) + (n-k+1)A(n-1, k+1)$
Question 3 Multiple Choice (Single Answer)

What is the initial condition for Eulerian numbers?

  1. $A(0, 0) = 1$
  2. $A(1, 0) = 1$
  3. $A(0, 1) = 1$
  4. $A(1, 1) = 1$
Question 4 Multiple Choice (Single Answer)

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. $F(x) = \frac{1+x}{1-xe^x}$
Question 5 Multiple Choice (Single Answer)

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. $A(n, k) \sim \frac{1}{k!}n^{k+2}$
Question 6 Multiple Choice (Single Answer)

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. $A(n, k) = \sum_{i=0}^k S(n+1, i+1)$
Question 7 Multiple Choice (Single Answer)

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. $A(n, k) = L(n+1, k+1)$
Question 8 Multiple Choice (Single Answer)

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. $A(2n+1, n+1) = C_n$
Question 9 Multiple Choice (Single Answer)

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. $A(2n+1, n+1) = M_{2n+1}$
Question 10 Multiple Choice (Single Answer)

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. $A(n, k) = N(n+1, k+1)$
Question 11 Multiple Choice (Single Answer)

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. $A(n, k) = D(n+1, k+1)$
Question 12 Multiple Choice (Single Answer)

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. $A(n, k) = S(n+1, k+1)$
Question 13 Multiple Choice (Single Answer)

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. $A(n, k) = G(n+1, k+1)$
Question 14 Multiple Choice (Single Answer)

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. $A(n, k) = B(n+1, k+1)$
Question 15 Multiple Choice (Single Answer)

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