Involutions, Derangements, and Higher-Order Permutation Theory Quiz

1) The number of permutations of a multiset with elements of sizes n₁, n₂,…, nₖ summing to n is:

N!

( \frac{n!}{n_1!n_2!\cdots n_k!} )

( \prod_{i=1}^k n_i! )

Nᵏ

Divide by factorials of repeated elements to count distinct permutations.

Explanation

Divide by factorials of repeated elements to count distinct permutations.

2) How many derangements (D_n) of a 6-element set?

265

720

⌊6!/e⌋

6! − 6

(D_6 = 265).

Explanation

(D_6 = 265).

3) Every permutation of an n-element set can be uniquely written as a product of disjoint cycles.

True

False

This is the fundamental cycle decomposition theorem.

Explanation

This is the fundamental cycle decomposition theorem.

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5) Match the Following

Transposition

Cycle decomposition

Permutation group

Parity of a permutation

Standard definitions of permutation concepts.

Explanation

Standard definitions of permutation concepts.

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6) Number of permutations of n elements with exactly one fixed point:

(D_n)

(n D_{n-1})

( \binom{n}{1} D_{n-1} )

( n!/e )

Choose which one point is fixed (n choices), derange the rest.

Explanation

Choose which one point is fixed (n choices), derange the rest.

7) If a permutation has an odd number of even-length cycles, it must be odd.

True

False

Parity depends on total transpositions, not cycle lengths alone.

Explanation

Parity depends on total transpositions, not cycle lengths alone.

8) How many 9-digit rearrangements of digits 1–9 have no digit in its original position?

9!

(D_9)

(9 D_8)

(9!/e)

This is exactly the definition of a derangement.

Explanation

This is exactly the definition of a derangement.

9) Which statements about permutations are true?

Every permutation in (S_n) has order ≤ n.

Number of permutations with a cycle type depends only on the cycle lengths.

Any permutation can be generated by transpositions.

(S_n) is abelian.

Cycle type determines count; transpositions generate (S_n); but (S_n) is non-abelian for n ≥ 3.

Explanation

Cycle type determines count; transpositions generate (S_n); but (S_n) is non-abelian for n ≥ 3.

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10) How many permutations of {1,…,n} have no cycles of length 1?

(D_n)

N!

( \sum_{k=0}^n (-1)^k \frac{n!}{k!} )

Both a and c

Both formulas represent the derangement number.

Explanation

Both formulas represent the derangement number.

11) In (S_n), any two disjoint cycles commute.

True

False

Disjoint cycles act on separate elements.

Explanation

Disjoint cycles act on separate elements.

12) How many permutations of n elements have exactly k cycles?

Stirling numbers of first kind (c_{n,k})

Nᵏ

(n!/k!)

Bell numbers

(c_{n,k}) count permutations with k cycles.

Explanation

(c_{n,k}) count permutations with k cycles.

13) Probability that a random permutation of n contains no 1-cycles or 2-cycles tends to:

0

(1/e)

(1/e^2)

Removing 1-cycles gives factor (1/e); removing 2-cycles gives another (1/e).

Explanation

Removing 1-cycles gives factor (1/e); removing 2-cycles gives another (1/e).

14) Match the Following

Descent number

Inversion number

Major index

Cycle index

Each statistic corresponds to a standard combinatorial measure.

Explanation

Each statistic corresponds to a standard combinatorial measure.

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15) Conditions guaranteeing two permutations are conjugate in (S_n):

Same order

Same number of cycles

Same cycle type

Same parity

Conjugacy in (S_n) depends only on cycle structure.

Explanation

Conjugacy in (S_n) depends only on cycle structure.

Involutions, Derangements, and Higher-Order Permutation Theory Quiz

1) The number of permutations of a multiset with elements of sizes n₁, n₂,…, nₖ summing to n is:

2)

What first name or nickname would you like us to use?

2) How many derangements (D_n) of a 6-element set?

3) Every permutation of an n-element set can be uniquely written as a product of disjoint cycles.

4) Total number of involutions (only 1- and 2-cycles) on n elements is:

5) Match the Following

6) Number of permutations of n elements with exactly one fixed point:

7) If a permutation has an odd number of even-length cycles, it must be odd.

8) How many 9-digit rearrangements of digits 1–9 have no digit in its original position?

9) Which statements about permutations are true?

10) How many permutations of {1,…,n} have no cycles of length 1?

11) In (S_n), any two disjoint cycles commute.

12) How many permutations of n elements have exactly k cycles?

13) Probability that a random permutation of n contains no 1-cycles or 2-cycles tends to:

14) Match the Following

15) Conditions guaranteeing two permutations are conjugate in (S_n):