An inversion in σ ∈ Sn is a pair (i, j) such that i < j...

An inversion in σ ∈ S_n is a pair (i, j) such that i < j and σ(i) > σ(j). Take σ, τ ∈ S_n and take i < j.

- Suppose (i, j) is not an inversion in τ. Show that (i, j) is an inversion in στ if and only if (τ(i), τ(j)) is an inversion in σ.

- Suppose (i, j) is an inversion in τ. Show that (i, j) is an inversion in στ if and only if (τ(j), τ(i)) is not an inversion in σ.

Expert Solution

Colby Messinger answered 1 day ago

3. Let S3 act on the set A={(i,j) : 1≤i,j≤3} by σ((i, j)) = (σ(i), σ(j))....

3. Let S3 act on the set A={(i,j) : 1≤i,j≤3} by σ((i, j)) = (σ(i), σ(j)). (a) Describe the orbits of this action. (b) Show this is a faithful action, i.e. that the permutation represen- tation φ:S3 →SA =S9 (c) For each σ ∈ S3, find the cycle decomposition of φ(σ) in S9.

Show that for all σ ∈ Sn we have sgn(σ) = sgn(σ−1). Does σ = (1,2,3,5,4)−1...

Show that for all σ ∈ Sn we have sgn(σ) = sgn(σ−1). Does σ = (1,2,3,5,4)−1 ∈ S13 belong to the alternating group A13? Justify your answer

3. Every element of Sn can be written as a product of disjoint cycles. If σ...

3. Every element of Sn can be written as a product of disjoint cycles. If σ = (i1 i2)(j1 j2)(k1 k2 k3 k4), with the cycles disjoint, we say that σ has cyclic structure ( )( )( ). (a) Find all possible cyclic structures in S7. Hint: There are 15. (b) Using part (a), find all possible orders in S7. (c) Find all possible orders in A7.

i j(i) i j(i) i j(i) i j(i) 0 -0.0499 7 -0.08539 13 0.144812 19 0.08021...

i j(i) i j(i) i j(i) i j(i) 0 -0.0499 7 -0.08539 13 0.144812 19 0.08021 1 0.107506 8 0.062922 14 -0.0499 20 -0.30103 2 -0.06719 9 -0.04444 15 -0.18366 21 -0.33834 3 -0.04717 10 0.219422 16 -0.02898 22 0.058373 4 -0.09176 11 0.083849 17 0.08021 23 0.79083 5 -0.25918 12 -0.02261 18 -0.14271 24 0.130254 6 0.055643 25 -0.10177 (3 points) Please encipher the following plaintext with Caesar Cipher and the encryption key of 10: TODAYISTUESDAY; (3 points) In...

Suppose there are n independent Gaussian r.v.s, X j ∼ N ( μ j , σ...

Suppose there are n independent Gaussian r.v.s, X j ∼ N ( μ j , σ 2 j ) for j = 1 , 2 , . . . , n , with possibly different means and variances (a) For any constants a j’s, find the MGF of a linear combination of these n independent Gaussian r.v.s, i.e., the MGF ofY=a1X1+a2X2+· · ·anXn(=Pnj=1ajXj) Hint: Since MX(t) =E(etX), it is the case that E(eatX) =MX(at), i.e., the MGF evaluated at at. (b)...

Prove that Sn is generated by {(i i + 1) I 1 < i < n - 1}.

(a) Prove that Sn is generated by the elements in the set {(i i+1) : 1≤i≤n}....

(a) Prove that Sn is generated by the elements in the set {(i i+1) : 1≤i≤n}. [Hint: Consider conjugates, for example (2 3) (1 2) (2 3)−1.] (b) ProvethatSn isgeneratedbythetwoelements(12)and(123...n) for n ≥ 3. (c) Prove that H = 〈(1 3), (1 2 3 4)〉 is a proper subgroup of S4.

Let L = {aibj | i ≠ j; i, j ≥ 0}. Design a CFG and...

Let L = {aibj | i ≠ j; i, j ≥ 0}. Design a CFG and a PDA for this language. Provide a direct design for both CFG and PDA (no conversions from one form to another allowed).

a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈...

a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈ {1, 2, 3, 4, 5}}. S5 will also act on this set. Consider the subgroup H = <(1, 2)(3, 4), (1, 3)(2, 4)>≤ S5. (a) Find the orbits of H in this action. Justify your answers. (b) For each orbit find the stabiliser one of its members. Justify your answers. action is this t.(i,j)=(t(i),t(j))

def longest(string): start=0;end=1;i=0; while i<len(string): j=i+1 while j<len(string) and string[j]>string[j-1]: j+=1 if end-start<j-i: #update if current...

def longest(string): start=0;end=1;i=0; while i<len(string): j=i+1 while j<len(string) and string[j]>string[j-1]: j+=1 if end-start<j-i: #update if current string has greater length than #max start and end end=j start=i i=j; avg=0 for i in string[start:end]: avg+=int(i) print('The longest string in ascending order is',string[start:end]) print('Teh average is',avg/(end-start)) s=input('Enter a string') longest(s) i need a definition and explanation of this code that how it works?

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