

Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation....
Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 < i S n 1 such that p, > Pi+1. How many permutations in Sn have exactly one descent?
Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 Pi+1. How many permutations in Sn have exactly one descent?
maybe use induction to prove?
Problem 2: Let p-p.Pn be a permutation considered in its one-line notation. An inversion in p is a pair 1 i<jS n such that j appears to the left of i in p (i.e., an out-of-order pair). Let inv(p) be the total number of inversions in p. Prove that PES where z is a variable.
Problem 2: Let p-p.Pn be a permutation considered in its one-line notation. An inversion in p is a pair 1...
Problem 6. Let P be an n × n permutation matrix with 1's on the anti-diagonal. Find det(P), Hint: How many exchange permutations are needed to implement P?
Problem 6. Let P be an n × n permutation matrix with 1's on the anti-diagonal. Find det(P), Hint: How many exchange permutations are needed to implement P?
Let w e Sbe a permutation which rearranges 8 objects identified with letters, altering their positions to become as in the lower line of what follows: [A B C D E F G H (F DAEH C B G a) Express w as a product of disjoint cycles. Is w an even permutation, or an odd permutation? What is its order? b) Calculate wy, w and w-2 as products of disjoint cycles. c) Does there exist TE Sg for which...
Let wE S7 be a permutation which rearranges 7 objects as follows, showing the result on the lower line 2 3 4 6 7 5 5 4 2 7 6 1 3 a) Express was a product of disjoint cycles representing how each object moves Is w an even permutation, or an odd permutation? What is its order? products of disjoint cycles b) Calculate w3, w5 and w' 2 as c) Does there exist T E S7 for which T-lwr...
(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X-n -2? (c) What is the probability that X-n-1? (d) What is the expectation of X? (Hint:...
Let f [n]n] be a permutation. A fixed point of f is an element x e [n] such that f(x)-x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X 2? (c) What is the probability that X--1? (d) What is the expectation of X? (Hint: As usual, express X as...
Please solve all in detail!
7. Recall that if o E Sn then P, is the nxn "permutation matrix” satisfying Col;(P.) = Co(j) for j = 1,...,n. (See the Tutorial notes for Feb. 24 for more information.) (a) Prove that if o, T E Sn then P.P, = Por. (Hint: it suffices to prove Col;(P.P.) = Col;(Por) for all j. Use the general fact that Col;(A) = Ae..] (b) Suppose o E Sn given in cycle notation is o =...
9. Let f be the following permutation in the symmetric group S9, written in two-line notation. 1 2 3 4 5 6 7 8 9 5 9 4 8 2 6 1 3 7 (a) Determine f3121 and explain why your answer is correct. (b) Determine ord(f) (c) Find a permutation p such that p-f
9. Let f be the following permutation in the symmetric group S9, written in two-line notation. 1 2 3 4 5 6 7 8 9...
Can you explain to me how this works? Specifically, how does
the permutation multiplication work. How does (1,3,4,6)(2,3,5)
become the 2 permutations multiplied together. I guess I am lost on
all of it.
4. Let T = (1,3,4,6)(2,3,5) in Ss. Find the index of <T> in So. S61 Solution: If we let H = (r), then we are looking for (S. : H) However, we cannot simply claim that H = 12 because the cycle decomposition for T is not...