Given X={1,2,....,n}, let us call a permutation τ of X an adjacency if it is a...
Given X={1,2,....,n}, let us call a permutation τ of X an adjacency if it is a transposition of the form (i i+1) for i < n. If i<j prove that (i j) is a product of an odd number of adjacencies.
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Given X={1,2,....,n}, let us call a permutation τ of X an
adjacency if it is a...
(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x...
(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 cons...
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...
Let be a permutation of {1,2,……n}.Let -1 be the (n-1)-tuple with one element from missing. Alice...
Let
be a permutation of {1,2,……n}.Let
-1 be the (n-1)-tuple with one element from
missing.
Alice shows Bob
-1[i] one by one in the increasing order of i from 1 to
(n-1).bob’s task is to compute the missing element from
-1 that is in
with very limited – O(log n) bits – of memory.
Design an algorithm to compute the missing element in this
memory-limited and access-limited model, i.e Alice can only show
each number to Bob once, and Bob...
Question 6: Let n 2 2 be an integer and let ai,a2,...,an be a permutation of...
Question 6: Let n 2 2 be an integer and let ai,a2,...,an be a permutation of the set (1, 2, . . . ,n). Define ao = 0 and an+1 = 0, and consider the sequence do, 1, d2, l3, . . . , Un, Un+1 A position i with 1 i n is called auesome, if ai > ai-1 and ai > ai+1. In words, i is awesome if the value at position i is larger than both its...
Solve all parts please 5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simp...
Solve all parts please
5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simple graph with n vertices is an n x n matrix with entries that are all 0 or 1. The entries on the diagonal are all 0, and the entry in the ih row and jth column is 1 if there is an edge between vertex i and vertex j and is 0 if there is not an edge between vertex...
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.,...
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...
1, and let σ be a permutation of {1, , n). Recall that for each integer m a) Let n 1, we denote ơ...
1, and let σ be a permutation of {1, , n). Recall that for each integer m a) Let n 1, we denote ơm--σ ο . . . o σ. Show that n times b) Let 21, and let be a permutation of..,n consisting of a unique cycle of length n. Deduce from the previous question that there exists i e (1,..., n) such that i +c() )+22(n1).
1, and let σ be a permutation of {1, , n). Recall...
Question 6: Let n 2 1 be an integer and let A[1...n] be an array that...
Question 6: Let n 2 1 be an integer and let A[1...n] be an array that stores a permutation of the set { 1, 2, . .. , n). If the array A s sorted. then Ak] = k for k = 1.2. .., n and, thus. TL k-1 If the array A is not sorted and Ak-i, where iメk, then Ak-서 is equal to the "distance" that the valuei must move in order to make the array sorted. Thus,...
1. Show that the set of rational numbers of the form m /n, where m, n...
1. Show that the set of rational numbers of the form m /n, where m, n E Z and n is odd is a subgroup of QQ under addition. 2. Let H, K be subgroups of a group G. Prove: H n K is a subgroup of G 3. Let G be an abelian group. Let S-aEG o(a) is finite . Show that S is a subgroup of G 4. What is the largest order of a permutation in S10?...
Let (di, d2,... .dk) (w1, w2,..., wx)-0 (mod n) be an error detection scheme for the...
Let (di, d2,... .dk) (w1, w2,..., wx)-0 (mod n) be an error detection scheme for the k-digit identification number did2 . . .dk, where 0 〈 di 〈 n. Prove that all transposition errors of two digits di and dj are detected if and only if gcd(ui-y,n) = 1 for i and j between 1 and k
(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...
Let
be a permutation of {1,2,……n}.Let
-1 be the (n-1)-tuple with one element from
missing.
Alice shows Bob
-1[i] one by one in the increasing order of i from 1 to
(n-1).bob’s task is to compute the missing element from
-1 that is in
with very limited – O(log n) bits – of memory.
Design an algorithm to compute the missing element in this
memory-limited and access-limited model, i.e Alice can only show
each number to Bob once, and Bob...
Question 6: Let n 2 2 be an integer and let ai,a2,...,an be a permutation of the set (1, 2, . . . ,n). Define ao = 0 and an+1 = 0, and consider the sequence do, 1, d2, l3, . . . , Un, Un+1 A position i with 1 i n is called auesome, if ai > ai-1 and ai > ai+1. In words, i is awesome if the value at position i is larger than both its...
Solve all parts please
5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simple graph with n vertices is an n x n matrix with entries that are all 0 or 1. The entries on the diagonal are all 0, and the entry in the ih row and jth column is 1 if there is an edge between vertex i and vertex j and is 0 if there is not an edge between vertex...
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...
1, and let σ be a permutation of {1, , n). Recall that for each integer m a) Let n 1, we denote ơm--σ ο . . . o σ. Show that n times b) Let 21, and let be a permutation of..,n consisting of a unique cycle of length n. Deduce from the previous question that there exists i e (1,..., n) such that i +c() )+22(n1).
1, and let σ be a permutation of {1, , n). Recall...
Question 6: Let n 2 1 be an integer and let A[1...n] be an array that stores a permutation of the set { 1, 2, . .. , n). If the array A s sorted. then Ak] = k for k = 1.2. .., n and, thus. TL k-1 If the array A is not sorted and Ak-i, where iメk, then Ak-서 is equal to the "distance" that the valuei must move in order to make the array sorted. Thus,...
1. Show that the set of rational numbers of the form m /n, where m, n E Z and n is odd is a subgroup of QQ under addition. 2. Let H, K be subgroups of a group G. Prove: H n K is a subgroup of G 3. Let G be an abelian group. Let S-aEG o(a) is finite . Show that S is a subgroup of G 4. What is the largest order of a permutation in S10?...
Let (di, d2,... .dk) (w1, w2,..., wx)-0 (mod n) be an error detection scheme for the k-digit identification number did2 . . .dk, where 0 〈 di 〈 n. Prove that all transposition errors of two digits di and dj are detected if and only if gcd(ui-y,n) = 1 for i and j between 1 and k
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