Question

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.

Answer 1

Given X = {1, 2,...n} also a permutation of x is called an adjacency if it is a transpo- sition of the form (i, i + 1) i < n. We have to prove if i < j, (i, j) is a Product of an odd number of adjacencies. For take j - i = n = > j = n + i = > (i, j) = (i, n + i) Use induction on n = j - 1. n = 1 = > (i, j) = (i, j + 1) is already on adjacency Assume that any transposition of n = k is a product of adjacencies Then For n = k + 1, (i, j) = (i, k + 1 + i) = (i, i + (k + 1)) = = (i, i + 1) (i + 1 i + (k + 1)) (i i + 1) Then the transposition (i + 1 i + (k + 1)) is with value k + 1 - 1 = k for which the result ius true. = > (i, j) = (i, i + (k + 1)) is a product of adjacencies = > (i, j) is a product of adjacencies for all n = j - i = > if i < j, (i, j) is a product of adjacencies Now we have any transposition is odd so the number of adjacencies that appears in the decomposition is also an odd. Hence proved