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u4 - Unit IV Determinants 1 Permutations Definition A...

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Unit IV: Determinants 1. Permutations Definition: A permutation of a set S is a function from S to S that is one-to-one and onto. If S has n elements, then there are n ! permutations of S . The notation π : { 1 , . . . , n } 7→ { k 1 . . . k n } refers to the permutation π of the integers 1 , . . . , n for which π (1) = k 1 , π (2) = k 2 , and so on, to π ( n ) = k n . Example: There are two permutations of the set { 1 , 2 } , namely the identity { 1 , 2 } 7→ { 1 , 2 } and the switch { 1 , 2 } 7→ { 2 , 1 } . There are six permutations of the set { 1 , 2 , 3 } . They are the identity { 1 , 2 , 3 } 7→ { 1 , 2 , 3 } , two circular shifts { 1 , 2 , 3 } 7→ { 2 , 3 , 1 } and { 1 , 2 , 3 } 7→ { 3 , 1 , 2 } , and three switches { 1 , 2 , 3 } 7→ { 2 , 1 , 3 } , { 1 , 2 , 3 } 7→ { 3 , 2 , 1 } , and { 1 , 2 , 3 } 7→ { 1 , 3 , 2 } . The composition σ π of two permutations σ and π is a permutation. Each permutation π has an inverse permutation π - 1 , defined so that π - 1 π = π π - 1 = the identity permutation. We will refer to a permutation that simply switches two values as a switch . (The math- ematical word is transposition .) Any permutation is a composition of switches. For in- stance, the circular shift { 1 , 2 , 3 } 7→ { 2 , 3 , 1 } is a composition of two switches, the switch { 1 , 2 , 3 } 7→ { 2 , 1 , 3 } of the first two values, followed by the switch { 1 , 2 , 3 } 7→ { 3 , 2 , 1 } of the first and third values. The inversion number of a permutation π : { 1 , . . . , n } 7→ { π (1) , . . . π ( n ) } is the number of pairs ( j, k ) such that j < k but π ( j ) > π ( k ). Example: The identity permutation of { 1 , 2 , 3 } has inversion number 0, the circular shifts have inversion number 2, and the switches have inversion numbers 1 or 3. For instance, the circular shift { 1 , 2 , 3 } 7→ { 2 , 3 , 1 } has inversion number 2, corresponding to the inversions π (1) = 2 > 1 = π (3) and π (2) = 3 > 1 = π (3). The switch { 1 , 2 , 3 } 7→ { 2 , 1 , 3 } has inversion number 1, corresponding to the inversion π (1) = 2 > 1 = π (2). Definition: An even permutation is a permutation whose inversion number is even, and an odd permutation is a permutation whose inversion number is odd. The parity of a per- mutation is even or odd according to whether its inversion number is even or odd. Definition: The sign of a permutation is +1 if the permutation is even and - 1 if the permutation is odd. We denote the sign of a permutation π by sgn( π ). Theorem 1. If we switch two values of a permutation, the sign changes. Proofsketch. If we switch the values of adjacent integers, say the values π ( j ) and π ( j +1), the inversion number of the new permutation changes by +1 or - 1, depending on whether π ( j ) < π ( j + 1) or π ( j ) > π ( j + 1). Switching two values of arbitrary integers, say π ( j ) and π ( k ) can be done through a succession of an odd number of switches of values of adjacent integers. Procedure. The theorem provides an easy way to determine the parity of a permutation. Count the number of switches you need to get from the identity permutation to π . If you can do it with an even number of switches, then π is even. If you can do it with an odd number of switches, then π is odd.
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