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Even permutation

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	Permutation - Wikipedia, the free encyclopedia
		A permutation is an ordered sequence containing each symbol from a set once and only once; neither (1, 2, 2, 3, 4, 5, 6) nor (1, 2, 4, 5, 6) are permutations.
		An even permutation is a permutation which can be expressed as the product of an even number of transpositions, and the identity permutation is an even permutation as it equals (1 2)(1 2).
		An odd permutation is a permutation which can be expressed as the product of an odd number of transpositions.
	en.wikipedia.org /wiki/Permutation (1407 words)

	Permutation article - Permutation permutation (music) mathematics change ringing alphabet string - What-Means.com (Site not responding. Last check: 2007-11-06)
		A permutation of the alphabet of 26 letters is a string of length 26 containing each letter just once; and it is clear that this definition works for any alphabet of N letters, with strings of length N.
		That is, a permutation is simply a sequence with no two elements the same, drawn from a fixed set of symbols, and of maximum length.
		This usage is obsolescent; within mathematics permutation without qualification is now generally taken to mean a bijective mapping from any given set X to itself, including the case where X is infinite.
	www.what-means.com /encyclopedia/Permutation (1562 words)

	Even and odd permutations - Wikipedia, the free encyclopedia
		An even permutation is one that can be produced by an even number of exchanges of two elements (these exchanges are called transpositions).
		Every permutation of odd order must be even; the converse is not true in general.
		Every permutation can be produced by a sequence of transpositions: with the first transposition we put the first element of the permutation in its proper place, the second transposition puts the second element right etc.
	en.wikipedia.org /wiki/Even_permutation (902 words)

	Permutations
		This would be the identity permutation where shell 1 stays in its position, shell 2 stays in its position and shell 3 also stays in its position.
		This is because (a) permutations are closed under the operation followed by, (b) since the operation is "followed by" it is associative, (c) the "leave everything where it is" permutation is the identity and (c) every permutation can be undone by "running the film backwards" thus all permutations have inverses.
		If you were given a permutation of the eggs and told to rearrange the eggs into that permutation you might begin with the first place in the egg carton, find the numbered egg that went into that place and traded the egg in the first place with the egg that belonged there.
	members.tripod.com /~dogschool/permutation.html (2315 words)

	[No title]
		permutation +------------------------------------------------------------ If S is a set, then a permutation of S is a bijective map from S into S. The intuition underlying the definition of a permutation is that a permutation determines a reordering of the elements in a list or the rearrangement of objects.
		Thus the identity permutation is an even permutation (since it is equal to the composition of any transposition with itself), and any transposition is an odd permutation (since it is equal to one transposition).
		That is, an odd permutation is the composition of an odd number of transpositions.
	www.math.harvard.edu /~knill/sofia/data/group.txt (4457 words)

	LSU EE 4720 Lecture Notes
		A permutation is a p, c shift permutation of size 2n, denoted Sp;c, if for all x 2 h2n i Sp;c(x) j xp + c (mod 2n); where c is a nonnegative integer and p is a nonnegative odd integer.
		The definition of a bitonic permutation will have a slight difference: The sequence is a sequence of integers, the integers are a permutation of hmn i, and the sequence when shifted is bitonic.
		9-8 * * 9-8 A permutation for which the sequence P (c), P (c + 1), P (c + 2); : : :; P (c + mn 1) is bitonic is called a bitonic permutation, where P is a permutation of hmi n, c is an integer, and arithmetic is modulo mn.
	www.ece.lsu.edu /ee7725/1997/lsli15.html (1612 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		A "permutation" of T is a bijection from T to itself.
		] [ n ] [ f(n) ] Furthermore, the inverse of the matrix of the permutation is the matrix of the inverse of the permutation: (b) P(f)^(-1) = P(f^(-1)).
		T is odd if and only if the number of cycles of even length in its cycle decomposition is odd.
	web.usna.navy.mil /~wdj/sm485_2.txt (1245 words)

	Determinants
		If this permutation is a sequence of an even number of transpositions, it is impossible to write this permutation as a sequence of an odd number of transpositions.
		If a permutation of n ordered elements can be expressed as an even number of transpositions, then it is called an even permutation.
		If a permutation of n ordered elements can be expressed as an odd number of transpositions, then it is called an odd permutation.
	www.ping.be /~ping1339/determ.htm (1885 words)

	PlanetMath: signature of a permutation
		"signature of a permutation" is owned by rspuzio.
		Cross-references: induction, identity map, clear, relation, totally ordered, odd, transposition, multiplicative group, homomorphism, permutation group, permutations, group, finite
		This is version 6 of signature of a permutation, born on 2003-02-26, modified 2004-10-31.
	planetmath.org /encyclopedia/SignatureOfAPermutation.html (118 words)

	Earliest Known Uses of Some of the Words of Mathematics (O)
		Odd and even are found in English in various Middle Age documents including Art of Nombrynge (around 1430) "Compt the nombre of the figures, and wete yf it be ode or even" (OED2).
		ODD PERMUTATION and EVEN PERMUTATION are found in "On the Relation between the Three-Parameter Groups of a Cubic Space Curve and a Quadric Surface," A. Coble, Transactions of the American Mathematical Society, Vol.
		Odd permutation is found in the fourth edition of Determinants (1906) by Laenas Gifford Weld, and may appear in earlier editions.
	members.aol.com /jeff570/o.html (2967 words)

	Elementary Row Operations
		Scaling a row scales the determinant, permuting rows negates the determinant if the permutation is odd, and row subtraction leaves the determinant unchanged.
		One permutation describes the term from det(c), and the other is the magic numbers we use to find a subterm within the expansion of this term.
		This permutation is joined to the term of det(b), and the result is the permutation that arranges rows and columns in det(c).
	www.mathreference.com /la-det,erop.html (1199 words)

	New Page 2 (Site not responding. Last check: 2007-11-06)
		This can be considered as the set of all permutations on the set {1, 2, 4}, and so this set of permutations has 6 elements, the same number as S
		Let c be a fixed odd permutation, and consider any odd permutation a.
		The inverse permutation c' is also odd, and hence c'a = b, which is an even permutation.
	cas.memphis.edu /rfaudree/hom44261.htm (199 words)

	43 QUINTILLION COMBINATIONS
		After an odd number of moves, the side cube is rotated through an odd permutation, whereas an even number of moves carries the side cube into an even permutation.
		Since a side cube cannot exhibit an odd and an even permutation simultaneously, there cannot be two paths back to start having odd and even lengths.
		If the corners are arranged in an odd permutation, then the sides will present an odd permutation (as a result of arranging the corners), and only odd permutations are accessible thereafter.
	www.eklhad.net /rubik/enumerat.html (1826 words)

	DeVill
		The blue one permutes the numbers in such a way, that you can solve that game, while the red one permutes so, that it is not possible to put the numbers in ascending order.
		This expression is non zero, and positive if and only if the permutation is even.) It is clear, that two even permutations done one after the other is an even permutation, and an even permutation and an odd permutation give an odd permutation.
		So if we start with an odd permutation and the empty space in the first row, than every time when the empty space is in the first row the permutation will be odd.
	www.cs.elte.hu /~devill/permjen.html (1620 words)

	[No title]
		If this configuration were obtainable starting from the permutation [3,5,2,4,1] (those with 7 passing considered at the beginning), it would mean that, starting from number 7 and adding or subtracting only even numbers we are able to get 0.
		If for a permutation the number of passing is even, we say that the permutation is of even class (or simply even), otherwise is of odd class (or odd).
		For instance the permutation [3,5,2,4,1] is an odd permutation, while the permutation [3,5,4,2,1] is even.
	www.dmi.units.it /~logar/duek2/sol_appr_engl.html (1472 words)

	Symmetric group - SmartyBrain Encyclopedia and Dictionary
		Since g can be written as a product of an odd number of transpositions, it is then called an odd permutation, whereas f is an even permutation.
		The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd.
		The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd.
	smartybrain.com /index.php/Symmetric_group (633 words)

	CSC 350: Fund. of Math. Computation --- Lab Assignment 03 (Site not responding. Last check: 2007-11-06)
		To compute the signature of a given permutation, all you have to do is count how many elements are out of place, as follows.
		For i from 2 to n do count how many element with index j < i are in fact greater than xi (that is, should have an index k larger than i) add this count to the total.
		If the grand total is odd, you have an odd permutation, use negative sign in your summation.
	einstein.cs.uri.edu /courses/spring2004/csc350/Assignments/lab03.html (338 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		A given permutation can be represented by % many different such transposition sequences, but the number of % such transpositions will always be odd or always be even.
		% If the number of transpositions is even or odd, the permutation is % said to be even or odd.
		% +1, the permutation is even, % -1, the permutation is odd.
	www.csit.fsu.edu /~burkardt/math2071/perm_sign.m (280 words)

	Alternating groups.
		If we insted would wish a cyclic permutation, (pqr...s) to be a set of alternations against a certain element, say a, then we
		This means that all permutations could be revritten as alternations, and that all these altenations could be done against one element.
		A1 gives us that a circular permutation involving n elements will be possible to do in n-1 alternations.
	hemsidor.torget.se /users/m/mauritz/math/alg/alt.htm (274 words)

	Teach/Me Data Analysis
		p is called odd or even, depending on whether the number of inversions is even or odd.
		Then its determinant (denoted with A) is the sum over the product of all permutations of all elements within any row, multiplied by either +1 or -1, depending on whether the respective permutation is even or odd.
		Note that determinants are only defined for square matrices, and that the determinant of a square matrix and its transposed are equal.
	www.vias.org /tmdatanaleng/cc_matrix_determinant.html (506 words)

	Introduction (Site not responding. Last check: 2007-11-06)
		The simplest non-trivial permutations on S map each element to itself save two, which are each mapped to the other.
		Permutations that are not even are odd, and every representation of an odd permutation as a composition of transpositions contains an odd number of transpositions.
		The set of all permutations on a given set form a group under the composition operation.
	www.mathpuzzle.com /characteristic.html (2118 words)

	Even and Odd Permutations (Site not responding. Last check: 2007-11-06)
		For instance, the permutation x = 23154 moves the element in position 1 to position 2, 2 to 3, and 3 back to 1, while elements in positions 4 and 5 are swapped.
		a transposition, the parity of the permutation is reversed.
		The even permutations form the kernel of the parity homomorphism, and are a normal subgroup in g.
	www.mathreference.com /grp,eop.html (317 words)

	Transpositions
		It can be shown that representation of a permutation as a product of cycles is unique up to their order.
		However, one quantity related to a permutation is invariant for its various representations as a product of transpositions.
		In other words, the product of two permutations (which is definitely another permutation) will be even iff the component permutations are either both even or both odd.
	www.cut-the-knot.org /do_you_know/pgroups.shtml (371 words)

	PlanetMath:
		odd integer (in even number) owned by mathcam
		odd number (in even number) owned by mathcam
		odd permutation (in signature of a permutation) owned by rspuzio
	planetmath.org /encyclopedia/O (698 words)

	Even and Odd Permutations (Site not responding. Last check: 2007-11-06)
		But it is true that if a permutation can be written as an even number of cycles, any representation will contain an even number of cycles.
		Thus the set of all permutations that are even form a subset of the full symmetric group.
		This can be proved by showing that the sliding operation is like a permutation group, and that the swapping of two blocks amounts to an odd permutation in that group, but the operation of sliding a block is an even permutation.
	mathcircle.berkeley.edu /BMC3/perm/node8.html (319 words)

	[No title]
		] A parity check in which the number of 0's or 1's in each word is expected to be odd; if the number is even, the check bit is 1, and if the number is odd, the check bit is 0.
		] A permutation that may be represented as the result of an odd number of transpositions.
		] A term of an atom or molecule for which the sum of the angular-momentum quantum numbers of all the electrons is odd, so that the states have odd parity; designated by a superscript o or u.
	www.accessscience.com /Dictionary/O/O3/DictO3.html (2557 words)

	15 Puzzle - mathematical implications of Sam Lloyd's mind game
		Of course, the mathematicians know the answer: the parity of the permutation (rearrangement) which represents the actual position is preserved during the moves.
		The order in which we choose the tiles to form the permutation respects the rule: every two consecutive tiles must be adjacent.
		Sam Lloyd knew that the 15-14 problem had no solution, because the start position was an odd permutation, while the end position was even (they differs by an inversion).
	www.holotronix.com /samlloyd15b.html (702 words)

	[No title]
		In each odd layer there is 1 of these orbits, consisting of 12 cubies, each of which can be in 2 states.
		In each odd layer there is on of these orbits, which has 6 cubies each of which can be in 4 rotation states.
		Note that the "number of positions" given is for both only one parity, that is, for both an even or odd number of quarter turns, and ignoring the all other orbits.
	akbar.marlboro.edu /~mahoney/cube/NxN.txt (3320 words)

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