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The inversions of this permutation using element-based notation are: (3, 1), (3, 2), (5, 1), (5, 2), and (5,4). In computer science and discrete mathematics , an inversion in a sequence is a pair of elements that are out of their natural order .
0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15 Each permutation in this sequence can be generated by concatenating two sequences of numbers: the previous permutation, with its values doubled, and the same sequence with each value increased by one.
When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form "where the ‘±’ signs are independent" or similar.
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An exchange matrix is the simplest anti-diagonal matrix.; Any matrix A satisfying the condition AJ = JA is said to be centrosymmetric.; Any matrix A satisfying the condition AJ = JA T is said to be persymmetric.
The reverse divisor properties of the first two of these numbers, 1089 and 2178, were mentioned by W. W. Rouse Ball in his Mathematical Recreations. [7] In A Mathematician's Apology , G. H. Hardy criticized Rouse Ball for including this problem, writing:
A graph and its transpose. In the mathematical and algorithmic study of graph theory, the converse, [1] transpose [2] or reverse [3] of a directed graph G is another directed graph on the same set of vertices with all of the edges reversed compared to the orientation of the corresponding edges in G.
At a certain distance from the mirror an inverted non-reversed image appears. A third type of non-reversing mirror was created by mathematics professor R. Andrew Hicks in 2009. It was created using computer algorithms to generate a "disco ball" like surface.