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The usual way to prove that there are n! different permutations of n objects is to observe that the first object can be chosen in n different ways, the next object in n − 1 different ways (because choosing the same number as the first is forbidden), the next in n − 2 different ways (because there are now 2 forbidden values), and so forth.
This table specifies the input permutation on a 64-bit block. The meaning is as follows: the first bit of the output is taken from the 58th bit of the input; the second bit from the 50th bit, and so on, with the last bit of the output taken from the 7th bit of the input.
The six possible inversions of a 4-element permutation. The following sortable table shows the 24 permutations of four elements (in the column) with their place-based inversion sets (in the p-b column), inversion related vectors (in the , , and columns), and inversion numbers (in the # column).
The inverse table inv represents the multiplicative inverse of a digit, that is, the value that satisfies d(j, inv(j)) = 0. The permutation table p applies a permutation to each digit based on its position in the number. This is actually a single permutation (1 5 8 9 4 2 7 0)(3 6) applied iteratively; i.e. p(i+j,n) = p(i, p(j,n)).
Again, the output is half the size of the input. Beginning with a 2 b-bit input word, the b−3 msbits are used for a shift amount, which is applied to the next-most-significant 2 b−1 +2 b−3 −1 bits, and the least significant 2 b−1 bits of the result are output. The low 2 b−1 −2 b−3 −b+4 bits are discarded.
A map of the 24 permutations and the 23 swaps used in Heap's algorithm permuting the four letters A (amber), B (blue), C (cyan) and D (dark red) Wheel diagram of all permutations of length = generated by Heap's algorithm, where each permutation is color-coded (1=blue, 2=green, 3=yellow, 4=red).
A Costas array may be represented numerically as an n×n array of numbers, where each entry is either 1, for a point, or 0, for the absence of a point. When interpreted as binary matrices, these arrays of numbers have the property that, since each row and column has the constraint that it only has one point on it, they are therefore also permutation matrices.
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way; Explained separately in a more accessible way: Combination; Permutation; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...
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