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Claim: If array A has length n, then permutations(n, A) will result in either A being unchanged, if n is odd, or, if n is even, then A is rotated to the right by 1 (last element shifted in front of other elements). Base: If array A has length 1, then permutations(1, A) will output A and stop, so A is unchanged. Since 1 is odd, this is what was ...
More generally, there are d! possible orders for a given array, one for each permutation of dimensions (with row-major and column-order just 2 special cases), although the lists of stride values are not necessarily permutations of each other, e.g., in the 2-by-3 example above, the strides are (3,1) for row-major and (1,2) for column-major.
In particular, the unsigned Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). [1] The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of ...
Their description of the algorithm used pencil and paper; a table of random numbers provided the randomness. The basic method given for generating a random permutation of the numbers 1 through N goes as follows: Write down the numbers from 1 through N. Pick a random number k between one and the number of unstruck numbers remaining (inclusive).
Multiplying a matrix M by either or on either the left or the right will permute either the rows or columns of M by either π or π −1.The details are a bit tricky. To begin with, when we permute the entries of a vector (, …,) by some permutation π, we move the entry of the input vector into the () slot of the output vector.
Permutations are calculated based on the number of samples; Block Permutations Blocks are batches of microarrays; for example for eight samples split into two groups (control and affected) there are 4!=24 permutations for each block and the total number of permutations is (24)(24)= 576. A minimum of 1000 permutations are recommended; [32] [36] [37]
Because bit-reversal permutations may be repeated multiple times as part of a calculation, it may be helpful to separate out the steps of the algorithm that calculate index data used to represent the permutation (for instance, by using the doubling and concatenation method) from the steps that use the results of this calculation to permute the ...
The types of permutations presented in the preceding two sections, i.e. permutations containing an even number of even cycles and permutations that are squares, are examples of so-called odd cycle invariants, studied by Sung and Zhang (see external links). The term odd cycle invariant simply means that membership in the respective combinatorial ...