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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.
The expected number of values within the permutation that are larger than all previous values is only / (+), smaller than its logarithmic value for unconstrained permutations. And the expected number of inversions is Θ ( n 3 / 2 ) {\displaystyle \Theta (n^{3/2})} , in contrast to its value of Θ ( n 2 ) {\displaystyle \Theta (n^{2})} for ...
A set S of finite binary words is balanced if for each n the subset S n of words of length n has the property that the Hamming weight of the words in S n takes at most two distinct values. A balanced sequence is one for which the set of factors is balanced. [12] A balanced sequence has complexity function at most n+1. [13]
The ! permutations of the numbers from 1 to may be placed in one-to-one correspondence with the ! numbers from 0 to ! by pairing each permutation with the sequence of numbers that count the number of positions in the permutation that are to the right of value and that contain a value less than (that is, the number of inversions for which is the ...
In a 1977 review of permutation-generating algorithms, Robert Sedgewick concluded that it was at that time the most effective algorithm for generating permutations by computer. [2] The sequence of permutations of n objects generated by Heap's algorithm is the beginning of the sequence of permutations of n+1 objects.
A sorting algorithm that takes a list and decides that because there is such a low probability that the list randomly occurred in its current permutation (a probability of 1/n!, where n is the number of elements), there must have been a reason for the list's order.
A main problem in permutation codes is to determine the value of (,), where (,) is defined to be the maximum number of codewords in a permutation code of length and minimum distance . There has been little progress made for 4 ≤ d ≤ n − 1 {\displaystyle 4\leq d\leq n-1} , except for small lengths.
Sorting a set of unlabelled weights by weight using only a balance scale requires a comparison sort algorithm. A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation (often a "less than or equal to" operator or a three-way comparison) that determines which of two elements should occur first in the final sorted list.