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An example of legal and illegal permutations can be better demonstrated by a specific problem such as balanced brackets (see Dyck language). A general problem is to count the number of balanced brackets (or legal permutations) that a string of m open and m closed brackets forms (total of 2m brackets). By legally arranged, the following rules apply:
C n is the number of permutations of {1, ..., n} that avoid the permutation pattern 123 (or, alternatively, any of the other patterns of length 3); that is, the number of permutations with no three-term increasing subsequence. For n = 3, these permutations are 132, 213, 231, 312 and 321.
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.
For a given set of n beads, all distinct, the number of distinct necklaces made from these beads, counting rotated necklaces as the same, is n! / n = (n − 1)!. This is because the beads can be linearly ordered in n ! ways, and the n circular shifts of such an ordering all give the same necklace.
Moreover, every Dyck string comes from a stack-sortable permutation in this way, and every two different stack-sortable permutations produce different Dyck strings. For this reason, the number of stack-sortable permutations of length n is the same as the number of Dyck strings of length 2n, the Catalan number
The sum of the numbers in the factorial number system representation gives the number of inversions of the permutation, and the parity of that sum gives the signature of the permutation. Moreover, the positions of the zeroes in the inversion table give the values of left-to-right maxima of the permutation (in the example 6, 8, 9) while the ...
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.
For n ≥ 0 and 0 ≤ k ≤ n, the rencontres number D n, k is the number of permutations of { 1, ..., n } that have exactly k fixed points. For example, if seven presents are given to seven different people, but only two are destined to get the right present, there are D 7, 2 = 924 ways this could happen.