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In combinatorial mathematics and theoretical computer science, a (classical) permutation pattern is a sub-permutation of a longer permutation.Any permutation may be written in one-line notation as a sequence of entries representing the result of applying the permutation to the sequence 123...; for instance the sequence 213 represents the permutation on three elements that swaps elements 1 and 2.
By contrast an increasing subsequence of a permutation is not necessarily contiguous: it is an increasing sequence obtained by omitting some of the values of the one-line notation. For example, the permutation 2453167 has the ascending runs 245, 3, and 167, while it has an increasing subsequence 2367.
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.
The largest clique in a permutation graph corresponds to the longest decreasing subsequence of the permutation that defines the graph (assuming the original non-permuted sequence is sorted from lowest value to highest). Similarly, the maximum independent set in a permutation
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 ...
likewise, an increasing subsequence in a permutation corresponds to an independent set of the same size in the corresponding permutation graph. the treewidth and pathwidth of permutation graphs can be computed in polynomial time; these algorithms exploit the fact that the number of inclusion minimal vertex separators in a permutation graph is ...
Arratia (1999) observes that, because the longest increasing subsequence of a random permutation has length (with high probability) approximately 2√n, it follows that a random permutation must have length at least k 2 /4 to have high probability of being a k-superpattern: permutations shorter than this will likely not contain the identity ...
It deals with the subsequences of a randomly uniformly drawn permutation from the set {,, …,}. The theorem makes a statement about the distribution of the length of the longest increasing subsequence in the limit.