Permutation pattern

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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. If π and σ are two permutations represented in this way (these variable names are standard for permutations and are unrelated to the number pi), then π is said to contain σ as a pattern if some subsequence of the entries of π has the same relative order as all of the entries of σ. For instance, permutation π contains the pattern 213 whenever π has three entries x, y, and z that appear within π in the order x...y...z but whose values are ordered as y < x < z, the same as the ordering of the values in the permutation 213. The permutation 32415 on five elements contains 213 as a pattern in several...

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