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Separable permutation
In combinatorial mathematics, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums.〔, p. 57.〕 Separable permutations may be characterized by the forbidden permutation patterns 2413 and 3142;〔; , Theorem 2.2.36, p. p.58.〕 they are also the permutations whose permutation graphs are cographs and the permutations that realize the series-parallel partial orders. It is possible to test in polynomial time whether a given separable permutation is a pattern in a larger permutation, or to find the longest common subpattern of two separable permutations.
==Definition and characterization==
define a separable permutation to be a permutation that has a ''separating tree'': a rooted binary tree in which the elements of the permutation appear (in permutation order) at the leaves of the tree, and in which the descendants of each tree node form a contiguous subset of these elements. Each interior node of the tree is either a positive node in which all descendants of the left child are smaller than all descendants of the right node, or a negative node in which all descendants of the left node are greater than all descendants of the right node. There may be more than one tree for a given permutation: if two nodes that are adjacent in the same tree have the same sign, then they may be replaced by a different pair of nodes using a tree rotation operation.
Each subtree of a separating tree may be interpreted as itself representing a smaller separable permutation, whose element values are determined by the shape and sign pattern of the subtree. A one-node tree represents the trivial permutation, a tree whose root node is positive represents the direct sum of permutations given by its two child subtrees, and a tree whose root node is negative represents the skew sum of the permutations given by its two child subtrees. In this way, a separating tree is equivalent to a construction of the permutation by direct and skew sums, starting from the trivial permutation.
As prove, separable permutations may also be characterized in terms of permutation patterns: a permutation is separable if and only if it contains neither 2413 nor 3142 as a pattern.〔
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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