Skew-merged permutation

In the theory of permutation patterns, a skew-merged permutation is a permutation that can be partitioned into an increasing sequence and a decreasing sequence. They were first studied by and given their name by.

Characterization
The two smallest permutations that cannot be partitioned into an increasing and a decreasing sequence are 3412 and 2143. was the first to establish that a skew-merged permutation can also be equivalently defined as a permutation that avoids the two patterns 3412 and 2143.

A permutation is skew-merged if and only if its associated permutation graph is a split graph, a graph that can be partitioned into a clique (corresponding to the descending subsequence) and an independent set (corresponding to the ascending subsequence). The two forbidden patterns for skew-merged permutations, 3412 and 2143, correspond to two of the three forbidden induced subgraphs for split graphs, a four-vertex cycle and a graph with two disjoint edges, respectively. The third forbidden induced subgraph, a five-vertex cycle, cannot exist in a permutation graph (see ).

Enumeration
For $$n=1,2,3,\dots$$ the number of skew-merged permutations of length $$n$$ is
 * 1, 2, 6, 22, 86, 340, 1340, 5254, 20518, 79932, 311028, 1209916, 4707964, 18330728, ....

was the first to show that the generating function of these numbers is
 * $$\frac{1-3x}{(1-2x)\sqrt{1-4x}},$$

from which it follows that the number of skew-merged permutations of length $$n$$ is given by the formula
 * $$\binom{2n}{n}\sum_{m=0}^{n-1}2^{n-m-1}\binom{2m}{m}$$

and that these numbers obey the recurrence relation
 * $$P_n=\frac{(9n-8)P_{n-1} - (26n-46)P_{n-2} + (24n-60)P_{n-3}}{n}.$$

Another derivation of the generating function for skew-merged permutations was given by.

Computational complexity
Testing whether one permutation is a pattern in another can be solved efficiently when the larger of the two permutations is skew-merged, as shown by.