Baxter permutation

In combinatorial mathematics, a Baxter permutation is a permutation which satisfies the following generalized pattern avoidance property:

There are no indices such that or.

Equivalently, using the notation for vincular patterns, a Baxter permutation is one that avoids the two dashed patterns and.
For example, the permutation in is not a Baxter permutation because, taking
, and, this permutation violates the first condition.
These permutations were introduced by Glen E. Baxter in the context of mathematical analysis.

Enumeration

For, the number of Baxter permutations of length is

1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, 2593341586, 16458756586,...

This is sequence in the OEIS. In general, has the following formula:
In fact, this formula is graded by the number of descents, runs and excedances|descents] in the permutations, i.e., there are
Baxter permutations in with descents.

Other properties

The number of alternating Baxter permutations of length is, the square of a Catalan number, and of length is
The number of doubly alternating Baxter permutations of length and is the Catalan number.
Baxter permutations are related to Hopf algebras, planar graphs, and tilings.

Motivation: commuting functions

Baxter introduced Baxter permutations while studying the fixed points of commuting continuous functions. In particular, if and are continuous functions from the interval to itself such that for all, and for finitely many
in, then:

the number of these fixed points is odd;
if the fixed points are then and act as mutually-inverse permutations on

and ;

the permutation induced by on uniquely determines the permutation induced by

on ;

under the natural relabeling,, etc., the permutation induced on is a Baxter permutation.