Baxter permutation

In combinatorial mathematics, a Baxter permutation is a permutation formula_1 which satisfies the following generalized pattern avoidance property:
Equivalently, using the notation for vincular patterns, a Baxter permutation is one that avoids the two dashed patterns 2-41-3 and 3-14-2.
For example, the permutation "σ" = 2413 in "S"4 (written in one-line notation) is "not" a Baxter permutation because, taking "i" = 1, "j" = 2 and "k" = 4, this permutation violates the first condition.
These permutations were introduced by Glen E. Baxter in the context of mathematical analysis. 
Enumeration.
For "n" = 1, 2, 3, ..., the number "an" of Baxter permutations of length "n" 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, "a""n" has the following formula:
In fact, this formula is graded by the number of descents in the permutations, i.e., there are formula_3 Baxter permutations in "S""n" with "k" – 1 descents.
Motivation: commuting functions.
Baxter introduced Baxter permutations while studying the fixed points of commuting continuous functions. In particular, if "f" and "g" are continuous functions from the interval [0, 1] to itself such that "f"("g"("x")) = "g"("f"("x")) for all "x", and "f"("g"("x")) = "x" for finitely many "x" in [0, 1], then: