Baxter permutation

In combinatorial mathematics, a Baxter permutation is a permutation $$\sigma \in S_n$$ 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 $$3-14-2$$.
 * There are no indices $$i<j<k$$ such that $$\sigma(j+1)<\sigma(i)<\sigma(k)<\sigma(j)$$ or $$\sigma(j)<\sigma(k)<\sigma(i)<\sigma(j+1)$$.

For example, the permutation $$\sigma=2413$$ 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 $$n = 1, 2, 3, \ldots$$, the number $$a_n$$ 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:



a_n \, = \,\sum_{k=1}^n \frac{\binom{n+1}{k-1}\binom{n+1}{k}\binom{n+1}{k+1}}{\binom{n+1}{1}\binom{n+1}{2}} .$$

In fact, this formula is graded by the number of descents in the permutations, i.e., there are $$\frac{\binom{n+1}{k-1}\binom{n+1}{k}\binom{n+1}{k+1}}{\binom{n+1}{1}\binom{n+1}{2}}$$ Baxter permutations in $$S_n$$ with $$k-1$$ descents.

Other properties
$$C_n C_{n+1}$$.
 * The number of alternating Baxter permutations of length $$2n$$ is $$(C_n)^2$$, the square of a Catalan number, and of length $$2n+1$$ is
 * The number of doubly alternating Baxter permutations of length $$2n$$ and $$2n+1$$ (i.e., those for which both $$\sigma$$ and its inverse $$\sigma^{-1}$$ are alternating) is the Catalan number $$C_n$$.
 * 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 $$f$$ 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: $$\{x_1,x_3, \ldots, x_{2k + 1} \} $$ and $$\{x_2, x_4,\ldots, x_{2k} \}$$; $$f$$ on $$\{ x_2<, x_4, \ldots, x_{2k}\}$$;
 * the number of these fixed points is odd;
 * if the fixed points are $$x_1 < x_2< \ldots < x_{2k + 1}$$ then $$f$$ and $$g$$ act as mutually-inverse permutations on
 * the permutation induced by $$f$$ on $$\{x_1, x_3, \ldots, x_{2k+1}\}$$ uniquely determines the permutation induced by
 * under the natural relabeling $$x_1\to 1$$, $$x_3\to 2$$, etc., the permutation induced on $$\{1, 2, \ldots, k + 1\}$$ is a Baxter permutation.