Baxter permutation

In combinatorial mathematics, a Baxter permutation is a permutation ${\displaystyle \sigma \in S_{n}}$ which satisfies the following generalized pattern avoidance property:

• There are no indices ${\displaystyle i<j<k}$ such that ${\displaystyle \sigma (j+1)<\sigma (i)<\sigma (k)<\sigma (j)}$ or ${\displaystyle \sigma (j)<\sigma (k)<\sigma (i)<\sigma (j+1)}$.

Equivalently, using the notation for vincular patterns, a Baxter permutation is one that avoids the two dashed patterns ${\displaystyle 2-41-3}$ and ${\displaystyle 3-14-2}$.

For example, the permutation ${\displaystyle \sigma =2413}$ in ${\displaystyle S_{4}}$ (written in one-line notation) is not a Baxter permutation because, taking ${\displaystyle i=1}$, ${\displaystyle j=2}$ and ${\displaystyle k=4}$, this permutation violates the first condition.

These permutations were introduced by Glen E. Baxter in the context of mathematical analysis.