In combinatorial mathematics, a Baxter permutation is a permutation
σ ∈
n
{\displaystyle \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
{\displaystyle 2-41-3}
and
3 − 14 − 2
For example, the permutation
σ = 2413
in
4
{\displaystyle S_{4}}
(written in one-line notation) is not a Baxter permutation because, taking
,
, this permutation violates the first condition.
These permutations were introduced by Glen E. Baxter in the context of mathematical analysis.
, the number
of Baxter permutations of length
This is sequence OEIS: A001181 in the OEIS.
In general,
has the following formula: In fact, this formula is graded by the number of descents in the permutations, i.e., there are
Baxter permutations in
descents.
Baxter introduced Baxter permutations while studying the fixed points of commuting continuous functions.
are continuous functions from the interval
for finitely many