Baxter Permutation

In combinatorial mathematics, a Baxter permutation is a permutation $\sigma \in S_n$ which satisfies the following generalized pattern avoidance property:
* There are no indices ''i'' < ''j'' < ''k'' such that ''σ''(''j'' + 1) < ''σ''(''i'') < ''σ''(''k'') < ''σ''(''j'') or ''σ''(''j'') < ''σ''(''k'') < ''σ''(''i'') < ''σ''(''j'' + 1).
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 ''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:

::$a_n \, = \,\sum_^n \frac .$

In fact, this formula is graded by the number of descents in the permutations, i.e., there are $\frac$ Baxter permutations in ''S''_''n'' with ''k'' – 1 descents.

Other properties

* The number of alternating Baxter permutations of length 2''n'' is (''C_n'')², the square of a Catalan number, and of length 2''n'' + 1 is ''C''_''n''''C''_''n''+1.
* The number of doubly alternating Baxter permutations of length 2''n'' and 2''n'' + 1 (i.e., those for which both ''σ'' and its inverse ''σ''⁻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 , 1to itself such that ''f''(''g''(''x'')) = ''g''(''f''(''x'')) for all ''x'', and ''f''(''g''(''x'')) = ''x'' for finitely many ''x'' in , 1 then:
* the number of these fixed points is odd;
* if the fixed points are ''x''₁ < ''x''₂ < ... < ''x''_{2''k'' + 1} then ''f'' and ''g'' act as mutually-inverse permutations on and ;
* the permutation induced by ''f'' on uniquely determines the permutation induced by ''f'' on ;
* under the natural relabeling ''x''₁ → 1, ''x''₃ → 2, etc., the permutation induced on is a Baxter permutation.

See also

* Enumerations of specific permutation classes

References

External links

* {{OEIS el, 1=A001181, 2=Number of Baxter permutations of length n

Permutation patterns