User:Dilks003/Baxter Numbers

In combinatorial mathematics, the Baxter numbers form a sequence of natural numbers that occur in various counting problems. They are named after the American mathematician Glen Baxter (1930–1983).

The nth Baxter number is given by


 * $$B_n = \sum_{k=0}^{n-1} \frac{\binom{n+1}{k}\binom{n+1}{k+1}\binom{n+1}{k+2}}{\binom{n+1}{1}\binom{n+1}{2}} $$

The first Baxter numbers for n = 0, 1, 2, 3, … are


 * 0, 1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, 2593341586, 16458756586, 105791986682, 687782586844, 4517543071924, 29949238543316, 200234184620736, 1349097425104912, 9154276618636016 …

Definition
The nth Baxter number is most often defined as the number of Baxter permutations of length n. A permutation is a Baxter permutation if it avoids the vincular patterns 2-41-3 and 3-14-2. This is to say, &pi; is a Baxter permutation if there do not exist indices i<j<j+1<k where &pi;(j+1)<&pi;(i)<&pi;(k)<&pi;(j) or &pi;(j)<&pi;(k)<&pi;(i)<&pi;(j+1). Equivalently, Baxter permutations are those that avoid the barred patterns 25 3 14 and 41 3 52.

History
Baxter permutations originally arose in the work of Glen Baxter, who was studying the fixed points of composites of continuous and commuting functions. The enumerative formula counting the number of Baxter permutations was given by Chung, Graham, Hoggatt Jr., and Kleiman by analytically solving a recurrence relation, and Mallows later confirmed that the summand corresponding to $$k$$ gave the number of Baxter permutations with $$k$$ ascents.

Other Objects
There are a number of other combinatorial objects which are known to be counted by the Baxter numbers. Some examples include diagonal rectangulations. (related to mosaic floorplans in the VLSI literature), pairs of twin binary trees, triples of non-intersecting lattice paths , $$3\times n$$ standard Young tableaux with no consecutive entries in any row and plane bipolar orientations

Extension of the Catalan Number
Baxter permutations of length $$n+1$$ with $$k$$ ascents are known to be in bijection with plane partitions that fit inside an $$k\times (n-k)\times 3$$ box. This can be seen as an extension of how the number of plane partitions inside a $$k\times (n-k)\times 2$$ box are counted by the Naryana number $$N(n,k)$$, and the number of plane partitions inside a $$k\times (n-k)\times 1$$ box are counted by the binomial coefficient $$\tbinom nk$$. If we fix $$n$$ and sum over all possible $$k$$, the binomial coefficients add up to $$2^n$$, the Narayana numbers add up to the Catalan number, and the number of Baxter permutations of length $$n+1$$ with $$k$$ ascents will add up to the total number of Baxter permutations, $$B_{n+1}$$. In this way, the Baxter number can be thought of as a natural extension of the Catalan number.