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 S₄ (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.^[1]

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 A001181 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}} .

^[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.^[3]

Other properties

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 C_nC_n+1.
The number of doubly alternating Baxter permutations of length 2n and 2n + 1 (i.e., those for which both σ and its inverse σ⁻¹ are alternating) is the Catalan number C_n.^[4]
Baxter permutations are related to Hopf algebras,^[5] planar graphs,^[6] and tilings.^[7]^[8]

References

↑ Baxter, Glen (1964), "On fixed points of the composite of commuting functions", Proceedings of the American Mathematical Society, 15: 851–855, doi:10.2307/2034894 .
↑ Chung, F. R. K.; Graham, R. L.; Hoggatt, V. E., Jr.; Kleiman, M. (1978), "The number of Baxter permutations" (PDF), Journal of Combinatorial Theory, Series A, 24 (3): 382–394, doi:10.1016/0097-3165(78)90068-7, MR 491652 .
↑ Dulucq, S.; Guibert, O. (1998), "Baxter permutations", Discrete Mathematics, 180 (1-3): 143–156, doi:10.1016/S0012-365X(97)00112-X, MR 1603713 .
↑ Guibert, Olivier; Linusson, Svante (2000), "Doubly alternating Baxter permutations are Catalan", Discrete Mathematics, 217 (1-3): 157–166, doi:10.1016/S0012-365X(99)00261-7, MR 1766265 .
↑ Giraudo, Samuele (2011), "Algebraic and combinatorial structures on Baxter permutations", 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, pp. 387–398, arXiv:1011.4288, MR 2820726 .
↑ Bonichon, Nicolas; Bousquet-Mélou, Mireille; Fusy, Éric (October 2009), "Baxter permutations and plane bipolar orientations", Séminaire Lotharingien de Combinatoire, 61A, Art. B61Ah, 29pp, arXiv:0805.4180, MR 2734180 .
↑ Korn, M. (2004), Geometric and algebraic properties of polyomino tilings, Ph.D. thesis, Massachusetts Institute of Technology .
↑ Ackerman, Eyal; Barequet, Gill; Pinter, Ron Y. (2006), "A bijection between permutations and floorplans, and its applications", Discrete Applied Mathematics, 154 (12): 1674–1684, doi:10.1016/j.dam.2006.03.018, MR 2233287 .

External links

Sequence A001181 in OEIS.

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Baxter permutation

Enumeration

Other properties

See also

References

External links