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Refined Enumeration of Permutations Sorted with Two Stacks and a D 8-Symmetry

Annals of Combinatorics volume 18, pages 199–232 (2014)Cite this article

Abstract

We study permutations that are sorted by operators of the form S ° α ° S, where S is the usual stack sorting operator introduced by Knuth and α is any D 8-symmetry obtained by combining the classical reverse, complement, and inverse operations. Such permutations can be characterized by excluded (generalized) patterns. Some conjectures about the enumeration of these permutations, refined with numerous classical statistics, have been proposed by Claesson, Dukes, and Steingrímsson. We prove these conjectures, and enrich one of them with a few more statistics. The proofs mostly rely on generating trees techniques, and on a recent bijection of Giraudo between Baxter and twisted Baxter permutations.

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Authors and Affiliations

  1. LaBRI, CNRS and Université de Bordeaux, 351, cours de la Libération, 33405, Talence cedex, France

    Mathilde Bouvel & Olivier Guibert

Authors
  1. Mathilde Bouvel
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  2. Olivier Guibert
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Correspondence to Mathilde Bouvel.

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Bouvel, M., Guibert, O. Refined Enumeration of Permutations Sorted with Two Stacks and a D 8-Symmetry. Ann. Comb. 18, 199–232 (2014). https://doi.org/10.1007/s00026-014-0219-8

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  • DOI: https://doi.org/10.1007/s00026-014-0219-8

Mathematics Subject Classification

  • 05A05
  • 05A15

Keywords

  • permutations
  • generalized patterns
  • stack sorting
  • symmetries of the square
  • enumeration
  • permutation statistics
  • Baxter permutations
  • generating trees

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