Anders Claesson ; Sergey Kitaev - Classification of bijections between 321- and 132-avoiding permutations

dmtcs:3594 - Discrete Mathematics & Theoretical Computer Science, January 1, 2008, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) - https://doi.org/10.46298/dmtcs.3594
Classification of bijections between 321- and 132-avoiding permutations

Authors: Anders Claesson 1; Sergey Kitaev 1

  • 1 The Mathematics Institute, Reyjavik University

It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs confirming this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and we show how they are related to each other (via "trivial'' bijections). Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also analyze how many permutation statistics (from a fixed, but large, set of statistics) each of the known bijections preserves, obtaining substantial extensions of known results. We also give a recursive description of the algorithmic bijection given by Richards in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of Simion and Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and it respects 11 statistics (the largest number of statistics any of the bijections respect).


Volume: DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
Section: Proceedings
Published on: January 1, 2008
Imported on: May 10, 2017
Keywords: pattern avoidance,Catalan structures,equidistribution,permutation statistics,bijection,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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Source : ScholeXplorer IsRelatedTo ARXIV math/0212328
Source : ScholeXplorer IsRelatedTo DOI 10.1016/j.jcta.2003.10.009
Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.math/0212328
  • math/0212328
  • 10.48550/arxiv.math/0212328
  • 10.1016/j.jcta.2003.10.009
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