Classification of bijections between 321- and 132-avoiding permutations

Anders Claesson, Sergey Kitaev

Research output: Contribution to conferencePaper

19 Citations (Scopus)

Abstract

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).

Conference

Conference20th International Conference on Formal Power Series & Algebraic Combinatorics
CountryChile
CityValparaiso
Period23/06/0827/06/08

Fingerprint

Bijection
Permutation
Statistics
Permutation Statistics
Bijective
Large Set
Trivial

Keywords

  • bijection
  • permutation statistics
  • equidistribution
  • pattern avoidance
  • Catalan structures

Cite this

Claesson, A., & Kitaev, S. (2008). Classification of bijections between 321- and 132-avoiding permutations. 495–506. Paper presented at 20th International Conference on Formal Power Series & Algebraic Combinatorics, Valparaiso, Chile.
Claesson, Anders ; Kitaev, Sergey. / Classification of bijections between 321- and 132-avoiding permutations. Paper presented at 20th International Conference on Formal Power Series & Algebraic Combinatorics, Valparaiso, Chile.12 p.
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Claesson, A & Kitaev, S 2008, 'Classification of bijections between 321- and 132-avoiding permutations' Paper presented at 20th International Conference on Formal Power Series & Algebraic Combinatorics, Valparaiso, Chile, 23/06/08 - 27/06/08, pp. 495–506.

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

2008. 495–506 Paper presented at 20th International Conference on Formal Power Series & Algebraic Combinatorics, Valparaiso, Chile.

Research output: Contribution to conferencePaper

TY - CONF

T1 - Classification of bijections between 321- and 132-avoiding permutations

AU - Claesson, Anders

AU - Kitaev, Sergey

PY - 2008

Y1 - 2008

N2 - 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).

AB - 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).

KW - bijection

KW - permutation statistics

KW - equidistribution

KW - pattern avoidance

KW - Catalan structures

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Claesson A, Kitaev S. Classification of bijections between 321- and 132-avoiding permutations. 2008. Paper presented at 20th International Conference on Formal Power Series & Algebraic Combinatorics, Valparaiso, Chile.