Classification of bijections between 321- and 132-avoiding permutations

Anders Claesson, Sergey Kitaev

Research output: Contribution to journalArticle

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 of 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 show how they are related to each other via "trivial" bijections. We classify the bijections according to statistics preserved (from a fixed, but large, set of statistics), obtaining substantial extensions of known results. Thus, we give a comprehensive survey and a systematic analysis of these bijections.

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 respects.
Language English B60d 30 Séminaire Lotharingien de Combinatoire 60 Published - 4 Nov 2008

Bijection
Permutation
Statistics
Bijective
Large Set
Trivial
Classify

Keywords

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

Cite this

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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 of 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 show how they are related to each other via {"}trivial{"} bijections. We classify the bijections according to statistics preserved (from a fixed, but large, set of statistics), obtaining substantial extensions of known results. Thus, we give a comprehensive survey and a systematic analysis of these bijections.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 respects.",
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In: Séminaire Lotharingien de Combinatoire, Vol. 60, B60d, 04.11.2008.

Research output: Contribution to journalArticle

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AU - Kitaev, Sergey

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