[Paper Review] Classification of bijections between 321- and 132-avoiding permutations

Anders Claesson, Sergey Kitaev|ArXiv.org|May 9, 2008

Advanced Combinatorial Mathematics9 references4 citations

TL;DR

This paper provides a comprehensive classification of all known bijections between 321- and 132-avoiding permutations, showing how they relate via trivial symmetries and identifying which statistics each preserves. It establishes that the Knuth-Richards bijection—equivalent to Simion-Schmidt and Krattenthaler’s maps—preserves 11 permutation statistics, the highest known number, and offers a recursive construction of this bijection.

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.

Motivation & Objective

• To systematically classify all published bijections between 321- and 132-avoiding permutations and determine their relationships.
• To identify which permutation statistics are preserved by each bijection, using a comprehensive set of 190 statistics derived from basic ones via symmetries.
• To demonstrate that the Knuth-Richards bijection preserves the maximum known number of 11 statistics, making it the most structure-preserving among known maps.
• To provide a recursive description of the Knuth-Richards bijection, linking it to earlier constructions by Knuth and Richards.
• To show that the bijection respects linearly independent statistics, ensuring maximal non-reduundancy in the preserved properties.

Proposed method

• Define a base set of 23 permutation statistics (e.g., asc, des, exc, lmax, peak) and generate 190 representative statistics by applying symmetries: reverse, complement, inverse, and their compositions.
• Use empirical equality (up to length 7) to collapse equivalent statistics, resulting in a final set of 190 distinct statistics for analysis.
• Apply recursive insertion of the largest element (n+1) to model bijections, tracking changes in statistics like lir, rir, slmax, and components.
• Establish bijection equivalences via trivial symmetries: e.g., showing that Knuth-Richards is equivalent to Simion-Schmidt and Krattenthaler via composition with reverse, complement, or inverse.
• Use Dyck path representations and tableau correspondence (via Robinson-Schensted insertion) to interpret and verify statistic preservation, especially for lir, lmax, and rmin.
• Prove equivalence of statistics under bijection by induction on permutation length, tracking active sites and component structure during recursive insertion.

Experimental results

Research questions

• RQ1 Which bijections between 321- and 132-avoiding permutations preserve the largest number of permutation statistics?
• RQ2 How are the known bijections related through trivial symmetries (reverse, complement, inverse) and their compositions?
• RQ3 Can the Knuth-Richards bijection be described recursively, and how does it relate to other well-known bijections like Simion-Schmidt or Krattenthaler?
• RQ4 What is the maximal set of linearly independent statistics preserved by any single bijection between these two classes?
• RQ5 How do statistics like lir, rir, slmax, and components behave under recursive insertion of the largest element in the bijection construction?

Key findings

• The Knuth-Richards bijection preserves 11 permutation statistics—the highest number known among any bijection between 321- and 132-avoiding permutations.
• This bijection is equivalent to the Simion-Schmidt and Krattenthaler bijections via compositions with trivial symmetries (reverse, complement, inverse).
• The bijection preserves the following 11 statistics: asc, des, exc, ldr, rdr, lir, rir, zeil, comp, lmax, rmax, and their symmetric variants via n-stat and m-stat.
• The recursive construction of the Knuth-Richards bijection is formally described, showing how insertion of n+1 at active sites preserves and transforms statistics like slmax and components.
• The statistics lir ≃ lmax and lir.i ≃ rmin are preserved via Dyck path interpretations, linking tableau structure to path slopes and return positions.
• The set of 190 statistics is closed under symmetries and empirical equality checks, ensuring no redundant or duplicate statistics are included in the analysis.