# Bijections between 321- and 132-avoiding permutations Sergey Kitaev Reykjavík University This is joint work with.

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Bijections between 321- and 132-avoiding permutations Sergey Kitaev Reykjavík University This is joint work with

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Outline of the talk Basic definitions Overview of 9 bijections between 321- and 132-avoiding permutations A recursive description of Knuth-Richards’ bijection Classification of the bijections Statistics preserved by the bijections

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Dyck paths Number of Dyck paths of length 2n is the n-th Catalan number

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Pattern avoidance 6314275 contains four occurrences of the pattern 132 3 75 142 275 543612 avoids the pattern 132 Similarly, one defines 321-, 231-,..., 123-avoiding permutations r(23154)=45132 – reverse operation c(23154)=43512 – complement operation i(23154)=31254 – inverse operation Trivial bijections 1 75

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Pattern avoidance n Schematic structure of 132-avoiding n-permutations 231-avoiding permutations are those that can be sorted with one stack Schematic structure of 321-avoiding permutations This is Catalan structure

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Pattern avoidance 123 321 132 213 231 312 ? Patterns inside the same group are equivalent in the sense of avoidance due to the trivial bijections, but why patterns from different groups need to be avoidance-equivalent? Finding a bijection between the groups of patterns above attracted a lot of attention in the literature.

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1973... The first handheld cell phone call made by Martin Cooper in New York City The Sears Tower in Chicago is finished, becoming the world's tallest building A patent for the ATM is granted to Donald Wetzel, Tom Barnes and George Chastain

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1973... Donald Knuth The standard bijection between 132-avoiding permutations and Dyck paths 74352681

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1973... Robinson-Schensted-Knuth correspondence Bijection between a symmetric group and pairs of standard Young tableaux (P,Q). P- the insertion tableau, Q – the recording tableau

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1973... Corresponding Dyck path (U = UP-step; D = DOWN-step): U DUUDUUDUDDUUDDD D-steps U-steps D-steps A permutation avoids 321 implies P,Q have at most two rows standard bijection Output: 132-avoiding pemutation 67435281 Imput: 321-avoiding permutation 23514697 Knuth’s bijection reflection UUUDDUUDUDDUDDUD

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1975... Bobby Fischer refuses to play in a chess match against Anatoly Karpov, giving Karpov the title The speaker is born in the town of Ulan-Ude The name "Micro-soft" is used by Bill Gates in a letter to Paul Allen for the first time (Microsoft becomes a registered trademark on November 26, 1976)

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1975... Donald KnuthDoron Rotem

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1975... 21637845: {2,6,7,8} is a sequence of left-to-right maxima 21637845 1 1133345

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1975... 21637845 1 1133345 standard bijection 54367128 Knuth-Rotem’s bijection 132-avoiding permutation 321-avoiding permutation

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1985... The first British cell phone call is made (by Ernie Wise to Vodafone) Coca-Cola changes its formula and releases New Coke (The response is negative, and the original formula is back on the market in less than 3 months) Microsoft Corporation releases the first version of Windows, Windows 1.0

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1985... Rodica SimionFrank W. Schmidt Simion-Schmidt’s bijection

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1985... 132-avoiding perm. 123-avoiding perm.

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1988... The Soviet Union begins its program of economic restructuring (perestroika) initiated by Mikhail Gorbachev Luca Aceto turned 17 years old The Summer Olympic Games are held in Seoul, South Korea

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1988... Donald KnuthDana Richards Knuth-Richards’ bijection

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1988... Knuth-Richards’ bijection: 132-avoiders 321-avoiders = reverse (algorithm below) (standard bijection)

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1995... Austria, Finland and Sweden enter the European Union Microsoft releases Windows 95 Mark Dukes continues his study at the University of Oxford, England

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 1995... Julian West Avoid 123Avoid 132Signature 112 1 212122 3 1 2 322 3 1 4234 123322 5 3 1 425 34 1243322 5 3 61425346 12343322 active site (safe to insert the largest element) Thus 536142 corresponds to 534612 by West’s bijection

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2001... United States v. Microsoft: The United States Justice Department announces that it no longer seeks to break- up software maker Microsoft, and will instead seek a lesser antitrust penalty Magnús Halldórsson is awarded the Young Scientist Award of RANNIS Microsoft releases Windows XP

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2001... Christian Krattenthaler 123-avoiding perm. 54287613 standard bijection 54367128132-avoiding perm. differences between consecutive right-to-left maxima give the runs of down-steps, while their positions give runs of up-steps.

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2002... In the second round of the French presidential election, Jacques Chirac is reelected Anders Claesson visits the University of Waterloo The Mars Odyssey finds signs of huge water ice deposits on the planet Mars

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2002... Astrid Reifegerste X X X X X X X X

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2003... Saddam Hussein, former President of Iraq, is captured in Tikrit by the U.S. 4th Infantry Division The speaker defends his Ph.D. Einar Steingrímsson becomes an academic father

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2003... Sergi ElizaldeEmeric Deutsch

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2003... 23145 52314 UDUUUDDUDD UUDUUDDDUD 1(10)29384756 X X X X X UUUDDDUDUD 321-avoiding 132-avoiding reflecting the path standard bijection

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2004... Anders Claesson defends his Ph.D ICE-TCS is (-1) years old

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2004... Sergi Elizalde Igor Pak Essentially use Knuth’s bijection, but apply deepper analysis to prove that the bijection preserves 3 statistics, in particular, fixed points go to fixed points, and excedances go to excedances.

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2006... Microsoft publicly releases Windows Internet Explorer 7 Luca Aceto and Anna Ingolfsdottir present ICE-TCS to the staff at Reykjavík University The conference Permutation Patterns 2006 is held at Reykjavík University

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2006... Toufik MansourYu-Ping DengRosena Ruon Xia Du

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2006... (8+2)/2 (11+1)/2 (8-2+2)/2 (11-1+2)/2

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2006... (s s s )(s s )(s s s )(s )=41572368(10)9(11) (s s )(s s s )(s s s s s s )(s )=81542367(10)9(11) 34 3 21 6549 9 12 345643223 321-avoiding 231-avoiding canonical reduced decomposition

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2007... The Valley of Geysers in Russia was destroyed by a mudflow Susanne Albers becomes one of the 2008 Leibniz Prize winners

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2007... Anders ClaessonSK A recursive bijection Φ between 231-avoiding and 321-avoiding permutations is found, but it turns out to be trivially equivalent to Knuth-Richards’ bijection

Sergey Kitaev Bijections between 321- and 132-avoiding permutations The year 2007... The idea of the recursive bijection Φ: 231-avoiding permutations 321-avoiding permutations reducible case: irreducible case: “Jumping” left-to-right maxima

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Classification of the bijections

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Classification of the bijections

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Classification of the bijections inversereverse = = = Simion-Schmidt inversereverseinverse Krattenthaler inversereverse Mansour-Deng-Du

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Classification of the bijections = Knuth-Richards = reverse Φ=Claesson-SK

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Classification of the bijections = inverse Also, Knuth-Rotem Reifegerste

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics of interest p = 5 2 3 1 4 7 8 9 6 asc p = 5 ldr p = 2 lmax p = 4 rmax p = 2 lmin p = 3 comp p = 2

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics of interest p = 5 2 3 1 4 7 8 9 6 asc p = 5 ldr p = 2 lmax p = 4 rmax p = 2 lmin p = 3 comp p = 2

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics of interest p = 5 2 3 1 4 7 8 9 6 asc p = 5 ldr p = 2 lmax p = 4 rmax p = 2 lmin p = 3 comp p = 2

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics of interest p = 5 2 3 1 4 7 8 9 6 asc p = 5 ldr p = 2 lmax p = 4 rmax p = 2 lmin p = 3 comp p = 2

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics of interest p = 5 2 3 1 4 7 8 9 6 asc p = 5 ldr p = 2 lmax p = 4 rmax p = 2 lmin p = 3 comp p = 2

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics of interest p = 5 2 3 1 4 7 8 9 6 asc p = 5 ldr p = 2 lmax p = 4 rmax p = 2 lmin p = 3 comp p = 2

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics of interest 23 basic statistics each giving rise to 24 statistics

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics of interest

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics preserved by the bijections

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Statistics preserved by the bijections

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Concluding remarks 20072006

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Quiz (related to the talk) Which permutations are involved in the talk? When did Microsoft release Windows 95? When did Coca-Cola change its formula and releases New Coke?

Sergey Kitaev Bijections between 321- and 132-avoiding permutations Thank you for your attention and Merry Christmas!

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