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Bijections between 321- and 132-avoiding permutations

This talk explores various bijections between 321 and 132 avoiding permutations, including Knuth's bijection and Richar's bijection. It also discusses the patterns and statistics preserved by these bijections.

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Bijections between 321- and 132-avoiding permutations

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

  2. 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 Bijections between 321- and 132-avoiding permutations

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

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

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

  6. Pattern avoidance 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? 132 213 231 312 ? 123 321 Finding a bijection between the groups of patterns above attracted a lot of attention in the literature. Bijections between 321- and 132-avoiding permutations

  7. The year 1973 ... • The first handheld cell phone call made by MartinCooper 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 Bijections between 321- and 132-avoiding permutations

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

  9. 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 Bijections between 321- and 132-avoiding permutations

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

  11. 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) Bijections between 321- and 132-avoiding permutations

  12. The year 1975 ... Donald Knuth Doron Rotem Bijections between 321- and 132-avoiding permutations

  13. The year 1975 ... 21637845: {2,6,7,8} is a sequence of left-to-right maxima 1 21637845 1 1 3 3 3 4 5 Bijections between 321- and 132-avoiding permutations

  14. The year 1975 ... 321-avoiding permutation 1 21637845 1 1 3 3 3 4 5 standard bijection 54367128 132-avoiding permutation Knuth-Rotem’s bijection Bijections between 321- and 132-avoiding permutations

  15. 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 Bijections between 321- and 132-avoiding permutations

  16. The year 1985 ... Rodica Simion Frank W. Schmidt Simion-Schmidt’s bijection Bijections between 321- and 132-avoiding permutations

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

  18. 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 Bijections between 321- and 132-avoiding permutations

  19. The year 1988 ... Donald Knuth Dana Richards Knuth-Richards’ bijection Bijections between 321- and 132-avoiding permutations

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

  21. 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 Bijections between 321- and 132-avoiding permutations

  22. The year 1995 ... active site (safe to insert the largest element) Julian West Thus 536142 corresponds to 534612 by West’s bijection Bijections between 321- and 132-avoiding permutations

  23. 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 Bijections between 321- and 132-avoiding permutations

  24. The year 2001 ... differences between consecutive right-to-left maxima give the runs of down-steps, while their positions give runs of up-steps. 123-avoiding perm. 54287613 standard bijection Christian Krattenthaler 54367128 132-avoiding perm. Bijections between 321- and 132-avoiding permutations

  25. 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 Bijections between 321- and 132-avoiding permutations

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

  27. 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 Bijections between 321- and 132-avoiding permutations

  28. The year 2003 ... Sergi Elizalde Emeric Deutsch Bijections between 321- and 132-avoiding permutations

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

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

  31. 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. Bijections between 321- and 132-avoiding permutations

  32. 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 Bijections between 321- and 132-avoiding permutations

  33. The year 2006 ... Toufik Mansour Yu-Ping Deng Rosena Ruon Xia Du Bijections between 321- and 132-avoiding permutations

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

  35. The year 2006 ... (s s s )(s s )(s s s )(s )=41572368(10)9(11) 4 3 6 5 4 9 3 2 1 321-avoiding canonical reduced decomposition (s s )(s s s )(s s s s s s )(s )=81542367(10)9(11) 3 2 4 3 2 6 5 4 3 9 2 1 231-avoiding Bijections between 321- and 132-avoiding permutations

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

  37. The year 2007 ... Anders Claesson SK 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 Bijections between 321- and 132-avoiding permutations

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

  39. Classification of the bijections Bijections between 321- and 132-avoiding permutations

  40. Classification of the bijections Bijections between 321- and 132-avoiding permutations

  41. Classification of the bijections inverse reverse Simion-Schmidt = inverse reverse inverse Krattenthaler = inverse reverse = Mansour-Deng-Du Bijections between 321- and 132-avoiding permutations

  42. Classification of the bijections = -1 Knuth-Richards = -1 reverse Φ=Claesson-SK Bijections between 321- and 132-avoiding permutations

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

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

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

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

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

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

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

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

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