Bijections between 321- and 132-avoiding permutations

Sergey KitaevReykjavík 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

nSchematic structure of132-avoiding n-permutations

231-avoiding permutations are those that can be sorted with one stack

Schematic structure of321-avoiding permutations

This is Catalan structure

Sergey Kitaev Bijections between 321- and 132-avoiding permutations

Pattern avoidance

123321

132213231312

?

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 aboveattracted 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-avoidingpermutations 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 Youngtableaux (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 DUUDUUD UDD UU DDD

D-steps

U-steps

U-steps

D-steps

A permutation avoids 321 implies P,Q have at most two rows

standard bijectionOutput: 132-avoiding pemutation 67435281

Imput: 321-avoiding permutation 23514697

Knuth’s bijectionreflection

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 Knuth Doron 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 1 1 3 3 3 4 5

Sergey Kitaev Bijections between 321- and 132-avoiding permutations

The year 1975 ...

21637845 1 1 1 3 3 3 4 5

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 Simion Frank 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 Knuth Dana 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 123 Avoid 132 Signature

1 1 2

1 2 12 22

3 1 2 3 12 322

3 1 42 34 12 3322

5 3 1 42 5 34 12 43322

5 3 6142 5346 12 343322

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

54367128 132-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

XX

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 Elizalde Emeric Deutsch

Sergey Kitaev Bijections between 321- and 132-avoiding permutations

The year 2003 ...

23145

52314

UDUUUDDUDD

UUDUUDDDUD

1(10)29384756

XX

XX

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 ElizaldeIgor Pak

Essentially use Knuth’s bijection, but apply deepper analysis toprove 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 Mansour Yu-Ping Deng Rosena 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)

343 2 1 6 5 4 9

91234564 3 223

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 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

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

reduciblecase:

irreduciblecase:

“Jumping” left-to-rightmaxima

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

inverse reverse

=

=

=

Simion-Schmidt

inverse reverse inverseKrattenthaler

inverse reverse

Mansour-Deng-Du

Sergey Kitaev Bijections between 321- and 132-avoiding permutations

Classification of the bijections

= -1

Knuth-Richards

= -1

reverse Φ=Claesson-SK

Sergey Kitaev Bijections between 321- and 132-avoiding permutations

Classification of the bijections

= inverse inverse

Also,

Knuth-RotemReifegerste

Sergey Kitaev Bijections between 321- and 132-avoiding permutations

Statistics of interest

p = 5 2 3 1 4 7 8 9 6asc 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 6asc 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 6asc 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 6asc 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 6asc 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 6asc 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!