Transcript Universal Cycles for combinatorial objects

Bijections between 321- and
132-avoiding permutations
Sergey Kitaev
Reykjavík University
This is joint work with
Outline of the talk
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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
1 75
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
Sergey Kitaev
Trivial
bijections
Bijections between 321- and 132-avoiding permutations
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
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
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?
123
321
?
132
213
231
312
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 ...
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The first handheld cell phone call made by Martin
Cooper in New York City
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The Sears Tower in Chicago is finished, becoming the
world's tallest building
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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 ...
The standard bijection between 132-avoiding
permutations and Dyck paths
Donald Knuth
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 ...
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):
U DU U D U U D U D D U U D D D
reflection
Knuth’s bijection UUUDDUUDUDDUDDUD
standard bijection
Output: 132-avoiding pemutation 67435281
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
The year 1975 ...
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Bobby Fischer refuses to play in a chess match against
Anatoly Karpov, giving Karpov the title
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The speaker is born in the town of Ulan-Ude
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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
Sergey Kitaev
Doron Rotem
Bijections between 321- and 132-avoiding permutations
The year 1975 ...
21637845: {2,6,7,8} is a sequence of left-to-right maxima
21637845
Sergey Kitaev
11133345
Bijections between 321- and 132-avoiding permutations
The year 1975 ...
321-avoiding permutation
21637845
11133345
standard bijection
54367128
132-avoiding permutation
Knuth-Rotem’s bijection
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
The year 1985 ...
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The first British cell phone call is made (by Ernie Wise to
Vodafone)
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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)
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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 ...
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The Soviet Union begins its program of economic
restructuring (perestroika) initiated by Mikhail Gorbachev
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Luca Aceto turned 17 years old
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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
= reverse
Sergey Kitaev
(algorithm below)
321-avoiders
(standard bijection)
Bijections between 321- and 132-avoiding permutations
The year 1995 ...
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Austria, Finland and Sweden enter the European Union
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Microsoft releases Windows 95
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Mark Dukes continues his study at the University of
Oxford, England
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
The year 1995 ...
active site (safe to insert the largest element)
Avoid 123 Avoid 132 Signature
Julian West
1
12
312
3 1 42
1
12
3 12
34 12
2
22
322
3322
5 3 1 42
5 3 6142
5 34 12
5346 12
43322
343322
Thus 536142 corresponds to 534612 by West’s bijection
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
The year 2001 ...
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United States v. Microsoft: The United States Justice
Department announces that it no longer seeks to breakup software maker Microsoft, and will instead seek a
lesser antitrust penalty
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Magnús Halldórsson is awarded the Young Scientist
Award of RANNIS
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Microsoft releases Windows XP
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
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
Christian Krattenthaler
standard bijection
54367128
Sergey Kitaev
132-avoiding perm.
Bijections between 321- and 132-avoiding permutations
The year 2002 ...
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In the second round of the French presidential election,
Jacques Chirac is reelected
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Anders Claesson visits the University of Waterloo
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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 ...
X
X
X
X
X
X
X
X
Astrid Reifegerste
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
The year 2003 ...
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Saddam Hussein, former President of Iraq, is captured in
Tikrit by the U.S. 4th Infantry Division
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The speaker defends his Ph.D.
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Einar Steingrímsson becomes an academic father
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
The year 2003 ...
Sergi Elizalde
Sergey Kitaev
Emeric Deutsch
Bijections between 321- and 132-avoiding permutations
The year 2003 ...
321-avoiding
23145
52314
X
132-avoiding
standard bijection
X
X
X
UDUUUDDUDD
X
reflecting the path
1(10)29384756
UUUDDDUDUD
Sergey Kitaev
UUDUUDDDUD
Bijections between 321- and 132-avoiding permutations
The year 2004 ...
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Anders Claesson defends his Ph.D
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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 ...
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Microsoft publicly releases Windows Internet Explorer 7
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Luca Aceto and Anna Ingolfsdottir present ICE-TCS to
the staff at Reykjavík University
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The conference Permutation Patterns 2006 is held at
Reykjavík University
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
The year 2006 ...
Toufik Mansour
Sergey Kitaev
Yu-Ping Deng
Rosena Ruon Xia Du
Bijections between 321- and 132-avoiding permutations
The year 2006 ...
(8+2)/2
(8-2+2)/2
Sergey Kitaev
(11+1)/2
(11-1+2)/2
Bijections between 321- and 132-avoiding permutations
The year 2006 ...
(s3s2s1)(s4s3)(s6s5s 4)(s9)=41572368(10)9(11)
321-avoiding
canonical reduced decomposition
(s3s2)(s4s3s2)(s6s5s 4s3s2s1)(s9)=81542367(10)9(11)
231-avoiding
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations
The year 2007 ...
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The Valley of Geysers in Russia was destroyed by a
mudflow
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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
reducible
case:
“Jumping” left-to-right
maxima
irreducible
case:
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
inverse
Krattenthaler
=
inverse
=
Sergey Kitaev
reverse
Mansour-Deng-Du
Bijections between 321- and 132-avoiding permutations
Classification of the bijections
-1
=
Knuth-Richards
=
-1
reverse
Sergey Kitaev
Φ=Claesson-SK
Bijections between 321- and 132-avoiding permutations
Classification of the bijections
Also,
=
Reifegerste
Sergey Kitaev
inverse
inverse
Knuth-Rotem
Bijections between 321- and 132-avoiding permutations
Statistics of interest
p = 5 2 3 1 4 7 8 9 6
Sergey Kitaev
lmax p = 4
asc p = 5
comp p = 2
lmin p = 3
rmax p = 2
ldr p = 2
Bijections between 321- and 132-avoiding permutations
Statistics of interest
lmax p = 4
asc p = 5
p = 5 2 3 1 4 7 8 9 6
Sergey Kitaev
comp p = 2
lmin p = 3
rmax p = 2
ldr p = 2
Bijections between 321- and 132-avoiding permutations
Statistics of interest
lmax p = 4
asc p = 5
p = 5 2 3 1 4 7 8 9 6
Sergey Kitaev
comp p = 2
lmin p = 3
rmax p = 2
ldr p = 2
Bijections between 321- and 132-avoiding permutations
Statistics of interest
lmax p = 4
asc p = 5
p = 5 2 3 1 4 7 8 9 6
Sergey Kitaev
comp p = 2
lmin p = 3
rmax p = 2
ldr p = 2
Bijections between 321- and 132-avoiding permutations
Statistics of interest
lmax p = 4
asc p = 5
p = 5 2 3 1 4 7 8 9 6
Sergey Kitaev
comp p = 2
lmin p = 3
rmax p = 2
ldr p = 2
Bijections between 321- and 132-avoiding permutations
Statistics of interest
lmax p = 4
asc p = 5
p = 5 2 3 1 4 7 8 9 6
Sergey Kitaev
comp p = 2
lmin p = 3
rmax p = 2
ldr p = 2
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
2006
Sergey Kitaev
2007
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!
Sergey Kitaev
Bijections between 321- and 132-avoiding permutations