Pattern avoidance in permutations and β(1,0)-trees Anders Claesson Sergey Kitaev Einar Steingrímsson Reykjavík University.

Pattern avoidance in permutations and β(1,0)-trees

Anders Claesson

Sergey Kitaev

Einar SteingrímssonReykjavík University

Outline of the talk

• Objects of interest and historical remarks– 2-stack sortable permutations– Avoiders and nonseparable permutations– β(1,0)-trees

• Statistics of interest

• Main results and bijections

• Open problems

Sorting with a stack

4 1 6 3 2 5

Numbers on stack mustincrease from top

Sorting with a stack

1 6 3 2 5

Numbers on stack mustincrease from top

4

Sorting with a stack

6 3 2 5

Numbers on stack mustincrease from top

4

1

Sorting with a stack

6 3 2 5

Numbers on stack mustincrease from top

4

1

Sorting with a stack

6 3 2 5

Numbers on stack mustincrease from top

1 4

Sorting with a stack

3 2 5

Numbers on stack mustincrease from top

6

1 4

Sorting with a stack

2 5

Numbers on stack mustincrease from top

6

1 4

3

Sorting with a stack

5

Numbers on stack mustincrease from top

6

1 4

32

Sorting with a stack

5

Numbers on stack mustincrease from top

6

1 4 2

3

Sorting with a stack

5

Numbers on stack mustincrease from top

6

1 4 2 3

Sorting with a stack

Numbers on stack mustincrease from top

6

1 4 2 3

5

Sorting with a stack

Numbers on stack mustincrease from top

6

1 4 2 3 5

Sorting with a stack

4 1 6 3 2 51 4 2 3 5 6 2 3 1

Theorem (Knuth):

A permutation is stack-sortable iff it avoids 2-3-1

2-stack-sortable (requires 2passes through the stack)

2-stack sortable (TSS) permutations

1)!(2n1)!(n

2(3n)!

Characterization of TSS permutations (West, 1990): ___

A permutation is TSS iff it avoids 2-3-4-1 and 3-5-2-4-1

Avoidance of 3-2-4-1 unless itis a part of a 3-5-2-4-1 pattern

Conjecture (West, 1990):

The number of TSS permutations is

Work related to TSS permutations

Zeilberger, 1992 the first proof of West’s conjecture

Dulucq, Gire, West, 1996

Goulden, West, 1996

Dulucq, Gire, Guibert, 1998

Bousquet-Mélou, 1998 enumeration of TSS perms subject to 5 statistics

8 classes of perms connecting TSS perms and nonseparable permutations

factorization linking TSS perms, rootednonseparable planar maps, and β(1,0)-trees

relations between rooted nonseparable planar maps and restricted permutations

Cori, Jacquard, Schaeffer, 1997 planar maps, β(1,0)-trees, TSS perms

Work related to TSS permutations

1)!2k-(2n1)!-(2kk)!-1(nk!

k)!-(2n1)!-k(n

1)!(2n1)!(n

2(3n)!

Theorem (Tutte, 1963): The number of rooted nonseparable planar maps on n+1 edges is

Theorem (Brown, Tutte, 1964): The number of rooted nonseparableplanar maps on n+1 edges with k vertices is

the number of TSS n-permswith k-1 ascents

Avoiders and nonseparable permutations

Avoiding 2-4-1-3 and 4-1-3-5-2 gives nonseparable permutations_

|nonseparable permutations| = |TSS permutations|

Avoiding 2-4-1-3 and 3-14-2 gives nonseparable permutations too!

Avoiders = avoiding 3-1-4-2 and 2-41-3 = reverse of nonseparable permutations

Properties of avoiders (avoiding 3-1-4-2 and 2-41-3)

Avoiders are closed under the following compositions: reverse○complement, inverse○reverse, inverse○complement

3 1 2 5 7 6 4 8the 3 (irreducible) components

reducible 8-avoider

8 9 7 5 3 4 6 1 2the 4 reverse components

Lemma: An n-avoider is irreducible iff n precedes 1

Properties of avoiders

Proposition: The number of n-avoiders with k componentsis equal to that with k reverse components

Proof

3 1 2 5 7 6 4 8

5 7 6

31 2

8

4

8

5 7 6 4

31 2

8 4 5 7 6 1 2 3

Properties of avoiders

Proposition: An n-avoider p is reverse irreducible iff either 1 precedes n (in p) or p contains 2-4-1-3 involving n and 1

Lemma: The following is true for avoiders:|1 precedes n precedes (n-1)| = |(n-1) precedes n precedes 1|

Corollary: For avoiders, |1 precedes n| = |(n-1) precedes n|

Properties of avoiders

Lemma: The following is true for avoiders:|1 precedes n precedes (n-1)| = |(n-1) precedes n precedes 1|

3 1 2 5 7 6 4

Proof

2 6 4 5 7 3 1

3 1 2

5 6 4

7

6 4 5

7

2 3 1

1 precedes 7

6 precedes 7

Properties of avoiders

Lemma: The following is true for avoiders:|1 precedes n precedes (n-1)| = |(n-1) precedes n precedes 1|

3 1 2 5 7 6 4

Proof

2 6 4 5 7 3 1

3 1 2

5 6 4

7

6 4 5

7

2 3 1

1 precedes 7

6 precedes 7

β(1,0)-trees

4

11

11 112

1 3

A leaf has label 1An internal non-root node has label ≤ sum of its children’s labelsThe root has label = sum of its children’s labels

A β(1,0)-tree is a labeled rooted plane tree such that

4

11

11 112

1 3

A leaf has label 1An internal non-root node has label ≤ sum of its children’s labelsThe root has label = sum of its children’s labels

A β(1,0)-tree is a labeled rooted plane tree such that

β(1,0)-trees

4

11

11 112

1 3

A leaf has label 1An internal non-root node has label ≤ sum of its children’s labelsThe root has label = sum of its children’s labels

A β(1,0)-tree is a labeled rooted plane tree such that

β(1,0)-trees

4

11

11 112

1 3

A leaf has label 1An internal non-root node has label ≤ sum of its children’s labelsThe root has label = sum of its children’s labels

A β(1,0)-tree is a labeled rooted plane tree such that

β(1,0)-trees

β(1,0)-trees and rooted nonseparable planar maps

Statistics of interest

4

11

11 112

1 3T =

p = 5 2 3 1 4 7 8 9 6

leaves T = 6

lsub T = 2

root T = 4

rpath T = 2

lpath T = 3

sub T = 2

1+asc p = 6

ldr p = 2

lmax p = 4

rmax p = 2

lmin p = 3

comp p = 2

4

11

11 112

1 3T =

p = 5 2 3 1 4 7 8 9 6

leaves T = 6

lsub T = 2

root T = 4

rpath T = 2

lpath T = 3

sub T = 2

1+asc p = 6

ldr p = 2

lmax p = 4

rmax p = 2

lmin p = 3

comp p = 2

Statistics of interest

4

11

11 112

1 3T =

p = 5 2 3 1 4 7 8 9 6

leaves T = 6

lsub T = 2

root T = 4

rpath T = 2

lpath T = 3

sub T = 2

1+asc p = 6

ldr p = 2

lmax p = 4

rmax p = 2

lmin p = 3

comp p = 2

Statistics of interest

4

11

11 112

3T =

p = 5 2 3 1 4 7 8 9 6

leaves T = 6

lsub T = 2

root T = 4

rpath T = 2

lpath T = 3

sub T = 2

1+asc p = 6

ldr p = 2

lmax p = 4

rmax p = 2

lmin p = 3

comp p = 2

1

Statistics of interest

4

11

11 112

1 3T =

p = 5 2 3 1 4 7 8 9 6

leaves T = 6

lsub T = 2

root T = 4

rpath T = 2

lpath T = 3

sub T = 2

1+asc p = 6

ldr p = 2

lmax p = 4

rmax p = 2

lmin p = 3

comp p = 2

Statistics of interest

4

11

11 112

3 3T =

p = 5 2 3 1 4 7 8 9 6

leaves T = 6

lsub T = 2

root T = 4

rpath T = 2

lpath T = 3

sub T = 2

1+asc p = 6

ldr p = 2

lmax p = 4

rmax p = 2

lmin p = 3

comp p = 2

1

label 1

Statistics of interest

T H

h

root T = k root H = m

rpath T = m

rpath H = k

leaves T

non-leaves T

sub T

rsub T

non-leaves H

leaves H

rsub H

sub H

1

1

1

The involution h

The involution h on plane rooted trees

A B

h(A)

h(B)

base casereducible case

h(A)A

irreducible case

hh h

Generating β(1,0)-trees

a

a

b

bc

ca b

a+b+c

c

indecomposable (irreducible) trees decomposable (reducible) tree

3 1

1

2

23

3

There is a 1-to-1 corr. between {1,..,k} x {β(1,0)-trees, n nodes, root=k}and {indecomposable β(1,0)-trees on n+1 nodes with 1 ≤ root ≤ k}

indecomposable (irreducible) trees: on therightmost path only the leaf has label 1

decomposable tree

1 1 11

1

1

1+1

+1

+11

+1

+11

+1

Generating β(1,0)-trees

Irreducible avoiders (the largest element precedes 1)

do nothing if it’s irreducible

Generating avoiders

Generating avoiders

Irreducible avoiders (the largest element precedes 1)

minimal elementto the left of

patterns to the left and to theright of are preserved

Example of bijection

There is a 1-to-1 corr. between {1,..,k} x {n-avoiders, lmax=k}and {irreducible (n+1)-avoiders with 1 ≤ lmax ≤ k}

Φ(1,123) = 4123; Φ(2,123) = 3412; Φ(3,123) = 2341

4

11

11 112

1 3

labels correspond to lmax

assign the empty word to each leaf

apply Φ at each leaf

join and repeat

There is a 1-to-1 corr. between {1,..,k} x {n-avoiders, lmax=k}and {irreducible (n+1)-avoiders with 1 ≤ lmax ≤ k}

Φ(1,123) = 4123; Φ(2,123) = 3412; Φ(3,123) = 2341

4

1,ε

2

1 3

labels correspond to lmax

assign the empty word to each leaf

apply Φ at each leaf

join and repeat1,ε1,ε 1,ε1,ε

1,ε

Example of bijection

There is a 1-to-1 corr. between {1,..,k} x {n-avoiders, lmax=k}and {irreducible (n+1)-avoiders with 1 ≤ lmax ≤ k}

Φ(1,123) = 4123; Φ(2,123) = 3412; Φ(3,123) = 2341

4

1,ε

2

1 3

labels correspond to lmax

assign the empty word to each leaf

apply Φ at each leaf

join and repeat1,ε1,ε 1,ε1,ε

1,ε 1= Φ (1,ε)

1 1

11 1

1

Example of bijection

There is a 1-to-1 corr. between {1,..,k} x {n-avoiders, lmax=k}and {irreducible (n+1)-avoiders with 1 ≤ lmax ≤ k}

Φ(1,123) = 4123; Φ(2,123) = 3412; Φ(3,123) = 2341

4

1,ε

2,12

13,123

labels correspond to lmax

assign the empty word to each leaf

apply Φ at each leaf

join and repeat1,ε1,ε 1,ε1,ε

1,ε 1= Φ (1,ε)

1 1

1 1 11

Example of bijection

There is a 1-to-1 corr. between {1,..,k} x {n-avoiders, lmax=k}and {irreducible (n+1)-avoiders with 1 ≤ lmax ≤ k}

Φ(1,123) = 4123; Φ(2,123) = 3412; Φ(3,123) = 2341

4

1,ε

2,12

3,123

labels correspond to lmax

assign the empty word to each leaf

apply Φ at each leaf

join and repeat1,ε1,ε 1,ε1,ε

1,ε 1= Φ (1,ε)

1 1

1 1 112311,2314

Example of bijection

There is a 1-to-1 corr. between {1,..,k} x {n-avoiders, lmax=k}and {irreducible (n+1)-avoiders with 1 ≤ lmax ≤ k}

Φ(1,123) = 4123; Φ(2,123) = 3412; Φ(3,123) = 2341

4

1,ε

2,12

3,123

labels correspond to lmax

assign the empty word to each leaf

apply Φ at each leaf

join and repeat1,ε1,ε 1,ε1,ε

1,ε 1= Φ (1,ε)

1 1

1 1 11231

2341

1,2314

52314

Example of bijection

More results

The first tuple has the same distribution on n-TSS permutationsas the second tuple has on n-avoiders:

( asc, rmax, comp’ )( asc, rmax, comp )

where the statistic comp’ can be defined using the decompositionof TSS permutations by Goulden and West

Theorem (Euler): For planar graphs n-e+f=2

Proof

Another proof

If p is a permutation then 1 + des p + asc p = |p|

For a tree T, leaves T + non-leaves T = all nodes T

(des p + 2) + (asc p+2) = (|p|+1)+2(# vertices) + (# faces) = (# edges)+2

More results

Application of our study

All β(0,1)-trees on k=2 edgesAll bicubic planar maps on 3k=6 edges

bipartite, all nodes of degree 3

Leaves have label 0.Root = 1 + sum of its childrenOther node ≤ 1 + sum of its children

Application of our study

Application of our study

Open problems

Conjecture: (asc, rmax, comp, ldr) is equidistributedon TSS permutations and avoiders

Conjecture: The following tuples of statistics are equidistributed on avoiders: (asc, comp, lmax, rmax) and (des, comp.r, rmax, lmax)

Describe a map (involution) on avoiders (not using other combinatorialobjects like the involution h and β(1,0)-trees) giving the equidistributionof (lmax,rmax) and (rmax, lmax) on avoiders

such an involution on permutationsis the operation of reverse

generalization: pattern between two leftmost lmax

Thank you!