pattern avoidance in permutations and β(1,0)-trees anders claesson sergey kitaev einar...
TRANSCRIPT
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 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
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
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