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

Pattern avoidance in permutations and β(1,0)-trees Anders Claesson Sergey Kitaev Einar Steingrímsson Reykjavík 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 must increase from top

Sorting with a stack 1 6 3 2 5 Numbers on stack must increase from top 4

Sorting with a stack 6 3 2 5 Numbers on stack must increase from top 1 4

Sorting with a stack 1 4 6 3 2 5 Numbers on stack must increase from top

Sorting with a stack 1 4 2 3 5 6 4 1 6 3 2 5 2 3 1 2-stack-sortable (requires 2 passes through the stack) Theorem (Knuth): A permutation is stack-sortable iff it avoids 2-3-1

2-stack sortable (TSS) permutations 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 it is 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 relations between rooted nonseparable planar maps and restricted permutations Dulucq, Gire, West, 1996 factorization linking TSS perms, rooted nonseparable planar maps, and β(1,0)-trees Goulden, West, 1996 Cori, Jacquard, Schaeffer, 1997 planar maps, β(1,0)-trees, TSS perms Dulucq, Gire, Guibert, 1998 8 classes of perms connecting TSS perms and nonseparable permutations enumeration of TSS perms subject to 5 statistics Bousquet-Mélou, 1998

Work related to TSS permutations Theorem (Tutte, 1963): The number of rooted nonseparable planar maps on n+1 edges is Theorem (Brown, Tutte, 1964): The number of rooted nonseparable planar maps on n+1 edges with k vertices is the number of TSS n-perms with 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(avoiding3-1-4-2and2-41-3) Avoiders are closed under the following compositions: reverse○complement, inverse○reverse, inverse○complement reducible8-avoider 3 1 2 5 7 6 4 8 the 3 (irreducible) components 8 9 7 5 3 4 6 1 2 the 4 reversecomponents Lemma: An n-avoider is irreducible iff n precedes 1

Properties of avoiders Proposition: The number of n-avoiders with k components is equal to that with k reverse components Proof 8 4 5 7 6 1 2 3 3 1 2 5 7 6 4 8 8 8 5 7 6 5 7 6 4 4 3 3 1 2 1 2

Properties of avoiders Lemma: The following is true for avoiders: |1 precedes n precedes (n-1)| = |(n-1) precedes n precedes 1| Proof 6 precedes 7 3 1 2 5 7 6 4 2 6 4 5 7 3 1 1 precedes 7 7 7 5 6 4 6 4 5 3 1 2 2 3 1

β(1,0)-trees 4 1 3 2 1 1 1 1 1 1 A β(1,0)-tree is a labeled rooted plane tree such that A leaf has label 1 An internal non-root node has label ≤ sum of its children’s labels The root has label = sum of its children’s labels

β(1,0)-trees 4 1 3 2 1 1 1 1 1 1 A β(1,0)-tree is a labeled rooted plane tree such that A leaf has label 1 Aninternal non-root nodehas label ≤ sum of its children’s labels The root has label = sum of its children’s labels

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

Statistics of interest root T = 4 4 leaves T = 6 T = sub T = 2 1 3 lpath T = 3 rpath T = 2 2 1 1 1 1 lsub T = 2 1 lmax p = 4 1 1+asc p = 6 comp p = 2 lmin p = 3 p= 5 2 3 1 4 7 8 9 6 rmax p = 2 ldr p = 2

Statistics of interest root T = 4 4 leaves T = 6 T = sub T = 2 1 3 lpath T = 3 rpath T = 2 1 1 1 2 1 lsub T = 2 1 1 lmax p = 4 1+asc p = 6 comp p = 2 lmin p = 3 p= 5 2 3 1 4 7 8 9 6 rmax p = 2 ldr p = 2

Statistics of interest root T = 4 4 label 1 leaves T = 6 T = sub T = 2 3 1 3 lpath T = 3 rpath T = 2 2 1 1 1 1 lsub T = 2 1 lmax p = 4 1 1+asc p = 6 comp p = 2 lmin p = 3 p= 5 2 3 1 4 7 8 9 6 rmax p = 2 ldr p = 2

The involution h root T = k root H = m T H h 1 rpath T = m rpath H = k 1 1 leaves T non-leaves H non-leaves T leaves H sub T rsub H rsub T sub H

The involution h on plane rooted trees h A B h(B) h A h(A) h h(A) reducible case irreducible case base case

Generating β(1,0)-trees a+b+c a c b a a c b b c indecomposable (irreducible) trees decomposable (reducible) tree 1 3 2 3 1 3 2 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}

Generating β(1,0)-trees decomposable tree 1 1 1 1 1 indecomposable (irreducible) trees: on the rightmost path only the leaf has label 1 1 +1 +1 +1 1 +1 +1 1 +1 1

Generating avoiders Irreducible avoiders (the largest element precedes 1) do nothing if it’s irreducible

Generating avoiders Irreducible avoiders (the largest element precedes 1) patterns to the left and to the right of are preserved minimal element to the left of

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 labels correspond to lmax 4 assign the empty word to each leaf 1 3 apply Φ at each leaf 2 1 1 1 1 join and repeat 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 labels correspond to lmax 4 assign the empty word to each leaf 1 3 apply Φ at each leaf 1,ε 1,ε 1,ε 2 1,ε join and repeat 1,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 labels correspond to lmax 4 assign the empty word to each leaf 1 3 1 1 1 1 apply Φ at each leaf 1,ε 1,ε 1,ε 2 1,ε join and repeat 1 1 1,ε 1,ε 1= Φ (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 labels correspond to lmax 4 assign the empty word to each leaf 3,123 1 1 1 1 1 apply Φ at each leaf 1,ε 1,ε 1,ε 2,12 1,ε join and repeat 1 1 1,ε 1,ε 1= Φ (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 labels correspond to lmax 4 assign the empty word to each leaf 1,2314 3,123 231 1 1 1 1 apply Φ at each leaf 1,ε 1,ε 1,ε 2,12 1,ε join and repeat 1 1 1,ε 1,ε 1= Φ (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 labels correspond to lmax 4 2341 52314 assign the empty word to each leaf 1,2314 3,123 231 1 1 1 1 apply Φ at each leaf 1,ε 1,ε 1,ε 2,12 1,ε join and repeat 1 1 1,ε 1,ε 1= Φ (1,ε)

More results The first tuple has the same distribution on n-TSS permutations as the second tuple has on n-avoiders: ( asc,rmax, comp’) ( asc, rmax, comp) where the statistic comp’ can be defined using the decomposition ofTSS permutations by Goulden and West

More results Theorem (Euler): For planar graphs n-e+f=2 If p is a permutation then 1 + des p + asc p = |p| Proof (des p + 2) + (asc p+2) = (|p|+1)+2 (# vertices) + (# faces) = (# edges)+2 Another proof For a tree T, leaves T + non-leaves T = all nodes T

Application of our study bipartite, all nodes of degree 3 Leaves have label 0. Root = 1 + sum of its children Other node≤ 1 + sum of its children All bicubic planar maps on 3k=6 edges All β(0,1)-trees on k=2 edges

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