Pattern avoidance, β(1,0)-trees, and 2-stack sortable permutations Anders Claesson Sergey Kitaev Einar Steingrímsson Reykjavík University.

Pattern avoidance, β(1,0)-trees, and 2-stack sortable permutations Anders Claesson Sergey Kitaev Einar Steingrímsson Reykjavík University

Permutation Patterns 2007 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

Permutation Patterns 2007 Sorting with a stack 4 1 6 3 2 5 Numbers on stack must increase from top

Permutation Patterns 2007 Sorting with a stack 1 6 3 2 5 Numbers on stack must increase from top 4

Permutation Patterns 2007 Sorting with a stack 6 3 2 5 Numbers on stack must increase from top 4 1

Permutation Patterns 2007 Sorting with a stack 6 3 2 5 Numbers on stack must increase from top 1 4

Permutation Patterns 2007 Sorting with a stack 3 2 5 Numbers on stack must increase from top 6 1 4

Permutation Patterns 2007 Sorting with a stack 2 5 Numbers on stack must increase from top 6 1 4 3

Permutation Patterns 2007 Sorting with a stack 5 Numbers on stack must increase from top 6 1 4 3 2

Permutation Patterns 2007 Sorting with a stack 5 Numbers on stack must increase from top 6 1 4 2 3

Permutation Patterns 2007 Sorting with a stack Numbers on stack must increase from top 6 1 4 2 3 5

Permutation Patterns 2007 Sorting with a stack 4 1 6 3 2 5 1 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 2 passes through the stack)

Permutation Patterns 2007 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

Permutation Patterns 2007 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, rooted nonseparable planar maps, and β(1,0)-trees relations between rooted nonseparable planar maps and restricted permutations Cori, Jacquard, Schaeffer, 1997planar maps, β(1,0)-trees, TSS perms

Permutation Patterns 2007 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

Permutation Patterns 2007 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

Permutation Patterns 2007 Properties of avoiders (of 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 8 the 3 (irreducible) components reducible 8-avoider 8 9 7 5 3 4 6 1 2 the 4 reverse components Lemma: An n-avoider is irreducible iff n precedes 1

Permutation Patterns 2007 Properties of avoiders Proposition: The number of n-avoiders with k components is equal to that with k reverse components Proof 3 1 2 5 7 6 4 8 5 7 6 3 1 2 8 4 8 5 7 6 4 3 1 2 8 4 5 7 6 1 2 3

Permutation Patterns 2007 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

Permutation Patterns 2007 β(1,0)-trees 4 1 1 1 1 112 13 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 A β(1,0)-tree is a labeled rooted plane tree such that

Permutation Patterns 2007 β(1,0)-trees 4 1 1 1 1 11 2 13 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 A β(1,0)-tree is a labeled rooted plane tree such that

Permutation Patterns 2007 β(1,0)-trees 4 1 1 1 1 112 13 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 A β(1,0)-tree is a labeled rooted plane tree such that

Permutation Patterns 2007 Statistics of interest 4 1 1 1 1 112 13 T = 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

Permutation Patterns 2007 Statistics of interest 4 1 1 1 1 11 2 13 T = 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

Permutation Patterns 2007 Statistics of interest 4 1 1 1 1 11 2 3 T = 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

Permutation Patterns 2007 Statistics of interest 4 1 1 1 1 1 12 1 3 T = 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

Permutation Patterns 2007 Statistics of interest 4 1 1 1 1 112 3 3 T = 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

Permutation Patterns 2007 Statistics of interest 2 1 1 111 2 3 2 T = p = 1 2 5 9 8 6 7 4 3 stem T = 2 zeil p = 3 lir p = 4 zeil.c p = 3 2 number of non-leaf nodes in the intersection of the leftmost and rightmost paths

Permutation Patterns 2007 Statistics of interest 2 1 1 111 2 3 2 T = p = 1 2 5 9 8 6 7 4 3 stem T = 2 zeil p = 3 lir p = 4 zeil.c p = 3 2 number of non-leaf nodes in the intersection of the leftmost and rightmost paths left increasing run

Permutation Patterns 2007 Statistics of interest 2 1 1 111 2 3 2 T = p = 1 2 5 9 8 6 7 4 3 stem T = 2 zeil p = 3 lir p = 4 zeil.c p = 3 2 number of non-leaf nodes in the intersection of the leftmost and rightmost paths left increasing run zeil applied to the complement of p

Permutation Patterns 2007 The involution h on β(1,0)-trees TH 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

Permutation Patterns 2007 Generating β(1,0)-trees a a b b c c a b a+b+c c indecomposable (irreducible) treesdecomposable (reducible) tree 3 1 1 2 2 3 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}

Permutation Patterns 2007 Generating β(1,0)-trees indecomposable (irreducible) trees: on the rightmost path only the leaf has label 1 decomposable tree 111 1 1 1 1 +1 1 1

Permutation Patterns 2007 Generating avoiders Irreducible avoiders (the largest element precedes 1) do nothing if it’s irreducible

Permutation Patterns 2007 Generating avoiders Irreducible avoiders (the largest element precedes 1) minimal element to the left of patterns to the left and to the right of are preserved

Permutation Patterns 2007 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 1 1 1 112 13 labels correspond to lmax assign the empty word to each leaf apply Φ at each leaf join and repeat

Permutation Patterns 2007 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 13 labels correspond to lmax assign the empty word to each leaf apply Φ at each leaf join and repeat 1,ε

Permutation Patterns 2007 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 13 labels correspond to lmax assign the empty word to each leaf apply Φ at each leaf join and repeat 1,ε 1 = Φ ( 1,ε) 1 1 1 1 1 1

Permutation Patterns 2007 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 1 3,123 labels correspond to lmax assign the empty word to each leaf apply Φ at each leaf join and repeat 1,ε 1 = Φ ( 1,ε) 11 1 11 1

Permutation Patterns 2007 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 repeat 1,ε 1 = Φ ( 1,ε) 11 1 11 1 231 1,2314

Permutation Patterns 2007 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 repeat 1,ε 1 = Φ ( 1,ε) 11 1 11 1 231 2341 1,2314 52314

Permutation Patterns 2007 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 1,ε 2,12 3,123 labels correspond to lmax assign the empty word to each leaf apply Φ at each leaf join and repeat 1,ε 1 = Φ ( 1,ε) 11 1 11 1 231 2341 1,2314 52314 4,523147896

Permutation Patterns 2007 Main results The first tuple has the same distribution on β(1,0)-trees with n+1 nodes as the second tuple has on n-avoiders: ( leaves, sub, lpath, rpath, stem.h.m, lsub, root ) ( asc, comp, lmin, rmax, lir, ldr, lmax ) ( leaves, sub, root, lsub, rsub, stem, rpath, stem.h.m ) ( des, ldr, lmin, zeil.c, comp, lir, lmax, zeil.c.r ) mirror image

Permutation Patterns 2007 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 of TSS permutations by Goulden and West

Permutation Patterns 2007 More results 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

Permutation Patterns 2007 Open problems Conjecture: (asc, rmax, comp, ldr) is equidistributed on 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 combinatorial objects like the involution h and β(1,0)-trees) giving the equidistribution of (lmax,rmax) and (rmax, lmax) on avoiders such an involution on permutations is the operation of reverse generalization: pattern between two leftmost lmax

Permutation Patterns 2007 Thank you for your attention!

Pattern avoidance, β(1,0)-trees, and 2-stack sortable permutations Anders Claesson Sergey Kitaev Einar Steingrímsson Reykjavík University.

Similar presentations

Presentation on theme: "Pattern avoidance, β(1,0)-trees, and 2-stack sortable permutations Anders Claesson Sergey Kitaev Einar Steingrímsson Reykjavík University."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Pattern avoidance, β(1,0)-trees, and 2-stack sortable permutations Anders Claesson Sergey Kitaev Einar Steingrímsson Reykjavík University.

Similar presentations

Presentation on theme: "Pattern avoidance, β(1,0)-trees, and 2-stack sortable permutations Anders Claesson Sergey Kitaev Einar Steingrímsson Reykjavík University."— Presentation transcript:

Similar presentations

About project

Feedback