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

2. 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

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

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

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

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

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

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

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

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

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

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

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

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

15. 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)

16. 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

17. 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

18. 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

19. 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

20. 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 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

21. 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

22. 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|

23. 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

24. 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

25. 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

26. 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 β(1,0)-trees

27. 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 β(1,0)-trees

28. 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 β(1,0)-trees

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

30. 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

31. 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 Statistics of interest

32. 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 Statistics of interest

33. 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 Statistics of interest

34. 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 Statistics of interest

35. 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 Statistics of interest

36. 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 The involution h

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

38. 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}

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

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

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

42. 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

43. 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 Example of bijection

44. 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 Example of bijection

45. 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 Example of bijection

46. 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 Example of bijection

47. 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 Example of bijection

48. 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

49. 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

50. 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 children Other node ≤ 1 + sum of its children

51. Application of our study

53. 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

54. Thank you!