Staircases, dominoes, and the growth rate of Av(1324)

Transcription

1 Staircases, dominoes, and the growth rate of Av(1324) Robert Brignall Joint work with David Bevan, Andrew Elvey Price and Jay Pantone TU Wien, 28th August 2017

2 Permutation containment Permutations in one-line notation: π = π(1) π(n) Pattern containment: σ π if there exists a subsequence of π(1) π(n) with the same relative ordering as σ. Containment is a partial order. Conversely, π avoids σ if σ π.

3 Permutation containment < Permutations in one-line notation: π = π(1) π(n) Pattern containment: σ π if there exists a subsequence of π(1) π(n) with the same relative ordering as σ. Containment is a partial order. Conversely, π avoids σ if σ π.

4 Permutation containment 102 Permutation class: a hereditary collection C, i.e. π C and σ π implies σ C. Principal classes characterised by avoiding one permutation: C = Av(β) = {permutations π : β π}.

5 Permutation containment 102 Permutation class: a hereditary collection C, i.e. π C and σ π implies σ C. Principal classes characterised by avoiding one permutation: C = Av(β) = {permutations π : β π}. Av(12) = {1, 21, 321,... } has 1 permutation of each length. has 1, 2, 5, 14, 42,... of lengths n = 1, 2, 3, 4, 5,....

6 Counting......precisely Generating function for a class C is the formal power series where C n = {π C : π = n}. f C (z) = z π = C n z n, π C n=1

7 Counting......precisely Generating function for a class C is the formal power series where C n = {π C : π = n}. f C (z) = z π = C n z n, π C n=1...vaguely For principal classes Av(β), the growth rate is n gr(av(β)) = lim Av(β) n. n Must exist due to Arratia (1999) and Marcus & Tardos (2004).

8 Counting Principal Classes State of knowledge, since 1997: β Av(β) n gr(av(β)) ( ) n n + 1 n 1342 (7n 2 3n 2) ( 1) 2413 n ( ) k+1 (2k 4)! n k + 2 ( 1) n k k!(k 2)! 2 k= n ( )( ) 2k n 2 3k k + 1 n 2kn k k (k + 1) k=0 (k + 2)(n k + 1) ?? Up to symmetries, this covers all Av(β) with β 4.

9 Exact enumeration of Av(1324) Not even God knows Av(1324) Doron Zeilberger, 2004

10 Exact enumeration of Av(1324) Not even God knows Av(1324) Doron Zeilberger, 2004 More recently, Conway & Guttman (2015) computed Av(1324) 36 =

11 Growth rate of Av(1324) Lower Upper 2004: Bóna : Bóna : Albert et al : Claesson, Jelínek & Steingrímsson : Bóna : Bóna : Bevan : Conway & Guttmann estimate gr(av(1324)) ± 0.01 An upper bound of follows from an unproven conjecture.

12 Growth rate of Av(1324) Lower Upper 2004: Bóna : Bóna : Albert et al : Claesson, Jelínek & Steingrímsson : Bóna : Bóna : Bevan 9.81 This work : Conway & Guttmann estimate gr(av(1324)) ± 0.01 An upper bound of follows from an unproven conjecture.

13 The staircase The infinite decreasing (, ) staircase: S = Proposition Av(1324) S

14 Gridding a 1324-avoider

15 Gridding a 1324-avoider p 1 p 1 uppermost 1 in a 213

16 Gridding a 1324-avoider p 1 p 2 p 1 uppermost 1 in a 213 p 2 leftmost 2 in a 132 consisting of points below p 1 divider

17 Gridding a 1324-avoider p 1 p 2 p 2 leftmost 2 in a 132 consisting of points below p 1 divider No points above p 1 and to the right of p 2

18 Gridding a 1324-avoider p 1 p 2 p 3 p 2 leftmost 2 in a 132 consisting of points below p 1 divider p 3 uppermost 1 in a 213 consisting of points to right of p 2 divider

19 Gridding a 1324-avoider p 1 p 2 p 3 p 3 uppermost 1 in a 213 consisting of points to right of p 2 divider No points to the left of p 2 and below p 3

20 Gridding a 1324-avoider p 1 p 2 p 3 p 4 p 3 uppermost 1 in a 213 consisting of points to right of p 2 divider p 4 leftmost 2 in a 132 consisting of points below p 3 divider

21 Gridding a 1324-avoider p 1 p 2 p 3 p 4 p 4 leftmost 2 in a 132 consisting of points below p 3 divider No points above p 3 and to the right of p 4

22 Gridding a 1324-avoider Terminates after no more than n/2 steps.

23 The greedy gridding of a large 1324-avoider Data provided by Einar Steingrímsson.

24 Where do I find 1324 in a staircase? Only in two adjacent cells, and only with two points in each cell.

25 Dominoes A domino is a gridded permutation in that avoids = =

26 Dominoes A domino is a gridded permutation in that avoids = = Important: / D (D = the set of dominoes)

27 Dominoes Theorem The number of n-point dominoes is A familiar sequence 2(3n + 3)!. gr(d) = 27/4. (n + 2)!(2n + 3)! Among other things, dominoes are equinumerous to West-2-stack-sortable permutations Rooted nonseparable planar maps Problem Find a bijection between dominoes and another combinatorial structure.

28 New upper bound for Av(1324) Ψ : Av n (1324) (, ) n D n

29 New upper bound for Av(1324) Ψ : Av n (1324) (, ) n D n

30 New upper bound for Av(1324) Ψ : Av n (1324) (, ) n D n Ψ is an injection. gr(av(1324)) 2 27/4 = 13.5

31 New lower bound for Av(1324) p elts q elts q elts p elts p elts q elts Begin with some dominoes and their symmetries.

32 New lower bound for Av(1324) p elts q elts q elts p elts p elts q elts Interleave with skew indecomposable components.

33 New lower bound for Av(1324) p elts q elts q 2 cpts q elts p elts p elts q elts q 2 cpts Interleave with skew indecomposable components.

34 New lower bound for Av(1324) p elts q elts q 2 cpts q elts p elts p 2 cpts p elts q elts q 2 cpts Interleave with skew indecomposable components.

35 New lower bound for Av(1324) p elts q elts q 2 cpts q elts p elts p 2 cpts p elts q elts q 2 cpts Analysis gives gr(av(1324))

36 New lower bound (2) Exploit finer structure of dominoes: gr(av(1324)) This is the root of a polynomial of degree 104, whose smallest coefficient has 86 digits.

37 Questions for the future Bijection between dominoes and something else Improvement on the (crude) upper bound Count trominoes?

38 Thanks!