Ascent sequences avoiding pairs of Lara Pudwell patterns - PowerPoint PPT Presentation

Ascent sequences avoiding pairs of patterns Ascent sequences avoiding pairs of Lara Pudwell patterns Introduction & History Pairs of Length 3 Patterns Unbalanced equivalences Lara Pudwell An Erd˝ os-Szekeres-like Theorem Other sequences faculty.valpo.edu/lpudwell Dyck paths Generating trees Onward... joint work with Andrew Baxter Permutation Patterns 2014 East Tennessee State University July 7, 2014

Ascent sequences Ascents avoiding pairs of patterns Lara Pudwell Introduction & History Definition Pairs of Length 3 An ascent in the string x 1 · · · x n is a position i such that Patterns Unbalanced equivalences x i < x i +1 . An Erd˝ os-Szekeres-like Theorem Other sequences Example: Dyck paths Generating trees Onward... 01024 01024 01024

Ascent sequences Ascents avoiding pairs of patterns Lara Pudwell Introduction & History Definition Pairs of Length 3 An ascent in the string x 1 · · · x n is a position i such that Patterns Unbalanced equivalences x i < x i +1 . An Erd˝ os-Szekeres-like Theorem Other sequences Example: Dyck paths Generating trees Onward... 01024 01024 01024 Definition asc ( x 1 · · · x n ) is the number of ascents of x 1 · · · x n . Example: asc (01024) = 3

Ascent sequences Ascent Sequences avoiding pairs of patterns Definition Lara Pudwell An ascent sequence is a string x 1 · · · x n of non-negative Introduction & History integers such that: Pairs of Length 3 ◮ x 1 = 0 Patterns Unbalanced equivalences ◮ x n ≤ 1 + asc ( x 1 · · · x n − 1 ) for n ≥ 2 An Erd˝ os-Szekeres-like Theorem Other sequences A n is the set of ascent sequences of length n Dyck paths Generating trees A 2 = { 00 , 01 } More examples: 01234, 01013 Onward... A 3 = { 000 , 001 , 010 , 011 , 012 } Non-example: 01024

Ascent sequences Ascent Sequences avoiding pairs of patterns Definition Lara Pudwell An ascent sequence is a string x 1 · · · x n of non-negative Introduction & History integers such that: Pairs of Length 3 ◮ x 1 = 0 Patterns Unbalanced equivalences ◮ x n ≤ 1 + asc ( x 1 · · · x n − 1 ) for n ≥ 2 An Erd˝ os-Szekeres-like Theorem Other sequences A n is the set of ascent sequences of length n Dyck paths Generating trees A 2 = { 00 , 01 } More examples: 01234, 01013 Onward... A 3 = { 000 , 001 , 010 , 011 , 012 } Non-example: 01024 Theorem (Bousquet-M´ elou, Claesson, Dukes, & Kitaev, 2010) |A n | is the n th Fishburn number (OEIS A022493). n |A n | x n = � � � (1 − (1 − x ) i ) n ≥ 0 n ≥ 0 i =1

Ascent sequences Patterns avoiding pairs of patterns Lara Pudwell Definition Introduction & The reduction of x = x 1 · · · x n , red ( x ), is the string obtained History by replacing the i th smallest digits of x with i − 1. Pairs of Length 3 Patterns Unbalanced equivalences Example: red (273772) = 021220 An Erd˝ os-Szekeres-like Theorem Other sequences Dyck paths Generating trees Onward...

Ascent sequences Patterns avoiding pairs of patterns Lara Pudwell Definition Introduction & The reduction of x = x 1 · · · x n , red ( x ), is the string obtained History by replacing the i th smallest digits of x with i − 1. Pairs of Length 3 Patterns Unbalanced equivalences Example: red (273772) = 021220 An Erd˝ os-Szekeres-like Theorem Other sequences Pattern containment/avoidance Dyck paths Generating trees Onward... a = a 1 · · · a n contains σ = σ 1 · · · σ m iff there exist 1 ≤ i 1 < i 2 < · · · < i m ≤ n such that red ( a i 1 a i 2 · · · a i m ) = σ . a B ( n ) = |{ a ∈ A n | a avoids B }| 001010345 contains 012, 000, 1102; avoids 210.

Ascent sequences Patterns avoiding pairs of patterns Lara Pudwell Definition Introduction & The reduction of x = x 1 · · · x n , red ( x ), is the string obtained History by replacing the i th smallest digits of x with i − 1. Pairs of Length 3 Patterns Unbalanced equivalences Example: red (273772) = 021220 An Erd˝ os-Szekeres-like Theorem Other sequences Pattern containment/avoidance Dyck paths Generating trees Onward... a = a 1 · · · a n contains σ = σ 1 · · · σ m iff there exist 1 ≤ i 1 < i 2 < · · · < i m ≤ n such that red ( a i 1 a i 2 · · · a i m ) = σ . a B ( n ) = |{ a ∈ A n | a avoids B }| 001010345 contains 012, 000, 1102; avoids 210. Goal Determine a B ( n ) for many of choices of B .

Ascent sequences Previous Work avoiding pairs of patterns ◮ Duncan & Steingr´ ımsson (2011) Lara Pudwell Introduction & Pattern σ { a σ ( n ) } n ≥ 1 OEIS History 001, 010 2 n − 1 Pairs of Length 3 A000079 Patterns 011, 012 Unbalanced equivalences 102 An Erd˝ os-Szekeres-like (3 n − 1 + 1) / 2 A007051 Theorem 0102, 0112 Other sequences Dyck paths 101, 021 Generating trees � 2 n 1 � A000108 Onward... 0101 n +1 n ◮ Mansour and Shattuck (2014) Callan, Mansour and Shattuck (2014) Pattern σ { a σ ( n ) } n ≥ 1 OEIS � n − 1 � n − 1 � C k 1012 A007317 k =0 k 1 − 4 x +3 x 2 0123 ogf: A080937 1 − 5 x +6 x 2 − x 3 8 pairs of length � 2 n 1 � A000108 4 patterns n +1 n

Ascent sequences Overview avoiding pairs of patterns Lara Pudwell Introduction & History ◮ 13 length 3 patterns Pairs of Length 3 Patterns 6 permutations, 000, 001, 010, 100, 011, 101, 110 � = 78 pairs Unbalanced equivalences ◮ � 13 An Erd˝ os-Szekeres-like Theorem 2 Other sequences ◮ at least 35 different sequences a σ,τ ( n ) Dyck paths Generating trees 16 sequences in OEIS Onward... ◮ 3 sequences from Duncan/Steingr´ ımsson ◮ 1 eventually zero ◮ 1 from pattern-avoiding set partitions ◮ 3 from pattern-avoiding permutations ◮ 1 sequence from Mansour/Shattuck (Duncan/Steingr´ ımsson conjecture)

Ascent sequences Unbalanced equivalences avoiding pairs of patterns Theorem Lara Pudwell a 010 , 021 ( n ) = a 010 ( n ) = a 10 ( n ) = 2 n − 1 Introduction & History Pairs of Length 3 Patterns Unbalanced equivalences An Erd˝ os-Szekeres-like Theorem Other sequences Dyck paths Generating trees Onward...

Ascent sequences Unbalanced equivalences avoiding pairs of patterns Theorem Lara Pudwell a 010 , 021 ( n ) = a 010 ( n ) = a 10 ( n ) = 2 n − 1 Introduction & History ◮ If σ contains 10, then a 010 ,σ = 2 n − 1 . Pairs of Length 3 Patterns Unbalanced equivalences An Erd˝ os-Szekeres-like Theorem Other sequences Dyck paths Generating trees Onward...

Ascent sequences Unbalanced equivalences avoiding pairs of patterns Theorem Lara Pudwell a 010 , 021 ( n ) = a 010 ( n ) = a 10 ( n ) = 2 n − 1 Introduction & History ◮ If σ contains 10, then a 010 ,σ = 2 n − 1 . Pairs of Length 3 Patterns Unbalanced equivalences Theorem An Erd˝ os-Szekeres-like Theorem Other sequences a 101 , 201 ( n ) = a 101 ( n ) = C n Dyck paths Generating trees Onward...

Ascent sequences Unbalanced equivalences avoiding pairs of patterns Theorem Lara Pudwell a 010 , 021 ( n ) = a 010 ( n ) = a 10 ( n ) = 2 n − 1 Introduction & History ◮ If σ contains 10, then a 010 ,σ = 2 n − 1 . Pairs of Length 3 Patterns Unbalanced equivalences Theorem An Erd˝ os-Szekeres-like Theorem Other sequences a 101 , 201 ( n ) = a 101 ( n ) = C n Dyck paths Generating trees ◮ 101-avoiders are restricted growth functions. Onward... ◮ If σ contains 201, then a 101 ,σ = C n .

Ascent sequences Unbalanced equivalences avoiding pairs of patterns Theorem Lara Pudwell a 010 , 021 ( n ) = a 010 ( n ) = a 10 ( n ) = 2 n − 1 Introduction & History ◮ If σ contains 10, then a 010 ,σ = 2 n − 1 . Pairs of Length 3 Patterns Unbalanced equivalences Theorem An Erd˝ os-Szekeres-like Theorem Other sequences a 101 , 201 ( n ) = a 101 ( n ) = C n Dyck paths Generating trees ◮ 101-avoiders are restricted growth functions. Onward... ◮ If σ contains 201, then a 101 ,σ = C n . Theorem a 101 , 210 ( n ) = 3 n − 1 +1 2

Ascent sequences Unbalanced equivalences avoiding pairs of patterns Theorem Lara Pudwell a 010 , 021 ( n ) = a 010 ( n ) = a 10 ( n ) = 2 n − 1 Introduction & History ◮ If σ contains 10, then a 010 ,σ = 2 n − 1 . Pairs of Length 3 Patterns Unbalanced equivalences Theorem An Erd˝ os-Szekeres-like Theorem Other sequences a 101 , 201 ( n ) = a 101 ( n ) = C n Dyck paths Generating trees ◮ 101-avoiders are restricted growth functions. Onward... ◮ If σ contains 201, then a 101 ,σ = C n . Theorem a 101 , 210 ( n ) = 3 n − 1 +1 2 ◮ Proof sketch: bijection with ternary strings with even number of 2s ımsson proof that a 102 ( n ) = 3 n − 1 +1 ◮ (Duncan/Steingr´ 2 uses bijection with same strings.)

Ascent sequences An Erd˝ os-Szekeres-like Theorem avoiding pairs of patterns Lara Pudwell Introduction & History Theorem Pairs of Length 3 Patterns  |A n | n ≤ 2 Unbalanced equivalences   An Erd˝ os-Szekeres-like  a 000 , 012 ( n ) = 3 n = 3 or n = 4 Theorem Other sequences  Dyck paths  0 n ≥ 5  Generating trees Onward... A 1 (000 , 012) = { 0 } A 2 (000 , 012) = { 00 , 01 } A 3 (000 , 012) = { 001 , 010 , 011 } A 4 (000 , 012) = { 0011 , 0101 , 0110 }