Pattern-avoiding ascent sequences An interesting equivalence - PowerPoint PPT Presentation

Pattern-avoiding ascent sequences Lara Pudwell Introduction & History Pattern-avoiding ascent sequences An interesting equivalence Generating Tree Counting Nodes Summary Lara Pudwell faculty.valpo.edu/lpudwell 2015 Joint Mathematics Meetings AMS Special Session on Enumerative Combinatorics January 11, 2015

Pattern-avoiding Ascent Sequences ascent sequences Lara Pudwell Definition Introduction & An ascent sequence is a string x 1 · · · x n of non-negative History integers such that: An interesting equivalence ◮ x 1 = 0 Generating Tree ◮ x n ≤ 1 + asc ( x 1 · · · x n − 1 ) for n ≥ 2 Counting Nodes Summary A n is the set of ascent sequences of length n A 2 = { 00 , 01 } More examples: 01234, 01013 A 3 = { 000 , 001 , 010 , 011 , 012 } Non-example: 01024

Pattern-avoiding Ascent Sequences ascent sequences Lara Pudwell Definition Introduction & An ascent sequence is a string x 1 · · · x n of non-negative History integers such that: An interesting equivalence ◮ x 1 = 0 Generating Tree ◮ x n ≤ 1 + asc ( x 1 · · · x n − 1 ) for n ≥ 2 Counting Nodes Summary A n is the set of ascent sequences of length n A 2 = { 00 , 01 } More examples: 01234, 01013 A 3 = { 000 , 001 , 010 , 011 , 012 } Non-example: 01024 Theorem |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

Pattern-avoiding Patterns ascent sequences Lara Pudwell Definition Introduction & History The reduction of x = x 1 · · · x n , red ( x ), is the string obtained An interesting by replacing the i th smallest digits of x with i − 1. equivalence Generating Tree Example: red (273772) = 021220 Counting Nodes Summary

Pattern-avoiding Patterns ascent sequences Lara Pudwell Definition Introduction & History The reduction of x = x 1 · · · x n , red ( x ), is the string obtained An interesting by replacing the i th smallest digits of x with i − 1. equivalence Generating Tree Example: red (273772) = 021220 Counting Nodes Pattern containment/avoidance Summary 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.

Pattern-avoiding Patterns ascent sequences Lara Pudwell Definition Introduction & History The reduction of x = x 1 · · · x n , red ( x ), is the string obtained An interesting by replacing the i th smallest digits of x with i − 1. equivalence Generating Tree Example: red (273772) = 021220 Counting Nodes Pattern containment/avoidance Summary 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.

Pattern-avoiding Patterns ascent sequences Lara Pudwell Definition Introduction & History The reduction of x = x 1 · · · x n , red ( x ), is the string obtained An interesting by replacing the i th smallest digits of x with i − 1. equivalence Generating Tree Example: red (273772) = 021220 Counting Nodes Pattern containment/avoidance Summary 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.

Pattern-avoiding Patterns ascent sequences Lara Pudwell Definition Introduction & History The reduction of x = x 1 · · · x n , red ( x ), is the string obtained An interesting by replacing the i th smallest digits of x with i − 1. equivalence Generating Tree Example: red (273772) = 021220 Counting Nodes Pattern containment/avoidance Summary 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 .

Pattern-avoiding Previous Work ascent sequences Lara Pudwell ◮ Duncan & Steingr´ ımsson (2011) Introduction & History Pattern b a b ( n ) OEIS An interesting 001, 010 2 n − 1 equivalence A000079 011, 012 Generating Tree 102 (3 n − 1 + 1) / 2 Counting Nodes A007051 0102, 0112 Summary 101, 021 � 2 n 1 � A000108 0101 n +1 n ◮ Mansour and Shattuck (2014) Callan, Mansour and Shattuck (2014) Patterns B a B ( n ) OEIS � n − 1 � n − 1 � C ℓ 1012 A007317 ℓ =0 ℓ 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

Pattern-avoiding Other sequences (Baxter & P.) ascent sequences Lara Pudwell Introduction & Patterns B OEIS a B ( n ) History 000,011 A000027 n An interesting � + 1 equivalence � n 011,100 A000124 � + n 2 Generating Tree � n 001,210 A000125 Counting Nodes 3 000,101 A001006 M n Summary 000,001 A000045 F n +1 001,100 A000071 F n +2 − 1 101,110 A001519 F 2 n − 1 100,101 A025242 (Generalized Catalan) 021,102 A116702 |S n (123 , 3241) | 102,120 A005183 |S n (132 , 4312) | 101,120 A116703 |S n (231 , 4123) | � n − 1 � n − 1 � C ℓ 201,210 A007317 ℓ =0 ℓ

Pattern-avoiding Binomial convolutions ascent sequences Lara Pudwell Introduction & History An interesting Theorem (P.) equivalence a 201 , 210 ( n ) = � n − 1 � n − 1 � C ℓ . Generating Tree ℓ =0 ℓ Counting Nodes Summary Theorem (Mansour & Shattuck) � n − 1 a 1012 ( n ) = � n − 1 � C ℓ . ℓ =0 ℓ Conjecture (Duncan & Steingr´ ımsson) a 0021 ( n ) = � n − 1 � n − 1 � C ℓ . ℓ =0 ℓ

Pattern-avoiding Binomial convolutions ascent sequences Lara Pudwell Introduction & History An interesting Theorem (P.) equivalence a 201 , 210 ( n ) = � n − 1 � n − 1 � C ℓ . Generating Tree ℓ =0 ℓ Counting Nodes Summary Theorem (Mansour & Shattuck) � n − 1 a 1012 ( n ) = � n − 1 � C ℓ . ℓ =0 ℓ / Conjecture (Duncan & Steingr´ / / / / / / / / / / / / ımsson) / Theorem (P.) a 0021 ( n ) = � n − 1 � n − 1 � C ℓ . ℓ =0 ℓ

Pattern-avoiding Wilf Equivalence ascent sequences Lara Pudwell Definition Introduction & Patterns σ and ρ are Wilf-equivalent if a σ ( n ) = a ρ ( n ) for History n ≥ 1. In this case, write: σ ∼ ρ . An interesting equivalence Generating Tree Example: 00 ∼ 01. Counting Nodes a 00 ( n ) = 1 (the strictly increasing sequence) Summary a 01 ( n ) = 1 (the all zeros sequence).

Pattern-avoiding Wilf Equivalence ascent sequences Lara Pudwell Definition Introduction & Patterns σ and ρ are Wilf-equivalent if a σ ( n ) = a ρ ( n ) for History n ≥ 1. In this case, write: σ ∼ ρ . An interesting equivalence Generating Tree Example: 00 ∼ 01. Counting Nodes a 00 ( n ) = 1 (the strictly increasing sequence) Summary a 01 ( n ) = 1 (the all zeros sequence). Known from Duncan/Steingr´ ımsson: All possible Wilf equivalences of length at most 4 are: 00 ∼ 01 10 ∼ 001 ∼ 010 ∼ 011 ∼ 012 102 ∼ 0102 ∼ 0112 101 ∼ 021 ∼ 0101 ∼ 0012 0021 ∼ 1012 (Duncan & Steingr´ ımsson / Mansour & Shattuck / P.)

Pattern-avoiding Binomial convolutions ascent sequences Lara Pudwell Introduction & Theorem (P.) History An interesting a 201 , 210 ( n ) = � n − 1 � n − 1 � C ℓ . equivalence ℓ =0 ℓ Generating Tree Counting Nodes Theorem (Mansour & Shattuck) Summary a 1012 ( n ) = � n − 1 � n − 1 � C ℓ . ℓ =0 ℓ / Conjecture (Duncan & Steingr´ / / / / / / / / / / / / ımsson) / Theorem (P.) a 0021 ( n ) = � n − 1 � n − 1 � C ℓ . ℓ =0 ℓ Proof scribble: generating tree → recurrence → system of functional equations → experimental solution → plug in for catalytic variables

Pattern-avoiding Generating Tree ascent sequences Lara Pudwell Introduction & History 0 An interesting equivalence 00 01 Generating Tree Counting Nodes 000 001 010 011 012 Summary 0000 0001 0010 0011 0012 0100 0101 0102 0110 0111 0112 0120 0121 0122 0123

Pattern-avoiding Generating Tree ascent sequences Lara Pudwell Introduction & History 0 An interesting equivalence 00 01 Generating Tree Counting Nodes 000 001 010 011 012 Summary 0000 0001 0010 0011 0012 0100 0101 0102 0110 0111 0112 0120 0121 0122 0123 Idea: Replace a with an ordered triple of statistics on a.

Pattern-avoiding Generating Tree ascent sequences Lara Pudwell 0 (0,2,0) Introduction & History An interesting 00 (0,1,1) 01 (1,3,0) equivalence Generating Tree 000 (0,1,1) 001 (1,1,2) 010 (0,1,2) 011 (1,2,1) 012 (2,4,0) Counting Nodes Summary

Pattern-avoiding Generating Tree ascent sequences Lara Pudwell 0 (0,2,0) Introduction & History An interesting 00 (0,1,1) 01 (1,3,0) equivalence Generating Tree 000 (0,1,1) 001 (1,1,2) 010 (0,1,2) 011 (1,2,1) 012 (2,4,0) Counting Nodes Summary ◮ root: (0 , 2 , 0) ◮ rules: ( j − 2 , j , 0) → ( j − 1 , j + 1 , 0) , ( i , i + 1 , j − 1 − i ) j − 2 i =0