Packing patterns in restricted permutations Lara Pudwell - PowerPoint PPT Presentation

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns in restricted permutations Lara Pudwell faculty.valpo.edu/lpudwell Special Session on Patterns in Permutations AMS Fall Southeastern Sectional Meeting University of Florida November 3, 2019 Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Notation ν ( ρ, π ) is the number of occurrences of ρ in π . max π ∈S n ν ( ρ, π ) d ( ρ ) = lim (packing density) � n � n →∞ | ρ | Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Notation ν ( ρ, π ) is the number of occurrences of ρ in π . max π ∈S n ν ( ρ, π ) d ( ρ ) = lim (packing density) � n � n →∞ | ρ | Known: d (12 · · · m ) = 1 (Pack 12 · · · m into 12 · · · n .) Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Notation ν ( ρ, π ) is the number of occurrences of ρ in π . max π ∈S n ν ( ρ, π ) d ( ρ ) = lim (packing density) � n � n →∞ | ρ | Known: d (12 · · · m ) = 1 (Pack 12 · · · m into 12 · · · n .) For all ρ ∈ S m , d ( ρ ) exists. Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Notation ν ( ρ, π ) is the number of occurrences of ρ in π . max π ∈S n ν ( ρ, π ) d ( ρ ) = lim (packing density) � n � n →∞ | ρ | Known: d (12 · · · m ) = 1 (Pack 12 · · · m into 12 · · · n .) For all ρ ∈ S m , d ( ρ ) exists. If ρ is layered, then max π ∈S n ν ( ρ, π ) is achieved by a layered π . √ d (132) = 2 3 − 3 ≈ 0 . 464 Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Notation ν ( ρ, π ) is the number of occurrences of ρ in π . Previous work: max π ∈S n ν ( ρ, π ) d ( ρ ) = lim � n � n →∞ | ρ | Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Notation ν ( ρ, π ) is the number of occurrences of ρ in π . Previous work: max π ∈S n ν ( ρ, π ) d ( ρ ) = lim � n � n →∞ | ρ | In this talk: max π ∈S n ( σ ) ν ( ρ, π ) max π ∈ A n ν ( ρ, π ) d σ ( ρ ) = lim d A ( ρ ) = lim � n � n � � n →∞ n →∞ | ρ | | ρ | A n is the set of alternating permutations , i.e. those that avoid consecutive 123 patterns and consecutive 321 patterns. Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns of length 3 max π ∈S n ( σ ) ν ( ρ, π ) Recall: d σ ( ρ ) = lim � n � n →∞ | ρ | ρ \ σ 123 132 213 231 312 321 - 123 1 √ 132 2 3 − 3 Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns of length 3 max π ∈S n ( σ ) ν ( ρ, π ) Recall: d σ ( ρ ) = lim � n � n →∞ | ρ | ρ \ σ 123 132 213 231 312 321 - 123 0 1 √ 132 0 2 3 − 3 Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns of length 3 max π ∈S n ( σ ) ν ( ρ, π ) Recall: d σ ( ρ ) = lim � n � n →∞ | ρ | ρ \ σ 123 132 213 231 312 321 - 123 0 1 1 1 1 1 1 √ 132 0 2 3 − 3 I n = 12 · · · n avoids σ ∈ S 3 \ { 123 } . Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns of length 3 max π ∈S n ( σ ) ν ( ρ, π ) Recall: d σ ( ρ ) = lim � n � n →∞ | ρ | ρ \ σ 123 132 213 231 312 321 - 123 0 1 1 1 1 1 1 √ √ √ 132 0 2 3 − 3 2 3 − 3 2 3 − 3 I n = 12 · · · n avoids σ ∈ S 3 \ { 123 } . Layered permutations avoid 231 and 312. Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns of length 3 max π ∈S n ( σ ) ν ( ρ, π ) Recall: d σ ( ρ ) = lim � n � n →∞ | ρ | ρ \ σ 123 132 213 231 312 321 - 123 0 1 1 1 1 1 1 √ √ √ 132 ? 0 ? 2 3 − 3 2 3 − 3 ? 2 3 − 3 I n = 12 · · · n avoids σ ∈ S 3 \ { 123 } . Layered permutations avoid 231 and 312. New: d 123 (132), d 213 (132), and d 321 (132) Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 ...and avoiding 123 Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 ...and avoiding 123 Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 ...and avoiding 123 � n − i � J i ⊕ J n − i has i copies of 2 132. ( J n = n · · · 21) Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 ...and avoiding 123 � n − i � J i ⊕ J n − i has i copies of 2 132. ( J n = n · · · 21) Maximized when i = ⌊ n 3 ⌋ . Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 ...and avoiding 123 � n − i � J i ⊕ J n − i has i copies of 2 132. ( J n = n · · · 21) Maximized when i = ⌊ n 3 ⌋ . Implies d 123 (132) = 4 9. Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 ...and avoiding 123 ...and avoiding 213 � n − i � J i ⊕ J n − i has i copies of 2 132. ( J n = n · · · 21) Maximized when i = ⌊ n 3 ⌋ . Implies d 123 (132) = 4 9. Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 ...and avoiding 123 ...and avoiding 213 � n − i � n − i � � J i ⊕ J n − i has i copies of I i ⊕ J n − i has i copies of 2 2 132. ( J n = n · · · 21) 132. Maximized when i = ⌊ n Maximized when i = ⌊ n 3 ⌋ . 3 ⌋ . Implies d 123 (132) = 4 Implies d 213 (132) = 4 9. 9. Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 and Avoiding 321 I a ⊕ ( I b ⊖ I c ) has a · b · c copies of 132. Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 and Avoiding 321 I a ⊕ ( I b ⊖ I c ) has a · b · c copies of 132. Replace initial I a with a 132-optimizer of length a to get more copies. Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 and Avoiding 321 I a ⊕ ( I b ⊖ I c ) has a · b · c copies of 132. Replace initial I a with a 132-optimizer of length a to get more copies. � √ √ � � � 2 − 1 3 3 3 Optimized when a = n , b = c = 4 − n . 2 4 √ 3 − 3 Implies d 321 (132) = 2 . Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Recap: max π ∈S n ( σ ) ν ( ρ, π ) d σ ( ρ ) = lim � n � n →∞ | ρ | ρ \ σ 123 132 213 231 312 321 - 123 0 1 1 1 1 1 1 √ √ √ √ 4 4 3 − 3 132 0 2 3 − 3 2 3 − 3 2 3 − 3 9 9 2 Packing patterns in restricted permutations Lara Pudwell

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Recap: max π ∈S n ( σ ) ν ( ρ, π ) d σ ( ρ ) = lim � n � n →∞ | ρ | ρ \ σ 123 132 213 231 312 321 - 123 0 1 1 1 1 1 1 √ √ √ √ 4 4 3 − 3 132 0 2 3 − 3 2 3 − 3 2 3 − 3 9 9 2 Or approximately... ρ \ σ 123 132 213 231 312 321 - 123 0 1 1 1 1 1 1 132 0 . 444 0 0 . 444 0 . 464 0 . 464 0 . 232 0 . 464 Packing patterns in restricted permutations Lara Pudwell