Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns in restricted permutations Lara Pudwell faculty.valpo.edu/lpudwell Rutgers Experimental Math Seminar March 5, 2020 Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Permutations Definition A permutation π of length n is an ordered list of the numbers 1 , 2 , . . . , n . S n is the set of all permutations of length n . π is often visualized by plotting the points ( i , π i ) in the Cartesian plane. 123 132 213 231 312 321 Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Permutation Constructions I n = 12 · · · n J n = n · · · 21 α β α β α ⊕ β α ⊖ β Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Permutation Patterns Definition π ∈ S n contains ρ ∈ S m as a pattern if there exist 1 ≤ i 1 < i 2 < · · · < i m ≤ n such that π i a < π i b iff ρ a < ρ b . If π doesn’t contain ρ , we say π avoids ρ and we write π ∈ S n ( ρ ). Example: π = 2314 ∈ S 4 (321). Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Pattern Avoidance Let s n ( ρ ) = |S n ( ρ ) | . Theorem For n ≥ 0, s n (12) = s n (21) = 1. S 8 (12) S 8 (21) Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Pattern Avoidance Symmetries s n (123) = s n (321) s n (132) = s n (213) = s n (231) = s n (312) Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Pattern Avoidance Theorem If ρ ∈ S 3 , then s n ( ρ ) = ( 2 n n ) n +1 . Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Pattern Avoidance Theorem If ρ ∈ S 3 , then s n ( ρ ) = ( 2 n n ) n +1 . A member of S 8 (123) A member of S 8 (132) Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Pattern Avoidance Theorem If ρ ∈ S 3 , then s n ( ρ ) = ( 2 n n ) n +1 . A member of S 8 (123) A member of S 8 (132) Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Patterns Definition π ∈ S n contains ρ ∈ S m as a pattern if there exist 1 ≤ i 1 < i 2 < · · · < i m ≤ n such that π i a < π i b iff ρ a < ρ b . π = 2314 contains... 1 copy of 123 Example: 2 copies of 213 1 copy of 231 Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Pattern Packing ν ( ρ, π ) is the number of occurrences of ρ in π . Given n and ρ , consider max π ∈S n ν ( ρ, π ) Example: n = 3 and ρ = 12 ν (12 , 123) = 3 ν (12 , 132) = 2 ν (12 , 213) = 2 ν (12 , 231) = 1 ν (12 , 312) = 1 ν (12 , 321) = 0 max π ∈S 3 ν (12 , π ) = 3 Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Pattern Packing ν ( ρ, π ) is the number of occurrences of ρ in π . Given n and ρ , consider max π ∈S n ν ( ρ, π ) Example: n = 3 and ρ = 12 ν (12 , 123) = 3 ν (12 , 132) = 2 ν (12 , 213) = 2 ν (12 , 231) = 1 ν (12 , 312) = 1 ν (12 , 321) = 0 max π ∈S 3 ν (12 , π ) = 3 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 Pattern Packing 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 Pattern Packing 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 Pattern Packing 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 Pattern Packing 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 . Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 Known: Since 132 is layered, then max π ∈S n ν (132 , π ) is achieved by a layered π . α π = α ⊕ J i � i � ν (132 , π ) = ν (132 , α ) + ( n − i ) · 2 Packing patterns in restricted permutations Lara Pudwell
Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing 132 Known: Since 132 is layered, then max π ∈S n ν (132 , π ) is achieved by a layered π . α π = α ⊕ J i � i � ν (132 , π ) = ν (132 , α ) + ( n − i ) · 2 √ � � ν (132 ,π ) 3 3 is maximized when i = 2 − n ≈ 0 . 634 n ( n 3 ) 2 √ Implies 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 Pattern Packing 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
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