Beautiful Bijections for Permutation Lara Pudwell Patterns - PowerPoint PPT Presentation

Beautiful Bijections for Permutation Patterns Beautiful Bijections for Permutation Lara Pudwell Patterns Pattern- Avoiding Permutations Strategy Lara Pudwell Beautiful Valparaiso University Bijections Compositions Dyck Paths Others? ...When All Else Fails Joint Mathematics Meetings Summary MAA Invited Paper Session on Clever Counting or Beautiful Bijection January 5, 2012

Outline Beautiful Bijections for Permutation Patterns Pattern-Avoiding Permutations 1 Lara Pudwell Pattern- Strategy 2 Avoiding Permutations Strategy Beautiful Bijections 3 Beautiful Compositions Bijections Compositions Dyck Paths Dyck Paths Others? Others? ...When All Else Fails ...When All Else Fails Summary Summary 4

Permutations Beautiful Bijections for Permutation Patterns Lara Pudwell Pattern- Avoiding A permutation of length n is an ordered list of the Permutations numbers { 1 , 2 , . . . , n } . Strategy There are n ! permutations of length n . Beautiful Bijections Example: the 6 permutations of { 1 , 2 , 3 } are 123, 132, Compositions Dyck Paths Others? 213, 231, 312, 321. ...When All Else Fails Summary

Graphs of Permutations Beautiful Bijections for Permutation Consider the permutation π = π 1 π 2 · · · π n as a function from Patterns { 1 , 2 , . . . , n } to { 1 , 2 , . . . , n } . Lara Pudwell Pattern- Avoiding Permutations Strategy Example, π = 51342 Beautiful Bijections Compositions π 1 = 5 Dyck Paths Others? π 2 = 1 ...When All Else Fails π 3 = 3 Summary π 4 = 4 π 5 = 2

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions Dyck Paths Others? ...When All Else Fails Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? ...When All Else Fails Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, ...When All Else Fails Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, 231, Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, 231, 312, Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, 231, 312, 321, Summary

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, 231, 312, 321, Summary 1342,

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, 231, 312, 321, Summary 1342, 4123,

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, 231, 312, 321, Summary 1342, 4123, 4132,

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, 231, 312, 321, Summary 1342, 4123, 4132, 4231,

Permutations Inside Permutations Beautiful Bijections for Permutation Containment/Avoidance Patterns π = π 1 · · · π n contains ρ = ρ 1 · · · ρ k as a pattern if there exist Lara Pudwell 1 ≤ i 1 < i 2 < · · · < i k ≤ n such that π i a < π i b if and only if Pattern- Avoiding ρ a < ρ b . Otherwise π avoids the pattern ρ . Permutations Strategy Picture definition: Beautiful π = 51342 contains Bijections Compositions 1, Dyck Paths Others? 12, 21, ...When All Else Fails 123, 132, 231, 312, 321, Summary 1342, 4123, 4132, 4231, 51342.