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Interval Avoidance in the Symmetric Group Isaiah Lankham UC Davis Fourth International Conference on Permutation Patterns Reykjav k University June 16, 2006 (joint work with Alexander Woo, UC Davis) Permutation Embeddings Interval

  1. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) } 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  2. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) } 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  3. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  4. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  5. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  6. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  7. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  8. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  9. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  10. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  11. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 ≻ 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  12. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 ≻ 3124 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  13. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 ≻ 3124 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  14. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 ≻ 3124 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  15. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 ≻ 3124 ≻ 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  16. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 ≻ 3124 ≻ 1324 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  17. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings The (Strong) Bruhat Order Definition (Inversion in σ ∈ S n ) An inversion is an embedding of the pattern 21 ∈ S 2 into σ . Definition (Length Function on S n ) ℓ ( σ ) = # { indices ( i 1 , i 2 ) | ( i 1 , i 2 ) is an inversion in σ } Example: ℓ ( 426153 ) = # { ( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) } Definition (Bruhat order on S n ) We say τ > σ in Bruhat order if τ can be transformed into σ by successively “undoing” inversions. Example. 3412 > 1324: 3412 ≻ 3142 ≻ 3124 ≻ 1324 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  18. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Bruhat Covering Relation Definition (Bruhat covering relation on S n ) We say σ ≺ τ in Bruhat order if σ = τ t for some transposition t ℓ ( σ ) = ℓ ( τ ) − 1 Equivalently: use transposition t to “undo” an embedding of 21 at positions i < k in τ such that ∄ index j for which i < j < k and τ i > τ j > τ k : ( i , τ i ) • • ( k , τ k ) 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  19. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Symmetry Properties of Bruhat Order Lemma (Bruhat order symmetries for σ, τ ∈ S n ) ⇒ σ − 1 < τ − 1 (Inverses) σ < τ = ⇒ τ r < σ r (Reverse) σ < τ = ⇒ τ c < σ c (Complement) σ < τ = ⇒ σ rc < τ rc (Reverse Complement) σ < τ = Examples: Starting with 1324 < 2341, 1324 − 1 = 1324 < 4123 = 2341 − 1 . 2341 r = 1432 < 4231 = 1324 r . 2341 c = 3214 < 4231 = 1324 c . 1324 rc = 1324 < 4123 = 2341 rc . 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  20. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Intervals in Bruhat Order Definition (Intervals in Bruhat order) Given σ, τ ∈ S n , [ σ, τ ] = { ω ∈ S n | σ ≤ ω ≤ τ } . 4321 Example. [ 1324 , 2341 ] : 4312 4231 3421 4132 4213 3412 2431 3241 4123 1432 2413 3142 2341 2341 3214 1423 1342 1342 2143 3124 2314 2314 1243 1324 1324 2134 1234 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  21. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  22. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. Example. [ 123 , 231 ] embeds into [ 1324 , 2341 ] : 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  23. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. Example. [ 123 , 231 ] embeds into [ 1324 , 2341 ] : 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  24. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. Example. [ 123 , 231 ] embeds into [ 1324 , 2341 ] : 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  25. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. Example. [ 123 , 231 ] embeds into [ 1324 , 2341 ] : 321 231 312 132 213 123 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  26. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. Example. [ 123 , 231 ] embeds into [ 1324 , 2341 ] : 321 231 312 132 213 123 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  27. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. Example. [ 123 , 231 ] embeds into [ 1324 , 2341 ] : 321 231 312 132 213 123 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  28. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. 4321 Example. [ 123 , 231 ] embeds into [ 1324 , 2341 ] : 4312 4231 3421 321 4132 4213 3412 2431 3241 231 312 4123 1432 2413 3142 2341 3214 132 213 1423 1342 2143 3124 2314 123 1243 1324 2134 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group 1234

  29. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Embeddings Intervals into Larger Intervals Definition (Interval Embedding) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, we say that [ π, ρ ] embeds into [ σ, τ ] if � π embeds into σ using same embedding ( i 1 , i 2 , . . . , i m ) ρ embeds into τ the intervals [ π, ρ ] and [ σ, τ ] are order-isomorphic. 4321 Example. [ 123 , 231 ] embeds into [ 1324 , 2341 ] : 4312 4231 3421 321 4132 4213 3412 2431 3241 231 312 4123 1432 2413 3142 2341 3214 132 213 1423 1342 2143 3124 2314 123 1243 1324 2134 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group 1234

  30. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings An Equivalent Definition of Interval Embeddings Lemma (Interval Embedding Characterization) Given π ≤ ρ ∈ S m and σ ≤ τ ∈ S n with m ≤ n, the interval [ π, ρ ] embeds into [ σ, τ ] iff σ i = τ i for i / ∈ { i 1 , i 2 , . . . , i m } (a common embedding) ℓ ( τ ) − ℓ ( σ ) = ℓ ( ρ ) − ℓ ( π ) Corollary Given any three of the permutations π , ρ , σ , and τ , the fourth is uniquely determine. Definition (Avoidance Set for an Interval) S n ([ π, ρ ]) = { τ ∈ S n | ∀ σ ∈ S n , [ π, ρ ] doesn’t embed into [ σ, τ ] } . 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  31. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Interval Embeddings & Avoidance Examples: If π = ρ , then S n ([ π, ρ ]) = S n ( ρ ) since the intervals [ π, ρ ] = { ρ } and [ σ, τ ] = { τ } are trivially order-isomorphic. 43512 “contains” [ 1324 , 3412 ] because the interval [ 1324 , 3412 ] embeds into [ 4 1325 , 4 3512 ] : ℓ ( 43512 ) − ℓ ( 41325 ) = 7 − 4 = 4 − 1 = ℓ ( 3412 ) − ℓ ( 1324 ) 426153 ∈ S n ([ 1324 , 3412 ]) because the interval [ 1324 , 3412 ] cannot embed into [ 1 2 43 5 6 , 4 2 61 5 3 ] : ℓ ( 426153 ) − ℓ ( 124356 ) = 8 − 1 > 4 − 1 = ℓ ( 3412 ) − ℓ ( 1324 ) “Universal” in characterizing singularities of Schubert varieties (A. Woo and A. Yong). 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  32. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. Connect point horizontally. Connect point vertically toward π . Shade between closest vertical “left-side down” and “right-side up” pairs of lines. 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  33. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. Connect point horizontally. Connect point vertically toward π . Shade between closest vertical “left-side down” and “right-side up” pairs of lines. 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  34. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. Connect point horizontally. Connect point vertically toward π . Shade between closest vertical “left-side down” and “right-side up” pairs of lines. 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  35. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. 4 Connect point horizontally. 3 Connect point vertically toward π . 2 Shade between closest vertical “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  36. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. 4 Connect point horizontally. 3 Connect point vertically toward π . 2 Shade between closest vertical “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  37. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. ◦ 4 Connect point horizontally. ◦ 3 Connect point vertically toward π . ◦ 2 Shade between closest vertical ◦ “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  38. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. ◦ 4 Connect point horizontally. ◦ 3 Connect point vertically toward π . ◦ 2 Shade between closest vertical ◦ “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  39. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. ◦ 4 Connect point horizontally. ◦ 3 Connect point vertically toward π . ◦ 2 Shade between closest vertical ◦ “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  40. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  41. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  42. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  43. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  44. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  45. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  46. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  47. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  48. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  49. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  50. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  51. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  52. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  53. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  54. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  55. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings A Geometric Form for Interval Pattern Containment For [ 2143 , 4231 ] : Algorithm (Forbidden Region for π ≤ ρ ) Graph π as circles, ρ as dots. • ◦ 4 Connect point horizontally. • ◦ 3 Connect point vertically toward π . ◦ • 2 Shade between closest vertical ◦ • “left-side down” and “right-side up” 1 pairs of lines. 0 0 1 2 3 4 Lemma Then a permutation τ ∈ S n “contains” [ π, ρ ] iff the forbidden region constructed above contains no “non-embedding” points. 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  56. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Forbidden Regions Examples: 43512 “contains” [ 1324 , 3412 ] because the Forbidden Region contains no “non-embedding” points. 426153 ∈ S n ([ 1324 , 3412 ]) because the Forbidden Region contains “non-embedding” points. • • ◦ 6 • • ◦ 5 • 5 • 4 • • ◦ 4 • • ◦ 3 ◦ • • 3 ◦ • • 2 • 2 ◦ • • 1 ◦ • • 1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 6 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  57. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  58. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : 4 3 2 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  59. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : 4 3 2 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  60. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : ◦ 4 ◦ 3 ◦ 2 ◦ 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  61. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : ◦ 4 ◦ 3 ◦ 2 ◦ 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  62. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : ◦ 4 ◦ 3 ◦ 2 ◦ 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  63. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  64. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  65. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  66. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  67. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  68. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  69. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  70. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  71. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  72. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  73. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  74. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ 4 ◦ • 3 • ◦ 2 ◦ • 1 0 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  75. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ 4 4 ◦ • 3 3 • ◦ 2 2 ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  76. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ 4 4 ◦ • 3 3 • ◦ 2 2 ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  77. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ ◦ 4 4 ◦ • ◦ 3 3 • ◦ ◦ 2 2 ◦ • ◦ 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  78. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ ◦ 4 4 ◦ • ◦ 3 3 • ◦ ◦ 2 2 ◦ • ◦ 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  79. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ ◦ 4 4 ◦ • ◦ 3 3 • ◦ ◦ 2 2 ◦ • ◦ 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  80. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ • ◦ 4 4 ◦ • ◦ • 3 3 • ◦ • ◦ 2 2 ◦ • ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  81. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ • ◦ 4 4 ◦ • ◦ • 3 3 • ◦ • ◦ 2 2 ◦ • ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  82. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ • ◦ 4 4 ◦ • ◦ • 3 3 • ◦ • ◦ 2 2 ◦ • ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  83. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ • ◦ 4 4 ◦ • ◦ • 3 3 • ◦ • ◦ 2 2 ◦ • ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  84. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ • ◦ 4 4 ◦ • ◦ • 3 3 • ◦ • ◦ 2 2 ◦ • ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  85. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ • ◦ 4 4 ◦ • ◦ • 3 3 • ◦ • ◦ 2 2 ◦ • ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

  86. Permutation Embeddings Interval Avoidance Overview of Bruhat Order Classification for Length Three Patterns Intervals & Embeddings Summary & Further Directions Geometric Interval Embeddings Examples of Strange Forbidden Regions For [ 1324 , 4231 ] : For [ 3412 , 4321 ] : • ◦ • ◦ 4 4 ◦ • ◦ • 3 3 • ◦ • ◦ 2 2 ◦ • ◦ • 1 1 0 0 0 1 2 3 4 0 1 2 3 4 4 th International Conference on Permutation Patterns Interval Avoidance in the Symmetric Group

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