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Sign Variation and Descents Sign Variation and Descents Aram Dermenjian Joint with: Nantel Bergeron and John Machacek York University 24 September 2020 A. Dermenjian York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020 Sign

  1. Sign Variation and Descents Projective sign vectors Let PV n be the set of sign vectors in RP n − 1 . In other words, for ω ∈ V n , then ω ∼ ω ′ iff ω = ω ′ or ω = − ω ′ and � V n \ { 0 } n � / ∼ ∼ PV n = = { ω ∈ V n : First non-zero entry of ω is + } . Example V 1 = { (+) , ( 0 ) , ( − ) } V 2 = { (+ , +) , (+ , 0 ) , (+ , − ) , ↓ ( 0 , +) , ( 0 , 0 ) , ( 0 , − ) , PV 1 = { (+) } ( − , +) , ( − , 0 ) , ( − , − ) } ↓ PV 2 = { (+ , +) , (+ , 0 ) , (+ , − ) , ( 0 , +) } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  2. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , − ) (+ , 0 ) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  3. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , +) (+ , − ) (+ , 0 ) (+ , 0 ) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  4. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , +) (+ , − ) (+ , 0 ) (+ , 0 ) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  5. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , +) (+ , − ) (+ , 0 ) ( 0 , +) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  6. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , +) (+ , − ) (+ , 0 ) ( 0 , +) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  7. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , +) (+ , − ) (+ , − ) (+ , 0 ) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  8. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , − ) (+ , − ) (+ , 0 ) (+ , 0 ) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  9. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , − ) (+ , − ) (+ , 0 ) (+ , 0 ) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  10. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , − ) (+ , 0 ) (+ , 0 ) ( 0 , +) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  11. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , − ) (+ , − ) (+ , 0 ) ( 0 , +) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  12. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , − ) (+ , − ) (+ , 0 ) ( 0 , +) ( 0 , +) ∼ ( 0 , − ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  13. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , − ) (+ , − ) (+ , 0 ) ( 0 , +) ( 0 , +) ∼ ( 0 , − ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  14. Sign Variation and Descents Ordering projective sign vectors Let P n denote the poset ( PV n , < ) where for ω, ω ′ ∈ PV n : ω ′ < ω ⇐ ⇒ ± ω ′ ⊆ ω in other words, if either ω ′ or − ω ′ is obtained from ω by replacing some components with 0. Example (+ , +) (+ , − ) (+ , 0 ) ( 0 , +) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  15. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . Example (+ , +) (+ , − ) (+ , 0 ) ( 0 , +) P 2 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  16. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . Example (+ , +) (+ , − ) (+ , 0 ) ( 0 , +) { ∅ } P 2 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  17. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . Example (+ , +) (+ , +) (+ , − ) (+ , − ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } (+ , 0 ) (+ , 0 ) ( 0 , +) ( 0 , +) { ∅ } P 2 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  18. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . Example (+ , +) (+ , +) (+ , − ) (+ , − ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } (+ , 0 ) (+ , 0 ) ( 0 , +) ( 0 , +) { ∅ } P 2 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  19. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . � � Example (+ , +) | (+ , 0 ) (+ , +) (+ , +) (+ , − ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } (+ , 0 ) (+ , 0 ) ( 0 , +) { ∅ } P 2 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  20. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . � � � � Example (+ , +) (+ , − ) | | (+ , 0 ) (+ , 0 ) (+ , +) (+ , − ) (+ , − ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } (+ , 0 ) (+ , 0 ) ( 0 , +) { ∅ } P 2 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  21. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . � � � � � � Example (+ , +) (+ , +) (+ , − ) | | | ( 0 , +) (+ , 0 ) (+ , 0 ) (+ , +) (+ , +) (+ , − ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } (+ , 0 ) ( 0 , +) ( 0 , +) { ∅ } P 2 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  22. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . � � � � � � � � Example (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) (+ , +) (+ , − ) (+ , − ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } (+ , 0 ) ( 0 , +) ( 0 , +) { ∅ } P 2 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  23. Sign Variation and Descents Order complex (of a poset) Simplicial complex ∆ - A collection of sets s.t. σ ∈ ∆ and τ ⊆ σ implies τ ∈ ∆ . The sets are called faces . Maximal sets are called facets . Order complex ∆( P ) of a poset P - Simplicial complex where faces are chains in P . � � � � � � � � Example (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) (+ , +) (+ , − ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } (+ , 0 ) ( 0 , +) { ∅ } P 2 ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  24. Sign Variation and Descents f -vector ∆ a d -dim simplicial complex. f i = number of i -dim faces f-vector is vector faces: f (∆) = ( f − 1 , f 0 , f 1 , . . . , f d ) . f (∆( P )) is number of elements in each row. Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) f (∆( P 2 )) = ( 1 , 4 , 4 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  25. Sign Variation and Descents h -vectors ∆ a d -dim simplicial complex with f (∆) = ( f − 1 , f 0 , . . . , f d ) . k � � d − i � ( − 1 ) k − i h k = f i − 1 . k − i i = 0 h-vector is vector of h k s: h (∆) = ( h 0 , h 1 , . . . , h d + 1 ) . Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) f (∆( P 2 )) = ( 1 , 4 , 4 ) h (∆( P 2 )) = ( 1 , 2 , 1 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  26. Sign Variation and Descents How can we find the h -vector? Theorem (Stanley 1992(?)) If a simplicial complex ∆ is Cohen-Macaulay, its h-vector has nonnegative entries. Theorem (Machacek 2019) The order complex ∆( P n ) is Cohen-Macaulay. Questions Is there a nice way to compute the h-vector of ∆( P n ) ? A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  27. Sign Variation and Descents Partitionable simplicial complex Conjecture (Stanley 1979, Garsia 1980; Counterexample Duval, Goeckner, Klivans, Martin 2016) Every Cohen-Macaulay simplicial complex is partitionable. Proposition (Stanley) If ∆ is partitionable, then the partitioning gives us the h-vector. A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  28. Sign Variation and Descents Partitionable A simplicial complex ∆ is partitionable if � ∆ = [ G i , F i ] where F i is a facet. Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  29. Sign Variation and Descents Partitionable A simplicial complex ∆ is partitionable if � ∆ = [ G i , F i ] where F i is a facet. Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  30. Sign Variation and Descents Partitionable A simplicial complex ∆ is partitionable if � ∆ = [ G i , F i ] where F i is a facet. Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  31. Sign Variation and Descents Partitionable A simplicial complex ∆ is partitionable if � ∆ = [ G i , F i ] where F i is a facet. Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  32. Sign Variation and Descents Partitionable A simplicial complex ∆ is partitionable if � ∆ = [ G i , F i ] where F i is a facet. Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  33. Sign Variation and Descents Partitionable Proposition (Stanley) Let ∆ be a partitionable simplicial complex with partitioning ∆ = ⊔ [ G i , F i ] where F i is a facet. Then � j : | G j | = i � | . h i (∆) = | Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) h (∆( P 2 )) = ( h 0 , h 1 , h 2 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  34. Sign Variation and Descents Partitionable Proposition (Stanley) Let ∆ be a partitionable simplicial complex with partitioning ∆ = ⊔ [ G i , F i ] where F i is a facet. Then � j : | G j | = i � | . h i (∆) = | Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) h (∆( P 2 )) = ( 1 , h 1 , h 2 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  35. Sign Variation and Descents Partitionable Proposition (Stanley) Let ∆ be a partitionable simplicial complex with partitioning ∆ = ⊔ [ G i , F i ] where F i is a facet. Then � j : | G j | = i � | . h i (∆) = | Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) h (∆( P 2 )) = ( 1 , 2 , h 2 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  36. Sign Variation and Descents Partitionable Proposition (Stanley) Let ∆ be a partitionable simplicial complex with partitioning ∆ = ⊔ [ G i , F i ] where F i is a facet. Then � j : | G j | = i � | . h i (∆) = | Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) h (∆( P 2 )) = ( 1 , 2 , 1 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  37. Sign Variation and Descents Main Theorem Let D n be a type D Coxeter group and let des B denote the type B descent set of an element π ∈ D n . Theorem (Bergeron, D., Machacek 2020BP) The order complex ∆( P n ) is partitionable. Moreover, h i (∆( P n )) = |{ π ∈ D n : | des B ( π ) | = i }| for each 0 ≤ i ≤ n. A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  38. Sign Variation and Descents Coxeter groups Type A n Type B n Type D n The elements in The elements in The elements in type A n Coxeter type B n Coxeter type D n Coxeter groups can be groups can be groups can be represented as represented as represented as permutations in signed even signed S n + 1 . permutations of S n . permutations of S n . 5 ¯ 723 ¯ 8 ¯ 5 ¯ 72 ¯ 3 ¯ 8 ¯ 57238146 ∈ A 7 146 ∈ B 8 146 ∈ D 8 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  39. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  40. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12345678 des A ( π ) = ∅ A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  41. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12345678 des A ( π ) = ∅ A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  42. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12345678 des A ( π ) = { 2 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  43. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12345678 des A ( π ) = { 2 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  44. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12345678 des A ( π ) = { 2 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  45. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12345678 des A ( π ) = { 2 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  46. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12345678 des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  47. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12345678 des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  48. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 12346578 des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  49. Sign Variation and Descents Descents - Type A For π = π 1 . . . π n + 1 ∈ A n let des A ( π ) denote the descent set of π . des A ( π ) = { i : π i > π i + 1 for 1 ≤ i ≤ n } Example π = 57238146 ∈ A 7 des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  50. Sign Variation and Descents Descents - Type B For π = π 1 . . . π n ∈ B n let des B ( π ) denote the descent set of π . des B ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = 0. Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ B 8 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  51. Sign Variation and Descents Descents - Type B For π = π 1 . . . π n ∈ B n let des B ( π ) denote the descent set of π . des B ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = 0. Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ B 8 To find descent, we add a 0 in front, and calculate like “normal”. 72 ¯ 3 ¯ 8 ¯ 05 ¯ 146 012345678 des B ( π ) = ∅ A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  52. Sign Variation and Descents Descents - Type B For π = π 1 . . . π n ∈ B n let des B ( π ) denote the descent set of π . des B ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = 0. Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ B 8 72 ¯ 3 ¯ 8 ¯ 05 ¯ 146 012345678 des B ( π ) = { 1 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  53. Sign Variation and Descents Descents - Type B For π = π 1 . . . π n ∈ B n let des B ( π ) denote the descent set of π . des B ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = 0. Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ B 8 72 ¯ 3 ¯ 8 ¯ 05 ¯ 146 012345678 des B ( π ) = { 1 , 3 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  54. Sign Variation and Descents Descents - Type B For π = π 1 . . . π n ∈ B n let des B ( π ) denote the descent set of π . des B ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = 0. Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ B 8 72 ¯ 3 ¯ 8 ¯ 05 ¯ 146 012345678 des B ( π ) = { 1 , 3 , 4 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  55. Sign Variation and Descents Descents - Type B For π = π 1 . . . π n ∈ B n let des B ( π ) denote the descent set of π . des B ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = 0. Example 72 ¯ 3 ¯ 8 ¯ π = 5 ¯ 146 ∈ B 8 des B ( π ) = { 1 , 3 , 4 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  56. Sign Variation and Descents Descents - Type D For π = π 1 . . . π n ∈ D n let des D ( π ) denote the descent set of π . des D ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = − π 2 . Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ D 8 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  57. Sign Variation and Descents Descents - Type D For π = π 1 . . . π n ∈ D n let des D ( π ) denote the descent set of π . des D ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = − π 2 . Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ D 8 To find descent, we add a 7 in front, and calculate like “normal”. 72 ¯ 3 ¯ 8 ¯ 75 ¯ 146 012345678 des D ( π ) = ∅ A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  58. Sign Variation and Descents Descents - Type D For π = π 1 . . . π n ∈ D n let des D ( π ) denote the descent set of π . des D ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = − π 2 . Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ D 8 72 ¯ 3 ¯ 8 ¯ 75 ¯ 146 012345678 des D ( π ) = { 0 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  59. Sign Variation and Descents Descents - Type D For π = π 1 . . . π n ∈ D n let des D ( π ) denote the descent set of π . des D ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = − π 2 . Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ D 8 72 ¯ 3 ¯ 8 ¯ 75 ¯ 146 012345678 des D ( π ) = { 0 , 1 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  60. Sign Variation and Descents Descents - Type D For π = π 1 . . . π n ∈ D n let des D ( π ) denote the descent set of π . des D ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = − π 2 . Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ D 8 72 ¯ 3 ¯ 8 ¯ 75 ¯ 146 012345678 des D ( π ) = { 0 , 1 , 3 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  61. Sign Variation and Descents Descents - Type D For π = π 1 . . . π n ∈ D n let des D ( π ) denote the descent set of π . des D ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = − π 2 . Example π = 5 ¯ 72 ¯ 3 ¯ 8 ¯ 146 ∈ D 8 72 ¯ 3 ¯ 8 ¯ 75 ¯ 146 012345678 des D ( π ) = { 0 , 1 , 3 , 4 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  62. Sign Variation and Descents Descents - Type D For π = π 1 . . . π n ∈ D n let des D ( π ) denote the descent set of π . des D ( π ) = { i : π i > π i + 1 for 0 ≤ i < n } where π 0 = − π 2 . Example 72 ¯ 3 ¯ 8 ¯ π = 5 ¯ 146 ∈ D 8 des D ( π ) = { 0 , 1 , 3 , 4 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  63. Sign Variation and Descents Main Theorem Theorem (Bergeron, D., Machacek 2020BP) The order complex ∆( P n ) is partitionable. Moreover, h i (∆( P n )) = |{ π ∈ D n : | des B ( π ) | = i }| for each 0 ≤ i ≤ n. Example � � � � � � � � (+ , +) (+ , − ) (+ , +) (+ , − ) | | | | ( 0 , +) ( 0 , +) (+ , 0 ) (+ , 0 ) h (∆( P 2 )) = ( 1 , 2 , 1 ) { ( 0 , +) } { (+ , +) } { (+ , − ) } { (+ , 0 ) } { ∅ } ∆( P 2 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  64. Sign Variation and Descents Permutations and maximal chains How do we associate permutations and maximal chains in our poset? Change π i to 0 inductively. (+ , + , + , + , + , + , + , +) (+ , 0 , 0 , + , 0 , + , 0 , +) (+ , + , + , + , 0 , + , + , +) (+ , 0 , 0 , + , 0 , + , 0 , 0 ) 57238146 ↔ (+ , + , + , + , 0 , + , 0 , +) ( 0 , 0 , 0 , + , 0 , + , 0 , 0 ) (+ , 0 , + , + , 0 , + , 0 , +) ( 0 , 0 , 0 , 0 , 0 , + , 0 , 0 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  65. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. 57238146 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  66. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. 57 | 238 | 146 min − − → A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  67. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57 | 238 | 146 min − − → A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  68. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57 | 238 | 146 min − − → (+ , 0 , 0 , + , 0 , + , 0 , 0 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  69. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57 | 238 | 146 min − − → (+ , 0 , 0 , + , 0 , + , 0 , 0 ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  70. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57238146 min (+ , 0 , 0 , + , 0 , + , 0 , 0 ) − − → des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  71. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57238146 min → (+ , 0 , 0 , + , 0 , + , 0 , 0 ) − − → des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  72. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57238146 min → 57 (+ , 0 , 0 , + , 0 , + , 0 , 0 ) − − → des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  73. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57238146 min → 57238 (+ , 0 , 0 , + , 0 , + , 0 , 0 ) − − → des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  74. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57238146 min → (+ , 0 , 0 , + , 0 , + , 0 , 0 ) − − → 57238146 des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  75. Sign Variation and Descents Permutations and chains How do we associate permutations and chains in our poset? Order 0s first Add increasing sequences inductively. (+ , + , + , + , 0 , + , 0 , +) 57238146 min → (+ , 0 , 0 , + , 0 , + , 0 , 0 ) − − → 57238146 des A ( π ) = { 2 , 5 } A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  76. Sign Variation and Descents Negatives? But how do we handle the negatives?! 5 ¯ 72 ¯ 3 ¯ 8 ¯ ? 146 ↔ A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  77. Sign Variation and Descents Negatives? But how do we handle the negatives?! 5 ¯ 72 ¯ 3 ¯ 8 ¯ ? 146 ↔ ( 57238146 , { 1 , 3 , 7 , 8 } ) A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  78. Sign Variation and Descents Sign Variations Sign vector ω ∈ V n = { + , 0 , −} n . var( ω ) = number of times ω changes sign i ∈ [ n ] is a sign flip of ω if there exists a j such that ω i − j ω i < 0 while ω i − k ω i = 0 for all 1 ≤ k < j . Example ω = (+ , + , − , − , − , − , + , − ) ⇒ var( ω ) = 3 (+ , + , − , − , − , 0 , + , − ) ↔ { 3 , 7 , 8 } 1 2 3 4 5 6 7 8 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

  79. Sign Variation and Descents Cyclic Sign Variations Sign vector ω ∈ V n = { + , 0 , −} n . cvar( ω ) = number of times ω changes sign, cyclically i ∈ [ n ] is a cyclic sign flip of ω if there exists a j such that ω i − j ω i < 0 while ω i − k ω i = 0 for all 1 ≤ k < j where ω i = ω i + n . Example ω = (+ , + , − , − , − , − , + , − ) ⇒ cvar( ω ) = 4 (+ , + , − , − , − , 0 , + , − ) ↔ { 1 , 3 , 7 , 8 } 1 2 3 4 5 6 7 8 A. Dermenjian – York Uni (Joint with: N. Bergeron, J. Machacek) 24 Sept 2020

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