Shaken braid arrangements and trees Olivier Bernardi - Brandeis - PowerPoint PPT Presentation

Shaken braid arrangements and trees Olivier Bernardi - Brandeis University 9 2 8 4 1 5 3 7 6 MIT, February 2016

Shaken braid arrangements and trees Olivier Bernardi - Brandeis University 9 2 8 4 1 5 3 7 6 4 1 5 6 9 7 2 3 8 MIT, February 2016

Hyperplane arrangements A hyperplane arrangement of dimension n is a finite collection of affine hyperplanes in R n . Example: x 2 x 1

Hyperplane arrangements A hyperplane arrangement of dimension n is a finite collection of affine hyperplanes in R n . Example: The hyperplanes cut the space into regions . x 2 7 regions x 1

Braid arrangement Def: The braid arrangement of dimension n has hyperplanes { x i − x j = 0 } for all 0 ≤ i < j ≤ n .

Braid arrangement Def: The braid arrangement of dimension n has hyperplanes { x i − x j = 0 } for all 0 ≤ i < j ≤ n . Example: n = 3 x 1 x 1 − x 3 = 0 x 1 − x 2 = 0 x 3 x 2 x 2 − x 3 = 0

Braid arrangement Def: The braid arrangement of dimension n has hyperplanes { x i − x j = 0 } for all 0 ≤ i < j ≤ n . Example: n = 3 x 1 x 1 − x 3 = 0 x 1 − x 2 = 0 n ! regions x 3 x 2 x 2 − x 3 = 0

Shaken braid arrangements Def: Fix S ⊂ Z finite. The S -shaken braid arrangement A S ( n ) ⊂ R n has hyperplanes { x i − x j = s } for all 0 ≤ i < j ≤ n , and all s ∈ S . We denote r S ( n ) = # regions of A S ( n ) .

Shaken braid arrangements Def: Fix S ⊂ Z finite. The S -shaken braid arrangement A S ( n ) ⊂ R n has hyperplanes { x i − x j = s } for all 0 ≤ i < j ≤ n , and all s ∈ S . We denote r S ( n ) = # regions of A S ( n ) . Example: S = { 0 , 1 } and n = 3 . x 1 x 1 − x 2 = 1 x 1 − x 2 = 0 r S (3) = 16 x 3 x 2

Known relations with trees [Athanasiadis, Postnikov, Stanley,. . . ] Let B ( n ) be the set of rooted binary trees with n labeled nodes. 9 8 2 4 1 5 3 7 6 (2 n )! |B ( n ) | = Cat ( n ) × n ! = ( n + 1)!

Known relations with trees [Athanasiadis, Postnikov, Stanley,. . . ] S = {− 1 , 0 , 1 } S = { 0 , 1 } S = {− 1 , 1 } S = { 1 } S = { 0 } Catalan Shi Semi-order Braid Linial T ∈B ( n ) T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. w w w v v v v v v v v u u u u u u u u u u u u > v u > v u < v u > v u > w u > w

Known relations with trees [Athanasiadis, Postnikov, Stanley,. . . ] S = {− 1 , 0 , 1 } S = { 0 , 1 } S = {− 1 , 1 } S = { 1 } S = { 0 } Catalan Shi Semi-order Braid Linial T ∈B ( n ) T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. w w w v v v v v v v v u u u u u u u u u u u u > v u > v u < v u > v u > w u > w “ Why? ” Ira Gessel

Known relations with trees [Athanasiadis, Postnikov, Stanley,. . . ] S = {− 1 , 0 , 1 } S = { 0 , 1 } S = {− 1 , 1 } S = { 1 } S = { 0 } Catalan Shi Semi-order Braid Linial T ∈B ( n ) T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. w w w v v v v v v v v u u u u u u u u u u u u > v u > v u < v u > v u > w u > w u n v v “ Why? ” u 2 u u Ira Gessel u 1 u > v u > v

Arrangements, trees, and discrete gas

Boxed trees • T ( m ) = set of rooted ( m +1) -ary trees with labeled nodes. 7 1 9 13 3 10 4 4 1 11 12 8 5 2 6

Boxed trees • T ( m ) = set of rooted ( m +1) -ary trees with labeled nodes. • The last node among the children of u is denoted cadet( u ) . • A cadet-sequence is any sequence of node ( v 1 , . . . , v k ) such that v i +1 = cadet( v i ) . 7 1 9 13 3 10 4 4 1 11 12 8 5 2 6

Boxed trees • T ( m ) = set of rooted ( m +1) -ary trees with labeled nodes. • The last node among the children of u is denoted cadet( u ) . • A cadet-sequence is any sequence of node ( v 1 , . . . , v k ) such that v i +1 = cadet( v i ) . • A m - boxed tree is a tree in T ( m ) decorated with boxes partitioning the nodes into cadet-sequences. 7 1 9 13 3 10 4 4 1 11 12 8 5 2 6

Main result Let S ⊂ Z . Let m = max( | s | , s ∈ S ) . Def: A S -boxed tree is a m -boxed tree such that each box satisfies ∀ i < j , if ( c i + c i +1 + · · · + c j − 1 ) ∈ S ∪ { 0 } then v i < v j , if − ( c i + c i +1 + · · · + c j − 1 ) ∈ S then v i > v j . m + 1 c j v k c i v j c 1 v 2 v i v 1

Main result Let S ⊂ Z . Let m = max( | s | , s ∈ S ) . Def: A S -boxed tree is a m -boxed tree such that each box satisfies ∀ i < j , if ( c i + c i +1 + · · · + c j − 1 ) ∈ S ∪ { 0 } then v i < v j , if − ( c i + c i +1 + · · · + c j − 1 ) ∈ S then v i > v j . Example: S = [ − a .. m ] with a ∈ { 0 , ..., m } a<c i ≤ m v k v 2 v i v 1 <v 2 < · · · <v k v 1

Main result Let S ⊂ Z . Let m = max( | s | , s ∈ S ) . Def: A S -boxed tree is a m -boxed tree such that each box satisfies ∀ i < j , if ( c i + c i +1 + · · · + c j − 1 ) ∈ S ∪ { 0 } then v i < v j , if − ( c i + c i +1 + · · · + c j − 1 ) ∈ S then v i > v j . Theorem: � ( − 1) n − # boxes , r S ( n ) = T ∈U S ( n ) where U S ( n ) is the set of S -boxed trees with n nodes,

Corollary Def: S is transitive if • If a, b / ∈ S , with ab > 0 , then a + b / ∈ S , • If a, b / ∈ S , with ab < 0 , then a − b / ∈ S , • If 0 , a / ∈ S , with a > 0 , then − a / ∈ S .

Corollary Def: S is transitive if • If a, b / ∈ S , with ab > 0 , then a + b / ∈ S , • If a, b / ∈ S , with ab < 0 , then a − b / ∈ S , • If 0 , a / ∈ S , with a > 0 , then − a / ∈ S . Examples: • Any subset of {− 1 , 0 , 1 } . • Any interval of integers containing 1 . • S such that [ − k ; k ] ⊆ S ⊆ [ − 2 k ; 2 k ] for some k .

Corollary Def: S is transitive if • If a, b / ∈ S , with ab > 0 , then a + b / ∈ S , • If a, b / ∈ S , with ab < 0 , then a − b / ∈ S , • If 0 , a / ∈ S , with a > 0 , then − a / ∈ S . Def: T S is set of trees in T ( m ) such that any v = cadet ( u ) satisfies Cond ( S ) : if # left-siblings ( v ) / ∈ S ∪ { 0 } then u < v , if − # left-siblings ( v ) / ∈ S then u > v . # left-siblings(v) v u

Corollary Def: S is transitive if • If a, b / ∈ S , with ab > 0 , then a + b / ∈ S , • If a, b / ∈ S , with ab < 0 , then a − b / ∈ S , • If 0 , a / ∈ S , with a > 0 , then − a / ∈ S . Def: T S is set of trees in T ( m ) such that any v = cadet ( u ) satisfies Cond ( S ) : if # left-siblings ( v ) / ∈ S ∪ { 0 } then u < v , if − # left-siblings ( v ) / ∈ S then u > v . Corollary: If S is transitive, then r S ( n ) = | T S ( n ) |

Corollary Def: T S is set of trees in T ( m ) such that any v = cadet ( u ) satisfies Cond ( S ) : if # left-siblings ( v ) / ∈ S ∪ { 0 } then u < v , if − # left-siblings ( v ) / ∈ S then u > v . Corollary: If S is transitive, then r S ( n ) = | T S ( n ) | Example: S = {− 2 , − 1 , 0 , 1 , 3 } v v Cond ( S ) : ⇒ u < v ⇒ u > v u u

Corollary Def: T S is set of trees in T ( m ) such that any v = cadet ( u ) satisfies Cond ( S ) : if # left-siblings ( v ) / ∈ S ∪ { 0 } then u < v , if − # left-siblings ( v ) / ∈ S then u > v . Corollary: If S is transitive, then r S ( n ) = | T S ( n ) | Example: If S = [ − m..m ] , then T S ( n ) = T ( m ) ( n ) . Catalan Semiorder If S [ − m..m ] \ { 0 } , then Cond(S)=“cadets with 0 left-siblings are less than parent”. Shi If S = [ − a..m ] with a ∈ { 0 , . . . , m } , then Cond(S)=“cadets with > a left-siblings are less than parent”. Linial If S [ − a..m ] \ { 0 } with a ∈ { 0 , . . . , m } , then Cond(S)=“cadets with 0 or > a left-siblings are less than parent”.

Proof of corollary. Locality: For S transitive a m -boxed tree is S -boxed if and only if ∀ i < j , if c i ∈ S ∪ { 0 } then v i < v i +1 , if − c i ∈ S then v i > v i +1 . v k c i v i + 1 v 2 v i v 1

Proof of corollary. Locality: For S transitive a m -boxed tree is S -boxed if and only if ∀ i < j , if c i ∈ S ∪ { 0 } then v i < v i +1 , if − c i ∈ S then v i > v i +1 . Remark: For v = cadet( u ) u, v satisfies Cond(S) ⇐ ⇒ u and v cannot be in same S -box.

Proof of corollary. Locality: For S transitive a m -boxed tree is S -boxed if and only if ∀ i < j , if c i ∈ S ∪ { 0 } then v i < v i +1 , if − c i ∈ S then v i > v i +1 . Remark: For v = cadet( u ) u, v satisfies Cond(S) ⇐ ⇒ u and v cannot be in same S -box. Sign-reversing involution: ( − 1) n − # boxes + � � ( − 1) n − # boxes r S ( n ) = T ∈U S , T ∈U S , satisfying Cond(S) not satisfying Cond(S) 0 |T S ( n ) | Merge/split box at v = cadet ( u ) Must have a different box not satisfying Cond(S). around each node.

Proof of Theorem x 1 9 8 2 4 1 5 3 7 6 x 3 x 2 Zaslavky formula Zaslavky formula Decomposition in runs + Mayers’ clusters + Mayers’ clusters 4 6 1 9 7 2 5 3 8 discrete gas model

� ( − 1) e + c − n | W S ( G ) | , Lemma 1: r S ( n ) = G =([ n ] ,E ) where e =#edges, c =#components, n =#vertices, and W S ( G ) = set of tuples ( x 1 , . . . , x n ) such that • ∀{ i, j } ∈ E with i < j , x i − x j ∈ S , • ∀ i ∈ [ n ] smallest in its component, x i = 0 .