s -weak order and s -permutahedra Cesar Ceballos Viviane Pons - PowerPoint PPT Presentation

Motivation The s -Weak order The s -Permutahedron and s -Associahedron s -weak order and s -permutahedra Cesar Ceballos – Viviane Pons Univ. of Vienna – LRI, Univ. Paris-Sud Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation The s -Weak order The s -Permutahedron and s -Associahedron Weak Order 4321 3421 4231 4312 321 3241 2431 3412 4213 4132 231 312 3214 2341 3142 2413 4123 1432 2314 3124 2143 1342 1423 213 132 2134 1324 1243 123 1234 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation The s -Weak order The s -Permutahedron and s -Associahedron Weak Order 4321 3421 4231 4312 321 3241 2431 3412 4213 4132 231 312 3214 2341 3142 2413 4123 1432 2314 3124 2143 1342 1423 213 132 2134 1324 1243 123 1234 2413 ∧ 4213 = 2413 2413 ∨ 4213 = 4231 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation The s -Weak order The s -Permutahedron and s -Associahedron From the Weak Order to the Tamari lattice 4321 3421 4231 4312 321 3241 2431 3412 4213 4132 231 312 3214 2341 3142 2413 4123 1432 2314 3124 2143 1342 1423 213 132 2134 1324 1243 123 1234 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation The s -Weak order The s -Permutahedron and s -Associahedron From the Weak Order to the Tamari lattice 4321 3421 321 3241 2431 231 3214 2341 1432 2314 2143 1342 213 132 2134 1324 1243 123 1234 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation The s -Weak order The s -Permutahedron and s -Associahedron From the Weak Order to the Tamari lattice Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation The s -Weak order The s -Permutahedron and s -Associahedron Weak order Permutahedron s -Weak order? s -Permutahedron? 4321 4312 4321 3421 3412 3421 4231 4312 4213 2431 3241 2431 3412 4213 4132 2413 2341 3214 3214 2341 3142 2413 4123 1432 2314 1432 1423 3124 2314 3124 2143 1342 1423 1342 2134 1324 1243 1324 2134 1243 1234 1234 Tamari lattice Associahedron ν -Tamari ν -Associahedron Pr´ eville-Ratelle, Viennot Ceballos, Padrol, Sarmiento Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation The s -Weak order The s -Permutahedron and s -Associahedron Weak order Permutahedron s -Weak order? s -Permutahedron? 3 2 1 3 3 2 2 1 1 4321 3 4312 3 4321 2 3421 2 1 1 3412 3 2 3421 4231 4312 4213 1 3 3 3 2431 2 3241 2431 3412 4213 4132 1 2 2 1 1 2413 3 2341 3214 2 3214 2341 3142 2413 4123 1432 1 3 2314 3 1432 2 1423 3124 1 2 2314 3124 2143 1342 1423 1 3 1342 3 2 1 2 2134 1324 1243 1324 1 3 2134 2 1243 1234 1234 1 Tamari lattice Associahedron ν -Tamari ν -Associahedron Pr´ eville-Ratelle, Viennot Ceballos, Padrol, Sarmiento Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice s -decreasing trees Let s be a sequence of n non-negative integers. An s -decreasing tree is a planar tree labeled with 1 . . . n such that each node i has s ( i ) + 1 children and labels are decreasing from root to leaves. s = (0 , 1 , 3 , 0 , 4 , 3) 1 2 3 4 5 6 6 4 5 2 3 1 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice How many trees? s = (0 , 1 , 3 , 0 , 4 , 3) Number of s -decreasing trees: Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice How many trees? s = (0 , 1 , 3 , 0 , 4 , 3) Number of s -decreasing trees: (1+3) 6 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice How many trees? s = (0 , 1 , 3 , 0 , 4 , 3) Number of s -decreasing trees: (1+3) × (1+3+4) 6 5 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice How many trees? s = (0 , 1 , 3 , 0 , 4 , 3) Number of s -decreasing trees: (1+3) × (1+3+4) × (1+3+4+0) 6 4 5 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice How many trees? s = (0 , 1 , 3 , 0 , 4 , 3) Number of s -decreasing trees: (1+3) × (1+3+4) × (1+3+4+0) × (1+3+4+0+3) 6 4 5 3 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice How many trees? s = (0 , 1 , 3 , 0 , 4 , 3) Number of s -decreasing trees: (1+3) × (1+3+4) × (1+3+4+0) × (1+3+4+0+3) × (1+3+4+0+3+1) 6 4 5 2 3 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice How many trees? s = (0 , 1 , 3 , 0 , 4 , 3) Number of s -decreasing trees: (1+3) × (1+3+4) × (1+3+4+0) × (1+3+4+0+3) × (1+3+4+0+3+1) 6 4 5 2 3 1 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice Permutations s = (1 , 1 , 1 , 1 , 1 , 1) Number of s -decreasing trees: 6! (1+1) × (1+1+1) × (1+1+1+1) × (1+1+1+1+1) × (1+1+1+1+1+1) 6 3 5 2 4 1 3 2 6 4 5 1 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice Tree-inversions For all b > a , we define 0 ≤ #( b , a ) ≤ s ( b ). c 6 a a #(b,a) = s(b) #(b,a) = 0 4 5 b 2 3 ... 1 a a a #(b,a) = 1,2,... #(6 , 5) = 3 #(6 , 4) = 1 #(6 , 3) = 3 #(6 , 2) = 1 #(6 , 1) = 1 #(5 , 4) = 0 #(5 , 3) = 2 #(5 , 2) = 0 #(5 , 1) = 0 #(4 , 3) = 0 #(4 , 2) = 0 #(4 , 1) = 0 #(3 , 2) = 0 #(3 , 1) = 0 #(2 , 1) = 1 Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice The s -weak order R , T , s -decreasing trees: R � T ⇔ ∀ b > a , # R ( b , a ) ≤ # T ( b , a ) Theorem The s -weak order is always a lattice. Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

Motivation s -decreasing trees The s -Weak order tree-inversions and lattice The s -Permutahedron and s -Associahedron s -Tamari lattice 3 2 1 3 3 3 2 2 2 1 1 1 3 3 3 3 3 2 2 2 2 1 2 1 1 1 1 3 3 3 2 2 1 2 1 1 3 3 3 3 3 3 3 2 2 1 2 2 1 1 2 1 2 2 1 1 1 3 3 3 3 2 2 2 2 1 1 1 1 3 3 3 3 3 2 1 2 2 1 1 2 1 2 1 3 3 3 3 3 2 2 1 2 1 2 1 2 1 1 3 3 3 2 2 2 1 1 1 (0 , 0 , 2) (0 , 1 , 2) (0 , 2 , 2) Cesar Ceballos – Viviane Pons s -weak order and s -permutahedra

s -weak order and s -permutahedra Cesar Ceballos Viviane Pons - PowerPoint PPT Presentation

Motivation The s -Weak order The s -Permutahedron and s -Associahedron s -weak order and s -permutahedra Cesar Ceballos Viviane Pons Univ. of Vienna LRI, Univ. Paris-Sud Cesar Ceballos Viviane Pons s -weak order and s -permutahedra

The facial weak order in hyperplane arrangements Aram Dermenjian 1,3 Christophe Hohlweg 1 , Thomas

The facial weak order in hyperplane arrangements Aram Dermenjian 1,3 Christophe Hohlweg 1 , Thomas

The facial weak order in hyperplane arrangements Aram Dermenjian 1,3 Christophe Hohlweg 1 , Thomas

The facial weak order and its lattice of quotients Aram Dermenjian Joint work with: Christophe

Facial Weak Order Aram Dermenjian Joint work with: Christophe Hohlweg (LACIM) and Vincent Pilaud

Facial Weak Order Aram Dermenjian Joint work with: Christophe Hohlweg (LACIM) and Vincent Pilaud

The Arithmetic of Coxeter Permutahedra Federico Ardila San Francisco State University

The weak Bruhat order on the symmetric group is Sperner Yibo Gao Joint work with: Christian

To the weak I became weak, that I might win the weak. I have become all things to all people,

f -vectors, descents sets and the weak order E. Swartz Cornell University, Ithaca, NY. Feb. 6,

WEAK INTERPOLATION PROPERTY over THE MINIMAL LOGIC Larisa Maksimova Sobolev Institute of

Classical and Weak Solutions to Local First Order Mean Field Games through Elliptic Regularity

Weak Cardinality Theorems for First-Order Logic Till Tantau Fakult at f ur Elektrotechnik

A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in

Logic as a Tool Chapter 3: Understanding First-order Logic 3.1 First-order structures and

Higher order corrections to Higgs production in Weak Boson Fusion Sophy Palmer Institute for

Logic as a Tool Chapter 3: Understanding First-order Logic 3.1 First-order structures and

Kolmogorov equations and weak order analysis for SPDEs with multiplicative noise Charles-Edouard

Weak-Signal Digital Modes Weak-Signal Digital Modes The weak-signal digimodes have been

Modelling and Verification Lecture 4 Weak bisimilarity and weak bisimulation games Properties of

New class of finite element methods: weak Galerkin methods Xiu Ye University of Arkansas at

Weak monadic second-order theory of one successor (WS1S) Presentation for Seminar on Decision

The weak-charged WIMP Shigeki Matsumoto (Kavli IPMU) The weak-charged WIMP, Majorana fermion with

The mathematics and metamathematics of weak analysis Fernando Ferreira Universidade de Lisboa