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

Flight Control Design Using Backstepping Ola Hrkegrd, T orkel Glad Linkping University

Un environnement de d emonstration universel Talk at CPR Guillaume Burel Wednesday March

Examination Approach High Yield Techniques Two types of neurologic examinations 1.

Joint Application of NMR and Scattering Haydyn Mertens PhD Methods combining SAXS with NMR

Mesh-Based Modeling Amal Delaunoy 3D Photography, ETH Zrich, May 2013 Schedule (tentative)

Software Technology Readiness for the Smart Grid Cristina Tugurlan, Harold Kirkham, David Chassin

Technologies for future mobile transport networks Pham Tien Dat 1 , Atsushi Kanno 1 , Naokatsu

Supernova limits on a light CP-even scalar and implications for the KOTO anomaly Yongchao Zhang