Viviane Pons Maˆıtresse de conf´erences, Univ. Paris-Saclay [email protected] – @PyViv
Permutahedron and Associahedron
Combinatorics and geometry
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 1 / 27 As vectors of Rn 123 → (1, 2, 3)
Permutahedron Construction from permutations
Permutations size 2: 12 21 size 3: 123 132 213 231 312 321 size 4: 1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 2 / 27 Permutahedron Construction from permutations
Permutations size 2: 12 21 size 3: 123 132 213 231 312 321 size 4: 1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321
As vectors of Rn 123 → (1, 2, 3)
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 2 / 27 x + y + z = 6
Permutahedron Construction from permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 x + y + z = 6
Permutahedron Construction from permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 x + y + z = 6
Permutahedron Construction from permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 x + y + z = 6
Permutahedron Construction from permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 x + y + z = 6
Permutahedron Construction from permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 x + y + z = 6
Permutahedron Construction from permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 x + y + z = 6
Permutahedron Construction from permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 x + y + z = 6
Permutahedron Construction from permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 Permutahedron Construction from permutations
x + y + z = 6
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 3 / 27 Permutahedron Construction from permutations
Using SageMath
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 4 / 27 Permutahedron Construction from permutations
For size 4
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 5 / 27 (12)(23)(12) = (23)(12)(23)
(23)(12) (12)(23)
(12) (23)
Id The the left weak order on permutations
Permutahedron The weak order
Looking at the“skeleton”of the polytope”
321
312 231
213 132
123
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 6 / 27 (12)(23)(12) = (23)(12)(23)
(23)(12) (12)(23)
(12) (23)
Id
Permutahedron The weak order
Looking at the“skeleton”of the polytope”
321
312 231
213 132
123 The the left weak order on permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 6 / 27 Permutahedron The weak order
Looking at the“skeleton”of the polytope”
321 (12)(23)(12) = (23)(12)(23)
312 231 (23)(12) (12)(23)
213 132 (12) (23)
123 Id The the left weak order on permutations
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 6 / 27 Permutahedron The weak order
The left and right weak orders
321 321
312 231 231 312
213 132 213 132
123 123 left weak order right weak order
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 7 / 27 2413 ∨ 4213 = 4231
2413 ∧ 4213 = 2413
Permutahedron The weak order Right 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 8 / 27 2413 ∨ 4213 = 4231
2413 ∧ 4213 = 2413
Permutahedron The weak order Right 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 8 / 27 2413 ∧ 4213 = 2413 2413 ∨ 4213 = 4231
Permutahedron The weak order Right 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 8 / 27 2413 ∧ 4213 = 2413 2413 ∨ 4213 = 4231
Permutahedron The weak order Right 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 8 / 27 2413 ∧ 4213 = 2413 2413 ∨ 4213 = 4231
Permutahedron The weak order Right 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 8 / 27 Permutahedron The weak order Right 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 8 / 27 Permutahedron The weak order Right 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 8 / 27 Permutahedron The weak order
As a reflection group
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 9 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 d = n − #parts
Permutahedron Faces Combinatorics of faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 Permutahedron Faces Combinatorics of faces
d = n − #parts
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 10 / 27 Permutahedron Faces
(image from V. Pilaud’s talk “The Associahedron and its friends”)
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 11 / 27 Permutahedron Faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 Permutahedron Faces
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 Permutahedron Faces
x1 + x2 + x3 = 6
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 Permutahedron Faces
x1 + x2 = 6 − x3
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 Permutahedron Faces
3 ≤ x1 + x2 ≤ 5
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 X |J| + 1 J ⊆ [n] → x ≥ j 2 j∈J
12|3 2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3 3|12 13|2 x1 + x3 ≥ 3 1|23 x1 ≥ 1
Permutahedron Faces
x1 + x2 ≥ 3
x1 + x2 ≤ 5
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 X |J| + 1 J ⊆ [n] → x ≥ j 2 j∈J
12|3 2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3 3|12 13|2 x1 + x3 ≥ 3 1|23 x1 ≥ 1
Permutahedron Faces
x1 + x2 ≥ 3
x3 ≥ 1
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 X |J| + 1 J ⊆ [n] → x ≥ j 2 j∈J
2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3
13|2 x1 + x3 ≥ 3 1|23 x1 ≥ 1
Permutahedron Faces
12|3 x1 + x2 ≥ 3
3|12 x3 ≥ 1
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3
13|2 x1 + x3 ≥ 3 1|23 x1 ≥ 1
Permutahedron Faces
X |J| + 1 J ⊆ [n] → x ≥ j 2 j∈J
12|3 x1 + x2 ≥ 3
3|12 x3 ≥ 1
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 23|1 x2 + x3 ≥ 3
13|2 x1 + x3 ≥ 3 1|23 x1 ≥ 1
Permutahedron Faces
X |J| + 1 J ⊆ [n] → x ≥ j 2 j∈J
12|3 x1 + x2 ≥ 3 2|13 x2 ≥ 1
3|12 x3 ≥ 1
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 13|2 x1 + x3 ≥ 3 1|23 x1 ≥ 1
Permutahedron Faces
X |J| + 1 J ⊆ [n] → x ≥ j 2 j∈J
12|3 x1 + x2 ≥ 3 2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3 3|12 x3 ≥ 1
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 1|23 x1 ≥ 1
Permutahedron Faces
X |J| + 1 J ⊆ [n] → x ≥ j 2 j∈J
12|3 x1 + x2 ≥ 3 2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3 3|12 x3 ≥ 1 13|2 x1 + x3 ≥ 3
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 Permutahedron Faces
X |J| + 1 J ⊆ [n] → x ≥ j 2 j∈J
12|3 x1 + x2 ≥ 3 2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3 3|12 x3 ≥ 1 13|2 x1 + x3 ≥ 3 1|23 x1 ≥ 1
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 12 / 27 Associahedron Tamari lattice
From the weak order to the Tamari lattice We define a surjection from permutations to binary trees which gives us a new lattice.
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 13 / 27 2 5
1 3
Associahedron Tamari lattice
Binary search tree insertion
4
15324 →
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 14 / 27 5
1 3
Associahedron Tamari lattice
Binary search tree insertion
4
15324 → 2
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 14 / 27 5
1
Associahedron Tamari lattice
Binary search tree insertion
4
15324 → 2
3
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 14 / 27 1
Associahedron Tamari lattice
Binary search tree insertion
4
15324 → 2 5
3
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 14 / 27 Associahedron Tamari lattice
Binary search tree insertion
4
15324 → 2 5
1 3
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 14 / 27 The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Associahedron Tamari lattice
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Associahedron Tamari lattice
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Associahedron Tamari lattice
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Associahedron Tamari lattice
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Associahedron Tamari lattice
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Associahedron Tamari lattice
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Associahedron Tamari lattice
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Associahedron Tamari lattice
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 Associahedron Tamari lattice
The Tamari lattice is a lattice on binary trees. It is a quotient lattice of the weak order.
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 15 / 27 Associahedron Tamari lattice
1234
2134 1324 1243
2314 3124 2143 1342 1423
3214 2341 2413 3142 1432 4123
3241 2431 4213 3412 4132
3421 4231 4312
4321
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 16 / 27 Associahedron Tamari lattice
More about the Tamari lattice Binary trees are counted by the Catalan numbers
1 2n n + 1 n . The Tamari lattice can be defined on many families of combinatorial objects, such as I Triangulations of regular polygons I Dyck paths I Ordered forests I certain pattern avoiding permutations (312 avoiding and 231 avoiding) I ways to parenthesize an expression (original definition of Tamari) I ...
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 17 / 27 Associahedron Tamari lattice
312 - avoiding permutations 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 18 / 27 Associahedron Tamari lattice
312 - avoiding permutations 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
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 18 / 27 4231 4312
3412 4213 4132
312 3142 2413 4123
3124 1423
Associahedron Tamari lattice
312 - avoiding permutations 4321
3421 321 3241 2431
231 3214 2341 1432
2314 2143 1342 213 132 2134 1324 1243
123 1234
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 18 / 27 4231 4312
3412 4213 4132
312 3142 2413 4123
3124 1423
Associahedron Tamari lattice
Dyck paths
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 18 / 27 Associahedron The Associahedron
The Associahedron – Stasheff polytope Different constructions I Loday 2004 I Billera-Filliman-Sturmfels 1990 I Gelfand-Kapranov-Zelevinsky 1994 I Chapoton-Fomin-Zelevinsky 2002 I Hohlweg-Lange 2007 I Ceballos-Santos-Ziegler 2011 I Hohlweg-Lange-Thomas 2012
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 19 / 27 1 4 1 4 5 18 1 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
( , , , , , , , )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 1 4 1 4 5 18 1 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
( , , , , , , , )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 4 1 4 5 18 1 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
(1, , , , , , , )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 1 4 5 18 1 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
(1, 4, , , , , , )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 4 5 18 1 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
(1, 4, 1, , , , , )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 5 18 1 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
(1, 4, 1, 4, , , , )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 18 1 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
(1, 4, 1, 4, 5, , , )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 1 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
(1, 4, 1, 4, 5, 18, , )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 8
Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
(1, 4, 1, 4, 5, 18, 1, )
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 Associahedron The Associahedron
Loday’s Associahedron i → (# of left leaves) × (# of right leaves)
6
5 8
4 7
2
1 3
(1, 4, 1, 4, 5, 18, 1, 8)
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 20 / 27 Associahedron The Associahedron
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 21 / 27 Associahedron The Associahedron
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 22 / 27 Associahedron The Associahedron
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 23 / 27 Associahedron The Associahedron
12|3 x1 + x2 ≥ 3 2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3 3|12 x3 ≥ 1 13|2 x1 + x3 ≥ 3 1|23 x1 ≥ 1
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 24 / 27 x1 + x3 ≥ 3
Associahedron The Associahedron
12|3 x1 + x2 ≥ 3 2|13 x2 ≥ 1 23|1 x2 + x3 ≥ 3 3|12 x3 ≥ 1 13|2 1|23 x1 ≥ 1
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 24 / 27 Associahedron More
More... On the weak order I Alain Lascoux and Marcel-Paul Schutzenberger,¨ Treillis et bases des groupes de Coxeter, Electron. J. Combin. 1996. I A. Bj¨orner, M.L. Wachs, Permutation statistics and linear extensions of posets, J. Combin. Theory Ser. A, 1991
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 25 / 27 Associahedron More
And more... Relations between the weak order and the Tamari lattice + Hopf algebras I F. Hivert, J.-C. Novelli, J.-Y. Thibon, The algebra of binary search trees, Theoret. Comput. Sci. 2005 I Andy Tonks, Relating the associahedron and the permutohedron, in: Proceedings of Renaissance Conferences, 1995 I Nathan Reading, Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A, 2005.
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 26 / 27 Associahedron More
And more... Generalizations of the Tamari lattice I Nathan Reading, Cambrian lattices, Adv. Math. 2006 I F. Bergeron and L.-F. Pr´eville-Ratelle. Higher trivariate diagonal harmonics via generalized Tamari posets. J. Comb. 2012. I L.-F. Pr´eville-Ratelle and Viennot X. An extension of Tamari lattices. DMTCS Proceedings, FPSAC, 2015 I V. Pilaud and V. Pons. Permutrees. Algebraic Combinatorics, 2018. I C. Ceballos and V. Pons. The s-weak order and s-permutahedra, FPSAC 2019.
Viviane Pons (Paris-Saclay) Permutahedron and Associahedron June 22, 2020 27 / 27