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Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A funny thing happened on the way toSteppenwolf Theatre. . .
from lattice paths to polytopes and Hopf algebras
T. Kyle Petersen
DePaul UniversityDepartment of Mathematical Sciences
MathFestMadison, WIAugust 2012
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Combinatorics, geometry, and an algebra of paths
1 Combinatorics: Walking to Steppenwolf
2 Geometry: Tamari poset/associahedron
3 Algebra: Loday-Ronco
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The problem
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The problem
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
C0 = 1 :
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
C0 = 1 :
C1 = 1 :
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
C0 = 1 :
C1 = 1 :
C2 = 2 :
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
C0 = 1 :
C1 = 1 :
C2 = 2 :
C3 = 5 :
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C4 = 14 :
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5?
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5?
Point of second-to-last return
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5?
Point of second-to-last return
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
C3 · C1
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
C3 · C1
+
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
C3 · C1
+
C4 · C0
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
C3 · C1
+
C4 · C0
= 1 · 14 + 1 · 5 + 2 · 2 + 5 · 1 + 14 · 1 = 42
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . .
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Also counts:
balanced parenthesizations (()(()))
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Also counts:
balanced parenthesizations (()(()))
planar binary trees
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Also counts:
balanced parenthesizations (()(()))
planar binary trees
noncrossing partitions:• •
••
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Also counts:
balanced parenthesizations (()(()))
planar binary trees
noncrossing partitions:• •
••
and about 200 other sets of things. . .
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Combinatorics, geometry, and an algebra of paths
1 Combinatorics: Walking to Steppenwolf
2 Geometry: Tamari poset/associahedron
3 Algebra: Loday-Ronco
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
→
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
→
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
→
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
→
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
n = 0
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
n = 0 n = 1
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
n = 0 n = 1 n = 2
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
n = 0 n = 1 n = 2 n = 3
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Tamari poset (Associahedron)
n = 4
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Combinatorics, geometry, and an algebra of paths
1 Combinatorics: Walking to Steppenwolf
2 Geometry: Tamari poset/associahedron
3 Algebra: Loday-Ronco
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Jean-Louis Loday: “the integers as molecules”
0 1 2 3 4
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A meditation
We can do arithmetic at the molecular level. Can we do it at theatomic level?
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The wedge operation
p ∨ q:
∨ =
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The wedge operation
p ∨ q:
∨ =
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The wedge operation
p ∨ q:
∨ =
(Recall the role the wedge played in the Catalan identity)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Decomposing paths
p = pℓ ∨ pr (unique decomposition)
= ∨
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Decomposing paths
p = pℓ ∨ pr (unique decomposition)
= ∨
= ∨ = ( ∨ ) ∨
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Decomposing paths
p = pℓ ∨ pr (unique decomposition)
= ∨
= ∨ = ( ∨ ) ∨
= ∨ = ∨ ( ∨ )
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Defining addition
Goal: define addition for paths,
p + q,
in a way that generalizes addition for integers
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Defining addition
Goal: define addition for paths,
p + q,
in a way that generalizes addition for integers
p + 0 = 0+ p = p
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Defining addition
Goal: define addition for paths,
p + q,
in a way that generalizes addition for integers
p + 0 = 0+ p = p
for p, q 6= 0, p + q = (p ⊣ q) ∪ (p ⊢ q)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Defining addition
Goal: define addition for paths,
p + q,
in a way that generalizes addition for integers
p + 0 = 0+ p = p
for p, q 6= 0, p + q = (p ⊣ q) ∪ (p ⊢ q)
now recursively,
p ⊣ q = pℓ ∨ (pr + q) and p ⊢ q = (p + qℓ) ∨ qr
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
1
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
1 1
+
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
1 1
+
2
=
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Split the plus sign: + = ⊣ ∪ ⊢
+ =(
⊣)
⋃
(
⊢)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Split the plus sign: + = ⊣ ∪ ⊢
+ =(
⊣)
⋃
(
⊢)
=(
∨(
+))
⋃
((
+)
∨)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Split the plus sign: + = ⊣ ∪ ⊢
+ =(
⊣)
⋃
(
⊢)
=(
∨(
+))
⋃
((
+)
∨)
=(
∨)
⋃
(
∨)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Split the plus sign: + = ⊣ ∪ ⊢
+ =(
⊣)
⋃
(
⊢)
=(
∨(
+))
⋃
((
+)
∨)
=(
∨)
⋃
(
∨)
=⋃
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
Want:
1
+
2
=
3
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
+ =
(
+
)
⋃
(
+
)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
+ =
(
+
)
⋃
(
+
)
=
(
⊣
)
⋃
(
⊢
)
⋃
(
⊣
)
⋃
(
⊢
)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
⊣ = ∨
(
+
)
=
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
⊣ = ∨
(
+
)
=
⊢ = ∨
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
⊣ = ∨
(
+
)
=
⊢ = ∨ =⋃
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Surprise!
Theorem
If a+ b = c (as nonnegative integers), then
a+ b = c
(as Tamari lattices)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A better way to add
Can we add two paths more simply?
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
\
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
�
�
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
�
�
/
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
�
�
�
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
Theorem (Loday-Ronco)
For any paths p and q,
p + q =⋃
p/q≤r≤p\q
r
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
Theorem (Loday-Ronco)
For any paths p and q,
p + q =⋃
p/q≤r≤p\q
r
There is also a multiplication that refines ordinary integermultiplication. . .
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
Theorem (Loday-Ronco)
For any paths p and q,
p + q =⋃
p/q≤r≤p\q
r
There is also a multiplication that refines ordinary integermultiplication. . . the Loday-Ronco algebra is the “free dendriformalgebra on one generator” and a “combinatorial Hopf algebra”
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Food for thought
We can now do arithmetic:
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Food for thought
We can now do arithmetic:
at the molecular level (whole numbers)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Food for thought
We can now do arithmetic:
at the molecular level (whole numbers)
at the atomic level (paths)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Food for thought
We can now do arithmetic:
at the molecular level (whole numbers)
at the atomic level (paths)
What about the subatomic level?
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
References
M. Aguiar and F. Sottile, Structure of the Loday-Ronco Hopfalgebra of trees. J. Algebra 295 (2006), 473–511.
J.-L. Loday, Dichotomy of the addition of natural numbers. inAssociahedra, Tamari Lattices and Related Structures,Progress in Math. 299, Birkhauser (2012). (alsoarXiv:1108.6238)
J.-L. Loday, Arithmetree. J. Algebra 258 (2002), 275–309.
J.-L. Loday and M. Ronco, Hopf algebra of the planar binarytrees. Adv. Math. 139 (1998), 293–309