A funny thing happened on the way to Steppenwolf...

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, geometry, and an algebra of paths

1 Combinatorics: Walking to Steppenwolf

2 Geometry: Tamari poset/associahedron

3 Algebra: Loday-Ronco

The problem

Counting “Clybourn paths”

Let Cn denote the number of paths of length 2n

C0 = 1 :

C1 = 1 :

C0 = 1 :

C1 = 1 :

C2 = 2 :

C0 = 1 :

C1 = 1 :

C2 = 2 :

C3 = 5 :

C4 = 14 :

Point of second-to-last return

C0 · C4

C1 · C3

C0 · C4

C1 · C3

C0 · C4

C1 · C3

C2 · C2

C0 · C4

C1 · C3

C2 · C2

C0 · C4

C1 · C3

C2 · C2

C3 · C1

C0 · C4

C1 · C3

C2 · C2

C3 · C1

C0 · C4

C1 · C3

C2 · C2

C3 · C1

C4 · C0

C0 · C4

C1 · C3

C2 · C2

C3 · C1

C4 · C0

= 1 · 14 + 1 · 5 + 2 · 2 + 5 · 1 + 14 · 1 = 42

Catalan numbers

C0 = 1, Cn+1 =n∑

CiCn−i

Catalan numbers

C0 = 1, Cn+1 =n∑

CiCn−i

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . .

Catalan numbers

C0 = 1, Cn+1 =n∑

CiCn−i

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1

, . . .

Catalan numbers

C0 = 1, Cn+1 =n∑

CiCn−i

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1

, . . .

Also counts:

balanced parenthesizations (()(()))

Catalan numbers

C0 = 1, Cn+1 =n∑

CiCn−i

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1

, . . .

Also counts:

planar binary trees

Catalan numbers

C0 = 1, Cn+1 =n∑

CiCn−i

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1

, . . .

Also counts:

planar binary trees

noncrossing partitions:• •

••

Catalan numbers

C0 = 1, Cn+1 =n∑

CiCn−i

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1

, . . .

Also counts:

planar binary trees

noncrossing partitions:• •

••

and about 200 other sets of things. . .

A local transformation

n = 0 n = 1

n = 0 n = 1 n = 2

n = 0 n = 1 n = 2 n = 3

Tamari poset (Associahedron)

Jean-Louis Loday: “the integers as molecules”

0 1 2 3 4

A meditation

We can do arithmetic at the molecular level. Can we do it at theatomic level?

The wedge operation

p ∨ q:

The wedge operation

p ∨ q:

The wedge operation

p ∨ q:

(Recall the role the wedge played in the Catalan identity)

Decomposing paths

p = pℓ ∨ pr (unique decomposition)

Decomposing paths

= ∨ = ( ∨ ) ∨

Decomposing paths

= ∨ = ( ∨ ) ∨

= ∨ = ∨ ( ∨ )

Defining addition

Goal: define addition for paths,

p + q,

in a way that generalizes addition for integers

Defining addition

p + q,

p + 0 = 0+ p = p

Defining addition

p + q,

p + 0 = 0+ p = p

for p, q 6= 0, p + q = (p ⊣ q) ∪ (p ⊢ q)

Defining addition

p + q,

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

Why 1+ 1 = 2

Split the plus sign: + = ⊣ ∪ ⊢

Why 1+ 1 = 2

Why 1+ 2 = 3

⊣ = ∨

Why 1+ 2 = 3

⊣ = ∨

⊢ =(

Why 1+ 2 = 3

⊣ = ∨

⊢ =(

⊣ = ∨

Why 1+ 2 = 3

⊣ = ∨

⊢ =(

⊣ = ∨

⊢ =(

Why 1+ 2 = 3

⊣ = ∨

⊢ =(

⊣ = ∨

⊢ = ∨

Why 1+ 2 = 3

⊣ = ∨

⊢ =(

⊣ = ∨

⊢ = ∨ =⋃

Surprise!

Theorem

If a+ b = c (as nonnegative integers), then

a+ b = c

(as Tamari lattices)

A better way to add

Can we add two paths more simply?

Adding paths and the Tamari poset

Theorem (Loday-Ronco)

For any paths p and q,

p + q =⋃

p/q≤r≤p\q

p + q =⋃

p/q≤r≤p\q

There is also a multiplication that refines ordinary integermultiplication. . .

p + q =⋃

p/q≤r≤p\q

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”

Food for thought

We can now do arithmetic:

Food for thought

at the molecular level (whole numbers)

Food for thought

at the atomic level (paths)

Food for thought

at the atomic level (paths)

What about the subatomic level?

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

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