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Action graphs and Catalan numbers
Julie Bergner
University of California, Riverside
March 4, 2015
Julie Bergner Action graphs and Catalan numbers
Catalan numbers
The Catalan numbers are a sequence of natural numbers
C0,C1,C2, . . .
given by the recursive formula
C0 = 1
Ck+1 =k∑
i=0
CiCk−i .
So, the first few Catalan numbers are:
C0 = 1
C1 = C0C0 = 1
C2 = C0C1 + C1C0 = 2
C3 = C0C2 + C1C1 + C2C0 = 5
C4 = C0C3 + C1C2 + C2C1 + C3C0 = 14.
Julie Bergner Action graphs and Catalan numbers
Catalan numbers
The Catalan numbers are a sequence of natural numbers
C0,C1,C2, . . .
given by the recursive formula
C0 = 1
Ck+1 =k∑
i=0
CiCk−i .
So, the first few Catalan numbers are:
C0 = 1
C1 = C0C0 = 1
C2 = C0C1 + C1C0 = 2
C3 = C0C2 + C1C1 + C2C0 = 5
C4 = C0C3 + C1C2 + C2C1 + C3C0 = 14.
Julie Bergner Action graphs and Catalan numbers
Where do Catalan numbers come from?
They are known to count at least 207 different kinds ofcombinatorial objects!
Ck is the number of ways of parenthesizing k + 1 variablespairwise.
(a)
(ab)
(ab)c a(bc)
(ab)(cd) a((bc)d) a(b(cd)) ((ab)c)d (a(bc))d
Ck is the number of planar rooted trees with k edges.
Julie Bergner Action graphs and Catalan numbers
Where do Catalan numbers come from?
They are known to count at least 207 different kinds ofcombinatorial objects!
Ck is the number of ways of parenthesizing k + 1 variablespairwise.
(a)
(ab)
(ab)c a(bc)
(ab)(cd) a((bc)d) a(b(cd)) ((ab)c)d (a(bc))d
Ck is the number of planar rooted trees with k edges.
Julie Bergner Action graphs and Catalan numbers
Where do Catalan numbers come from?
They are known to count at least 207 different kinds ofcombinatorial objects!
Ck is the number of ways of parenthesizing k + 1 variablespairwise.
(a)
(ab)
(ab)c a(bc)
(ab)(cd) a((bc)d) a(b(cd)) ((ab)c)d (a(bc))d
Ck is the number of planar rooted trees with k edges.
Julie Bergner Action graphs and Catalan numbers
Directed graphs
Our goal is to give a new way to get the Catalan numbers, using acertain family of directed graphs.
Recall that a directed graph has vertices and edges, and the edgeshave a direction. We denote them by an arrow.
• // • //
��
•
• //
??~~~~~~~•
A path in a directed graph is a sequence of edges with compatibledirection.
In this talk, we’ll include “trivial paths”, which stay at a singlevertex.
Julie Bergner Action graphs and Catalan numbers
Labeled directed graphs
We will also assume that every vertex of our directed graph islabeled by a natural number.
•0 // •1 //
��
•3
•2 //
>>||||||||•3
Julie Bergner Action graphs and Catalan numbers
Action graphs
In recent work with Hackney on group actions on groups, we founda sequence of labeled directed graphs which begins as follows.
A0 : •0 A1 : •0 // •1
A2 : A3 : •3 •3
•2 •2 •2
OO
•2
OO
•0
OO
// •1
OO
•3 •0
OO
��
oo // •1
OO
��•3 •3
Julie Bergner Action graphs and Catalan numbers
How are these graphs defined?
Begin with the graph A0 with one vertex, labeled by 0, and noedges.
Inductively, we build Ak+1 from Ak by looking at paths in Ak .
For every directed path in Ak , add a new edge starting at thesource of that path and ending at a a new vertex labeled by k + 1.
Julie Bergner Action graphs and Catalan numbers
Counting new vertices
How many new vertices does each graph Ak have?
Theorem (Alvarez-B-Lopez, Definition 208?)
The number of new vertices in Ak is given by Ck .
The idea is the proof is to see that the edges coming out of thevertex labeled by 1 are formed just as the arrows coming out of thevertex labeled by 0 at the previous step, and so forth.
Julie Bergner Action graphs and Catalan numbers
Counting new vertices
How many new vertices does each graph Ak have?
Theorem (Alvarez-B-Lopez, Definition 208?)
The number of new vertices in Ak is given by Ck .
The idea is the proof is to see that the edges coming out of thevertex labeled by 1 are formed just as the arrows coming out of thevertex labeled by 0 at the previous step, and so forth.
Julie Bergner Action graphs and Catalan numbers
Counting new vertices
How many new vertices does each graph Ak have?
Theorem (Alvarez-B-Lopez, Definition 208?)
The number of new vertices in Ak is given by Ck .
The idea is the proof is to see that the edges coming out of thevertex labeled by 1 are formed just as the arrows coming out of thevertex labeled by 0 at the previous step, and so forth.
Julie Bergner Action graphs and Catalan numbers
Comparison with planar rooted trees
Often with new approaches to the Catalan numbers, it is nice toconstruct a direct comparison to one of the other known ways toobtain them.
Theorem (Alvarez-B-Lopez)
There is a one-to-one correspondence between new vertices in Ak
and planar rooted trees with k edges.
The idea is to assemble the planar rooted trees in such a way thatwe obtain the action graph.
Julie Bergner Action graphs and Catalan numbers
But where did action graphs come from?
In joint work with Hackney, we wanted to find a diagrammaticapproach to group actions on other groups.
Let’s start by looking at what this would mean for groups.
Consider the following labeled directed graphs:
•0 •0 //•1 •0 //•1 //•2 · · ·
Let [n] = (•0 → · · · → •n).
Assign to each [n] a set Xn.
Julie Bergner Action graphs and Catalan numbers
We want to impose the following conditions:
X0 = ∗, a set with one element.
X1 is any set.
X2∼= X1 × X1.
More generally, Xn∼= X1 × · · · × X1︸ ︷︷ ︸
n
.
But what does this structure tell us?
Julie Bergner Action graphs and Catalan numbers
Start with the set X1.
If X2∼= X1 × X1, we can “compose” or “multiply” elements in X1
with one another.
There is a map X0 → X1 which specifies an “identity element”.
The fact that X3∼= X1 × X1 × X1 tells us that the composition is
associative.
Thus, we get a monoid structure.
Can also incorporate inverses with an additional condition.
Julie Bergner Action graphs and Catalan numbers
Groups acting on groups
Recall that a group G acts on a set X if there exists a function
G × X → X
satisfying some conditions.
The idea is that an element of the group takes each element of theset to another element of the set.
Here, we want to assume that X is also a group, so that we have agroup acting on another group.
We’d like to find a way to describe this kind of structure viadiagrams.
Julie Bergner Action graphs and Catalan numbers
Consider the following two directed graphs.
1 y z
∗ 0
p
OO
wg //
p•gOO
x
p•idx
OO
The first acts on the second.
We can define two families of graphs, where the first acts on thesecond.
Some of the graphs in the second family give the action graphs.
Julie Bergner Action graphs and Catalan numbers
Further work
There are other algebraic structures that can be described bydiagrams: abelian monoids/groups, categories, operads.
These structures are of interest up to homotopy, when welook at diagrams of spaces.
Each of these structures can also be considered with a groupaction.
Are there more examples?
Do the diagrams we get have other interesting patterns?
Julie Bergner Action graphs and Catalan numbers
References
Julia E. Bergner and Philip Hackney, Reedy categories whichencode the notion of category actions, to appear in Fund.Math., preprint available at math.AT/1207.3467.
Gerardo Alvarez, Julia E. Bergner, and Ruben Lopez, Actiongraphs and Catalan numbers, preprint available atmath.CO/1503.00044.
Julie Bergner Action graphs and Catalan numbers