Action graphs and Catalan numbers

Transcription

1 University of California, Riverside March 4, 2015

2 Catalan numbers The Catalan numbers are a sequence of natural numbers C 0, C 1, C 2,... given by the recursive formula C 0 = 1 C k+1 = k C i C k i. i=0

3 Catalan numbers The Catalan numbers are a sequence of natural numbers C 0, C 1, C 2,... given by the recursive formula C 0 = 1 k C k+1 = C i C k i. i=0 So, the first few Catalan numbers are: C 0 = 1 C 1 = C 0 C 0 = 1 C 2 = C 0 C 1 + C 1 C 0 = 2 C 3 = C 0 C 2 + C 1 C 1 + C 2 C 0 = 5 C 4 = C 0 C 3 + C 1 C 2 + C 2 C 1 + C 3 C 0 = 14.

4 Where do Catalan numbers come from? They are known to count at least 207 different kinds of combinatorial objects!

5 Where do Catalan numbers come from? They are known to count at least 207 different kinds of combinatorial objects! C k is the number of ways of parenthesizing k + 1 variables pairwise. (ab)c (a) (ab) a(bc) (ab)(cd) a((bc)d) a(b(cd)) ((ab)c)d (a(bc))d

6 Where do Catalan numbers come from? They are known to count at least 207 different kinds of combinatorial objects! C k is the number of ways of parenthesizing k + 1 variables pairwise. (ab)c (a) (ab) a(bc) (ab)(cd) a((bc)d) a(b(cd)) ((ab)c)d (a(bc))d C k is the number of planar rooted trees with k edges.

7 Directed graphs Our goal is to give a new way to get the Catalan numbers, using a certain family of directed graphs. Recall that a directed graph has vertices and edges, and the edges have a direction. We denote them by an arrow A path in a directed graph is a sequence of edges with compatible direction. In this talk, we ll include trivial paths, which stay at a single vertex.

8 Labeled directed graphs We will also assume that every vertex of our directed graph is labeled by a natural number

9 Action graphs In recent work with Hackney on group actions on groups, we found a sequence of labeled directed graphs which begins as follows. A 0 : 0 A 1 : 0 1 A 2 : A 3 :

10 How are these graphs defined? Begin with the graph A 0 with one vertex, labeled by 0, and no edges. Inductively, we build A k+1 from A k by looking at paths in A k. For every directed path in A k, add a new edge starting at the source of that path and ending at a a new vertex labeled by k + 1.

11 Counting new vertices How many new vertices does each graph A k have?

12 Counting new vertices How many new vertices does each graph A k have? Theorem (Alvarez-B-Lopez, Definition 208?) The number of new vertices in A k is given by C k.

13 Counting new vertices How many new vertices does each graph A k have? Theorem (Alvarez-B-Lopez, Definition 208?) The number of new vertices in A k is given by C k. The idea is the proof is to see that the edges coming out of the vertex labeled by 1 are formed just as the arrows coming out of the vertex labeled by 0 at the previous step, and so forth.

14 Comparison with planar rooted trees Often with new approaches to the Catalan numbers, it is nice to construct a direct comparison to one of the other known ways to obtain them. Theorem (Alvarez-B-Lopez) There is a one-to-one correspondence between new vertices in A k and planar rooted trees with k edges. The idea is to assemble the planar rooted trees in such a way that we obtain the action graph.

15 But where did action graphs come from? In joint work with Hackney, we wanted to find a diagrammatic approach to group actions on other groups. Let s start by looking at what this would mean for groups. Consider the following labeled directed graphs: Let [n] = ( 0 n ). Assign to each [n] a set X n.

16 We want to impose the following conditions: X 0 =, a set with one element. X 1 is any set. X 2 = X1 X 1. More generally, X n = X1 X }{{} 1. n But what does this structure tell us?

17 Start with the set X 1. If X 2 = X1 X 1, we can compose or multiply elements in X 1 with one another. There is a map X 0 X 1 which specifies an identity element. The fact that X 3 = X1 X 1 X 1 tells us that the composition is associative. Thus, we get a monoid structure. Can also incorporate inverses with an additional condition.

18 Groups acting on groups Recall that a group G acts on a set X if there exists a function satisfying some conditions. G X X The idea is that an element of the group takes each element of the set to another element of the set. Here, we want to assume that X is also a group, so that we have a group acting on another group. We d like to find a way to describe this kind of structure via diagrams.

19 Consider the following two directed graphs. 0 The first acts on the second. 1 y z p p g w g x p id x We can define two families of graphs, where the first acts on the second. Some of the graphs in the second family give the action graphs.

20 Further work There are other algebraic structures that can be described by diagrams: abelian monoids/groups, categories, operads. These structures are of interest up to homotopy, when we look at diagrams of spaces. Each of these structures can also be considered with a group action. Are there more examples? Do the diagrams we get have other interesting patterns?

21 References Julia E. Bergner and Philip Hackney, Reedy categories which encode the notion of category actions, to appear in Fund. Math., preprint available at math.at/ Gerardo Alvarez, Julia E. Bergner, and Ruben Lopez, Action graphs and Catalan numbers, preprint available at math.co/