Permutations, Pattern Avoidance, and Catalan Numbers

In this talk, we will discuss an interesting connection between certain permutations of the symmetric group, pattern avoidance, and Catalan numbers. Along the way, we will explore visual representations of permutations in terms string diagrams and heaps, as well as touch on an open problem involving Coxeter groups.

This talk was given by my undergraduate research student Molly Green (Northern Arizona University) on March 1, 2014 at the Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at Mesa Community College, Mesa, AZ.

Transcript

1. Permutations, Pattern Avoidance, and Catalan Numbers Molly Green Supervised by D.C. Ernst Northern Arizona University Mathematics & Statistics Department Southwestern Undergraduate Mathematics Research Conference March 1, 2014 Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 1 / 18
2. What is a Catalan number? Most of you have heard of the Fibonacci numbers, they look like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . The Fibonacci numbers show up a lot in nature, from the arrangement of a pine cone, to the family tree of honeybees! Well, Catalan numbers are pretty cool, too! They provide solutions to a series of combinatorial problems, such as counting the how many parentheses are correctly matched, as well as the number of permutations there are in a special case. Definition For n ≥ 0, the Catalan numbers are defined by Cn = 1 n + 1 2n n = (2n)! (n + 1)!n! The first few Catalan numbers are: 1, 1, 2, 5, 14, 132, 429, 1430, . . . Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 2 / 18
3. Applications of Catalan numbers ()((()())) Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 3 / 18
4. A special type of group Here is a classic example of a group. Definition The symmetric group, Sym(n), is the collection of bijections from {1, 2, . . . , n} to {1, 2, . . . , n} where the operation is function composition (left ← right). Each element rearranges a string of n objects; called a permutation. One example is taking 1 2 3 and rearranging it to 2 1 3 . In this case, the permutation is “swap first and second.” Things to think about: • What is the identity permutation? • Given a permutation, what is its inverse? • How do you compose two permutations? Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 4 / 18
5. String diagrams & cycle notation Two ways of representing elements from Sym(n) are via permutation diagrams and via cycle notation. Example Here are some examples from Sym(5). α = q q q q q q q q q q = (1 2 3 4 5) β = q q q q q q q q q q = (2 4 3) σ = q q q q q q q q q q = (1 3)(2 5 4) γ = q q q q q q q q q q = (1 5) Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 5 / 18
6. String diagrams & cycle notation (continued) Let’s try composing. Example Using diagrams: αβ = β q q q q q q q q q q α q q q q q q q q q q = q q q q q q q q q q βα = α q q q q q q q q q q β q q q q q q q q q q = q q q q q q q q q q Using cycle notation: αβ = (1 2 3 4 5)(2 4 3) = (1 2 5) βα = (2 4 3)(1 2 3 4 5) = (1 4 5) We see that compositions of permutations do not necessarily commute (order matters). Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 6 / 18
7. String diagrams & cycle notation (continued) However, sometimes permutations do commute. Example βγ = (2 4 3)(1 5) = γ q q q q q q q q q q β q q q q q q q q q q = q q q q q q q q q q γβ = (1 5)(2 4 3) = β q q q q q q q q q q γ q q q q q q q q q q = q q q q q q q q q q So, β and γ commute. We’ve stumbled upon the following general fact. Theorem Compositions of disjoint cycles commute. Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 7 / 18
8. Example Consider the following products in S4 : (1 2)(3 4)(2 3)(1 2)(2 3) and (3 4)(2 3)(1 2). It turns out that these are both expressions for the element (1 4 3 2). This is easily verified by just composing each expression. It will be useful for us to have methods for converting one expression into another. It turns out that we only need three “tools.” Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 8 / 18
9. The adjacent 2-cycles There are some ”simple” cycles we can use to build all other permutations. Definition In Sym(n), the adjacent 2-cycles are defined as follows. s1 = (1 2), s2 = (2 3), s3 = (3 4), . . . , sn−1 = (n − 1 n) We use s1 to mean ”swap position 1 with position 2”. Similarly, we use si to mean ”swap position i with position i + 1.” Example The adjacent 2-cycles in Sym(5) are s1, s2, s3 and s4 , or (1 2), (2 3), (3 4), and (4 5). Theorem Every permutation in Sym(n) can be written as a composition of the adjacent 2-cycles. That is, the adjacent 2-cycles generate Sym(n). Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 9 / 18
10. The adjacent 2-cycles (continued) It is important to note that there are potentially many different ways to express a given permutation as a product of adjacent 2-cycles, but we only need a few tools to get from one expression for a permutation to another. Theorem The symmetric group sn is generated by the adjacent 2-cycles subject only to the following relations. 1. (si )2 = (i i + 1)2 = (1) (2-cycles have order two) 2. si sj = sj si → (i i + 1)(j j + 1) = (j j + 1)(i i + 1), where |i − j| > 1 (disjoint cycles commute) 3. si sj si = sj si sj → (i i + 1)(i + 1 i + 2)(i i + 1) = (i + 1 i + 2)(i i + 1)(i + 1 i + 2), where |i − j| = 1 (braid relation) Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 10 / 18
11. Examples and Proofs Picture proof of 1 Order two of 2-cycles: (si )2 = si q q q q si q q q q = q q q q Picture proof of 2 Disjoint cycles commute: si sj = sj q q q q q q q q si q q q q q q q q = q q q q q q q q = si q q q q q q q q sj q q q q q q q q = sj si Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 11 / 18
12. Picture proof of 3 Braid relation: si si+1 si = si q q q q q q si+1 q q q q q q si q q q q q q = q q q q q q = si+1 q q q q q q si q q q q q q si+1 q q q q q q = si+1 si si+1 Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 12 / 18
13. Example Let’s play with an example. s1 s3 s2 s1 s2 = s1 s3 s1 s2 s1 = s1 s3 s1 s2 s1 = s3 s1 s1 s2 s1 = s3 s1 s1 s2 s1 = s3 s2 s1 If we express a permutation as a product of adjacent 2-cycles in the most efficient way possible (i.e., there is not a way to write the product with fewer factors), then we call the expression a reduced expression. Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 13 / 18
14. Equivalence classes & Heaps Example There are 11 reduced expressions for (1 3 5 4) that split into 2 “classes”: s1 s2 s1 s4 s3 s1 s2 s4 s1 s3 s1 s4 s2 s1 s3 s1 s2 s4 s3 s1 s1 s4 s2 s3 s1 s4 s1 s2 s3 s1 s4 s1 s2 s1 s3 s2 s1 s2 s4 s3 s2 s1 s4 s2 s3 s2 s4 s1 s2 s3 s4 s2 s1 s2 s3 We can visually represent each class using heaps. 1 3 2 4 1 1 3 2 2 4 Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 14 / 18
15. Equivalence classes & Heaps (continued) Example Let’s look at an example that has only one heap. (13)(254) = s2 s4 s1 s3 s2 can be represented by 2 1 3 2 4 This is special because it is a unique heap. Oh, by the way, remember those Catalan numbers from the second slide?... Theorem The nth Catalan number corresponds to the number of permutations in Sym(n) with unique heaps. These are the permutations with no opportunity to apply a braid relation. Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 15 / 18
16. The longest element Now, let’s look at the other end of the spectrum. Definition The longest element, denoted by w0 , in Sym(n) is the (unique) element having maximal “length”. The string diagram for w0 is of the form (n = 5 here): r r r r r r r r r r The number of reduced expressions for w0 is known. But what we don’t know is: Open problem How many heaps does the longest element in the Sym(n) have? For a given n, we could work really hard to figure out the answer (the bigger n is, the harder we’d have to work). But what we want is a general solution. A (good) solution would either be a function of n or a recurrence relation. Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 16 / 18
17. Example of heaps for the longest word In Sym(4), the longest element is (1 4)(2 3). In this case, there are 8 distinct heaps. Example 1 1 3 3 2 2 1 1 3 3 2 2 1 1 1 2 2 3 3 1 3 2 2 3 1 1 3 2 2 2 3 3 1 2 2 2 1 1 3 2 2 2 3 3 1 2 2 2 Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 17 / 18
18. Closing remarks Remark According to the On-Line Encyclopedia of Integer Sequences, the number of heaps of the longest element follows this sequence: 1, 1, 2, 8, 62, 908, 24698, 1232944, 112018190, 18410581880, 5449192389984. What is this sequence?!! It’s not known. Acknowledgement I would like to thank: • Organizers of SUnMaRC. • Northern Arizona University • NAU Department of Mathematics and Statistics • Dana C. Ernst • All of you! Molly Green Permutations, Pattern Avoidance, and Catalan Numbers 18 / 18