Double Factorial Binomial Coefficients Mitsuki Hanada Submitted in Partial Fulfillment of the Prerequisite for Honors in the Wellesley College Department of Mathematics under the advisement of Alexander Diesl May 2021 c 2021 Mitsuki Hanada ii Double Factorial Binomial Coefficients Abstract Binomial coefficients are a concept familiar to most mathematics students. In particular, n the binomial coefficient k is most often used when counting the number of ways of choosing k out of n distinct objects. These binomial coefficients are well studied in mathematics due to the many interesting properties they have. For example, they make up the entries of Pascal's Triangle, which has many recursive and combinatorial properties regarding different columns of the triangle. Binomial coefficients are defined using factorials, where n! for n 2 Z>0 is defined to be the product of all the positive integers up to n. One interesting variation of the factorial is the double factorial (n!!), which is defined to be the product of every other positive integer up to n. We can use double factorials in the place of factorials to define double factorial binomial coefficients (DFBCs). Though factorials and double factorials look very similar, when we use double factorials to define binomial coefficients, we lose many important properties that traditional binomial coefficients have. For example, though binomial coefficients are always defined to be integers, we can easily determine that this is not the case for DFBCs. In this thesis, we will discuss the different forms that these coefficients can take. We will also focus on different properties that binomial coefficients have, such as the Chu-Vandermonde Identity and the recursive relation illustrated by Pascal's Triangle, and determine whether there exists analogous results for DFBCs. Finally, we will generalize some of our results regarding the form of DFBCs to m-factorial binomial coefficients for arbitrary m 2 Z>0. Mitsuki Hanada iii Acknowledgements I would like to thank my advisor, Professor Diesl, for guiding me through this journey of a thesis, as well as all the wonderful professors I have had at Wellesley who have shaped me into the mathematician I am today. In particular, I thank Professor Hirschhorn, who has supported me from Math 120 in my first semester to my first graduate level course, Professor Lange and her Math 309 class which helped me discover new mathematical interests, and Professor McAskill of the Physics Department for helping me explore mathematics through the lense of physics. I would also like to thank Professor Chan, Professor Voli´c,and Professor Battat for being a part of my thesis committee and reading through this thesis. I thank my parents who have always let me pursue my intellectual interests and my sisters who let me ramble about math and pretend to understand. I am so lucky to have such a supportive and loving family. I also thank my quarantine roommate Woo Young, for helping me work through many of the proofs and mathematical arguments in this thesis, as well as dealing with my general thesis/grad school application stress. Finally, I would like to thank all of the friends I have made throughout my years at Wellesley. I will always cherish our late night problem set sessions in Clapp and Freeman 4th floor kitchen. iv Double Factorial Binomial Coefficients Contents Abstract ii Acknowledgements iii 1. Background1 1.1. Binomial Coefficients1 1.2. Double Factorials and Double Factorial Binomial Coefficients (DFBCs)2 2. Different forms of DFBCs3 2.1. n even, k odd6 2.2. n odd7 3. Combinatorial Interpretation of DFBCs (Chu-Vandermonde identity)9 4. Pascal's Triangle 13 4.1. Introducing Pascal's Triangle and its properties 13 4.2. Finding similar triangles for certain DFBCs 15 5. Generalization to m-Factorial Binomial Coefficients 19 References 26 Mitsuki Hanada 1 1. Background We will start this thesis by introducing our topic of interest: double factorial binomial coefficients. However, before we do this we must define some basic concepts that help us build the definition of double factorial binomial coefficients. 1.1. Binomial Coefficients. We will first define binomial coefficients and some of their interesting properties. The factorial is a well-known concept in mathematics, where the factorial of a positive integer n denoted (n!) is defined: n! := n(n − 1)(n − 2) ··· (2)(1) and 0! := 1 for n = 0. Recall that these factorials carry some combinatorial meaning: for a nonnegative integer n, it follows that n! denotes the number of permutations of n distinct objects. Though there are many different ways of defining binomial coefficients, the definition we give below involves factorials. n Definition 1.1. For n; k 2 Z≥0 such that 0 ≤ k ≤ n, the binomial coefficient (denoted k ) is defined n n! := : k k!(n − k)! Most people who have taken an introductory combinatorics course will be familiar with binomial coefficients. The binomial coefficient has many inherent combinatorial properties, some that are apparent through alternative definitions. We will give one alternative definition that highlights some of these properties. Definition 1.2. For n; k 2 Z≥0 such that 0 ≤ k ≤ n, the binomial coefficient is defined to be the coefficient of the term xkyn−k in the polynomial expansion n X n (x + y)n = xkyn−k: k k=0 When we choose k many of the (x + y) terms on the left hand side to take the x value from and take the y value from the remaining n − k terms, the resulting product is xkyn−k. n k n−k Therefore the coefficient k of x y must denote the number of ways we can choose k many of the n total (x + y) terms. This is one of the most well known combinatorial properties of the binomial coefficients: n Lemma 1.3. The binomial coefficient k denotes the number of ways to choose k many of the n total distinct objects. n Since we have that k counts the number of cases, this implies that these binomial coef- ficients will always be integer values. This is in fact true and another well known property of binomial coefficients. n Lemma 1.4. For any n; k 2 Z≥0 such that 0 ≤ k ≤ n, it follows that k 2 Z>0: 2 Double Factorial Binomial Coefficients There are many different ways of proving this statement using combinatorial, algebraic, or number theory properties. (Many of them are mentioned in [2].) We will mention one method of proving this in Section2. We can see another well known property of binomial coefficients from both of these defi- nitions. Lemma 1.5. Binomial coefficients are symmetric: for n; k such that 0 ≤ k ≤ n, the equation n n k = n−k always holds. Proof. This follows from the fact that both definitions we have given depend symmetrically on both k and n − k. Using Definition 1.1, we can observe the following: n n! n! n = = = : k k!(n − k)! (n − (n − k))!(n − k)! n − k 1.2. Double Factorials and Double Factorial Binomial Coefficients (DFBCs). Now that we have discussed different properties of binomial coefficients, we will define our topic of interest: double factorial binomial coefficients. The definition of double factorial binomial coefficients utilizes double factorials instead of the standard factorial. Definition 1.6. The double factorial of a non-negative integer n (denoted n!!) is defined 8 n(n − 2)(n − 4) ··· (4)(2) for n even, n > 0 <> n!! := n(n − 2)(n − 4) ··· (3)(1) for n odd :>1 for n = 0: These double factorials have many interesting properties. In [3], the authors discuss how certain double factorials have an enumerative combinatorial meaning. For example, the two colored permutation of the set [n] = f1; 2; : : : ; ng for positive integer n is defined to be a permutation of [n] where each of the entries are colored by one of two colors. The even double factorial (2n)!! = 2n(n)! denotes the number of possible two colored permutations of [n]. Another example mentioned is for the odd double factorials (2n − 1)!!, which represents the number of matchings in the set [2n] = f1; 2;:::; 2ng, where a matching of [2n] is defined to be a partition of [2n] into n unordered pairs. There are also other combinatorial interpretations of double factorials, like those mentioned in [1]. We can also simplify the double factorial n!! using a \standard" factorial for certain choices of n: Lemma 1.7. For an even positive integer 2m, we have (2m)!! = 2m(m)!. Proof. This comes from simple manipulation of the the terms. Note that we can write (2m)!! = (2m)(2m − 2)(2m − 4) ··· (2): Mitsuki Hanada 3 Note that there are m many terms on the right hand side. If we factor out a factor of 2 from each of them, we get (2m)!! = (2m)(2(m − 1))(2(m − 2)) ··· (2) = 2m(m)(m − 1)(m − 2) ··· 1 = 2m(m!): Lemma 1.7 will be useful to us later when we are trying to prove results about prime factors of certain double factorials. Now using these double factorials and the definition of the binomial coefficients, we can finally define double factorial binomial coefficients. Definition 1.8. For n; k 2 Z≥0 such that 0 ≤ k ≤ n, the double factorial binomial coefficient n (DFBC) (denoted k ) is defined n n!! := : k k!!(n − k)!! n n Recall from Lemma 1.5 that binomial coefficients are symmetric: that is, k = n−k . By definition it is evident that this property also holds for DFBCs. Lemma 1.9. Double factorial binomial coefficients are symmetric: for any n; k such that n n 0 ≤ k ≤ n, the equation k = n−k always holds.