A LATTICE POINT ENUMERATION APPROACH TO PARTITION IDENTITIES

Transcription

1 A LATTICE POINT ENUMERATION APPROACH TO PARTITION IDENTITIES A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree Master of Arts In Mathematics by Nguyen Hong Le San Francisco, California August 2010

2 Copyright by Nguyen Hong Le 2010

3 CERTIFICATION OF APPROVAL I certify that I have read Integer Partitions by George E Andrews and Kimmo Eriksson and that in my opinion this work meets the criteria for approving a thesis submitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University Matthias Beck Professor of Mathematics Federico Ardila Professor of Mathematics Serkan Hosten Professor of Mathematics

4 A LATTICE POINT ENUMERATION APPROACH TO PARTITION IDENTITIES Nguyen Hong Le San Francisco State University 2010 In this paper, we present a novel method to find generating functions of partition identities Our method is based on integer-point enumeration in polyhedra We show how lattice-point enumeration can be applied to partition identity theorems that were proved using MacMahon s Ω-operator, and establish the full generating functions of these theorems In addition to introducing our new method, we establish connections between the different mathematic areas of Geometric Combinatorics and Number Theory I certify that the Abstract is a correct representation of the content of this thesis Chair, Thesis Committee Date

5 ACKNOWLEDGMENTS I would like to thank my awesome advisor, Dr Matthias Beck for his direction, assistance and guidance I also wish to thank my committee members Dr Serkan Hosten and Dr Federico Ardila for serving on my Thesis Committee Thanks are also due to Ngan Le, my sister, for her assistance Special thanks should be given to my parents who supported me in many forms Finally, words alone cannot express the thanks I owe to my husband, Kennard Ngo, for his encouragement, unconditional support and love throughout my Master s career v

6 TABLE OF CONTENTS 1 Introduction 1 11 Leibniz and Euler 1 12 Goal of This Paper 2 2 Partition Functions and Ω Theorems 3 21 Partition Functions 3 22 Generating Functions 4 23 Ω Theorems k-gon Partitions Partitions with Difference Conditions Partitions with Higher Order Difference Conditions Partitions With Mixed Difference Conditions 10 3 Polyhedra The Language of Cones Integer-Point Transforms for Rational Cones 13 4 Geometric Proofs of Ω Theorems Integer-Sided Triangles of Perimeter n Geometric Proof of Theorem Geometric Proof of Theorem Geometric Proofs of Theorem vi

7 45 Geometric Proof Of Theorem Bibliography 39 vii

8 LIST OF FIGURES 31 The simplicial cone K = {(0, 0) + λ 1 ( 2, 3) + λ 2 (1, 1) : λ 1, λ 2 0} The cone K and its fundamental parallelogram The cone K = {(λ 1 (0, 1, 1)+λ 2 (1, 1, 1)+λ 3 (1, 1, 2) : λ 1, λ 3 0 and λ 2 > 0} 19 viii

9 Chapter 1 Introduction 11 Leibniz and Euler According to [2], Leibniz ( ) was the first person who was asking a question about the number of partitions of integers Leibniz observed that there are three partitions of 3 (3, 2+1, and 1+1+1), five partitions of 4, seven partitions of 5, and eleven partitions of 6 These beginnings opened the field of partitions On September 4, 1740, Naude ( ) wrote Euler ( ) to ask how many partitions there are of 50 into seven distinct parts The correct answer is 522 [2] However, it is not likely to be obtained by writing out all the ways of adding seven distinct positive integers to get 50 To solve this problem Euler introduced generating functions, arguably the most important innovation in the history of partitions We can find the use of generating functions in the theory of partitions in [3, Ch 13], [4, Chs 1 1

10 2 and 2], [6, Ch 5], and later in this paper 12 Goal of This Paper This paper presents a novel method for finding generating functions for various forms of partitions In Chapter 2 we introduce the definitions of these objects and highlight the utility of generating functions We also give some theorems of partition identities which Andrews et al proved in [5] and [7] using the Ω-operator [8] Our method is based on integer-point enumeration in polyhedra [9] Chapter 3 provides the mathematical background for this method We start with the language of cones in term of the affine structure of R d and integer-point transforms for rational cones We introduce theorems and a lemma which help us to obtain the generating functions for the Ω theorems of Chapter 2 The main results of this paper appear in Chapter 4 We will reprove and extend the theorems of Chapter 2 using our lattice-point enumeration approach In particular, Chapter 4 provides the full generating functions of the theorems of Chapter 2 The motivation for this paper is to shed new lights on known theorems We hope that the method that we have used will establish further connections between geometric combinatorics and number theory

11 Chapter 2 Partition Functions and Ω Theorems 21 Partition Functions A partition of a positive integer n ( or a partition of weight n ) is a non-increasing sequence λ = (λ 1, λ 2,, λ k ), where the λ is are non-negative integers such that k λ i = n The λ is are the parts of the partition λ i=1 Example 21 The partitions of 5 are: (5), (4, 1), (3, 2), (3, 1, 1), (2, 2, 1), (2, 1, 1, 1), and (1, 1, 1, 1, 1) In particular, the number of partitions of 5 is 7 There are various forms of partitions For example, we can have partitions of at most k parts, partitions into odd parts, partitions into distinct parts, and so on Example 22 The partitions of 5 into odd parts are: (5), (3, 1, 1), and (1, 1, 1, 1, 1) Thus, the number of partitions of 5 into odd parts is 3 3

12 4 22 Generating Functions Generating functions form a tool to deal with partitions Let {a k } k=0 be an infinite sequence The generating function of the sequence a k is a function F (x) expressed as the formal power series F (x) = a k x k k=0 Example 23 A key generating function is the one for the constant sequence 1, 1, 1, 1, 1,, namely F (x) = x k = 1 1 x k=0 Generating functions are very useful in that the degree of each monomial keeps track of the position in the sequence while the coefficient provides the actual value of the term Then if we play by the rules of either formal or analytic power series, we may be able to derive results The following is one of the examples of the power of generating functions provided in [9, p 3] Consider the classic example of the Fibonacci sequence f k, named after Leonardo Pisano Fibonacci ( ) and defined by the recursion f 0 = 0, f 1 = 1, and f k+2 = f k+1 + f k for k 0 This gives the sequence {f k } k=0 = (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ) Let F (z) = k 0 f k z k

13 5 We embed both sides of the recursion identity into their generating functions: f k+2 z k = (f k+1 + f k )z k = f k+1 z k + f k z k (21) k 0 k 0 k 0 k 0 Then the left-hand side of (21) is f k+2 z k = 1 f z 2 k+2 z k+2 = 1 f z 2 k z k = 1 (F (z) z), z2 k 0 k 0 k 2 and the right-hand side of (21) is f k+1 z k + k 0 k 0 f k z k = 1 F (z) + F (z) z So (21) can be restated as 1 z (F (z) z) = 1 F (z) + F (z) 2 z Solving for F (z) we obtain F (z) = z 1 z z 2 F (z) has the following partial fraction expansion F (z) = z 1 z z = 1/ z 1/ z 2 2

14 6 Now, we use the well known geometric series from Example 23 with x = z and x = z, respectively We obtain ( z F (z) = 1 z z = z k 0 ( 1 5 = k ) k 1 5 ) k ( k ( 1 5 z 2 ) k ) k This provides the desired closed-form expression for the Fibonacci sequence ( f k = ) k ( ) k 5 2 The rational function obtained using the properties of geometric series is called a rational generating function Often we will jump back and forth from generating functions to rational generating functions Furthermore, we can obtain interesting results about partitions by using generating functions Throughout this paper we will see the utility of generating functions in establishing relations between various partition functions 23 Ω Theorems In this section, we introduce some partition theorems which Andrews et al established using the Ω-operator in [5] and [7] First, we introduce some definitions

15 7 231 k-gon Partitions Definition 21 [7, Definition 2] As the set of non-degenerate k-gon partitions into positive parts we define τ k := {(a 1,, a k ) Z k : 1 a 1 a 2 a k and a a k 1 > a k } As the set of non-degenerate k-gon partitions of n into positive parts we define υ k (n) := {(a 1,, a k ) τ k : a a k = n} The corresponding cardinality is denoted by t k (n) := υ k (n) The term non-degenerate refers to the restriction to strict inequality, ie to a 1 + a a k 1 > a k Definition 22 [7, Definition 3] For an integer k 0, let T k (q) := n k t k (n)q n, and S k (x 1,, x k ) := x a1 1 x a k (a 1,,a k ) τ k k

16 8 Theorem 21 [7, Theorem 1] Let k 3 and X i = x i x k for 1 i k Then S k (x 1,, x k ) = X 1 (1 X 1 )(1 X 2 ) (1 X k ) X 1X k 2 k 1 1 X k (1 X k 1 )(1 X k 2 X k )(1 X k 3 Xk 2) (1 X 1X k 2 k ) 232 Partitions with Difference Conditions Theorem 22 [5, Theorem 31] Let m (n) denote the number of partitions of n into 2m + 1 nonnegative parts n = a 1 + a 2 + a a 2m+1, where the parts are listed in non-increasing order and additionally a 1 a 2 a 3 + a 4 0, a 3 a 4 a 5 + a 6 0, a 2m 3 a 2m 2 a 2m 1 + a 2m 0, a 2m 1 a 2m a 2m+1 0

17 9 Then m (n) equals the number of partitions of n into parts that are either 2m and even or of the form (j + 1)(2m + 1 j) with 0 j m 233 Partitions with Higher Order Difference Conditions For the following theorem, we will need to define triangular numbers A triangular number is the number of dots we need to make triangles These are the first few triangular numbers The following image is provided in [1] The way to get these numbers without drawing pictures is to add up all the numbers that come before a certain number For example, the tenth triangular number is By this method, a triangular number is, equivalently, the sum of the natural numbers from 1 to n, and the nth triangular number, T n, is T n = (n 1) + n = n(n + 1) 2 ( ) n + 1 = 2 Theorem 23 [5, Theorem 41] Let p 2 (n) denote the number of partitions of n of the form a 1 + a a s (s arbitrary) where the first differences are nonnegative, ie, a i a i+1 0 for 1 i s 1, and the the second differences are nonnegative, ie, a i 2a i+1 + a i+2 0 for 1 i s 1 (assuming a s+1 = 0) Then p 2 (n) equals

18 10 the number of partitions of n into triangular numbers 234 Partitions With Mixed Difference Conditions Theorem 24 [5, Theorem 51] Let p ± (m, n) denote the number of partitions of n of the form a 1 + a a m, wherein a i a i+1 0 for 1 i m 1, while a i 2a i+1 + a i+2 0 for 1 i m 1 (with a m+1 = 0) Then p ± (m, n) equals the number of partitions of n of the form (m j)(m + j + 1)/2 where 0 j m Note that Theorems constituted the main content of [5] and [7] In Chapter 4, we will prove and extend the above theorems using lattice-point enumeration

19 Chapter 3 Polyhedra 31 The Language of Cones Let a R d and b R Then a hyperplane is a set of the form {x R d : a 1 x 1 +a 2 x 2 + +a d x d = b} and a halfspace is a set of the form {x R d : a x b} A pointed cone K R d is a set of the form K = {v + λ 1 w 1 + λ 2 w λ m w m : λ 1, λ 2,, λ m 0}, where v, w 1, w 2,, w m R d are such that there exists a hyperplane H for which H K = {v} The vector v is called the apex of K and each w i is called a generator K is said to be rational if all of its generators and apex are rational The dimension of K is the dimension of the affine space spanned by K; if K is 11

20 12 of dimension d, we call it a d-cone A d-cone K is said to be simplicial if it has exactly d linearly independent generators Figure 31: The simplicial cone K = {(0, 0) + λ 1 ( 2, 3) + λ 2 (1, 1) : λ 1, λ 2 0} Figure 31 shows the hyperplanes 3x + 2y = 0, x y = 0 and the cone K is created by the intersection of two halfspaces 3x + 2y 0 and x y 0 We say that the hyperplane H = {x R d : a x = b} is a supporting hyperplane of the pointed d-cone K if K lies entirely on one side of H, that means, K {x R d : a x b} or K {x R d : a x b}

21 13 A face of K is a set of the form K H, where H is a supporting hyperplane of K The (d 1)-dimensional faces are called facets and the 1-dimensional faces are called edges and the apex of K is its unique 0-dimensional face 32 Integer-Point Transforms for Rational Cones For a cone K R n, let σ K (z) = σ K (z 1, z 2,, z d ) = m K Z d z m, with the usual monomial notation z m = z m 1 1 z m 2 2 z mn The generating function σ K lists all integer points in K in a special form: not as a list of vectors, but as a formal sum of monomials For example, the integer point (3, 4) would be listed as the monomial z 3 1z 4 2 We call σ K the integer-point transform of K; the function σ K also goes by the name moment generating function or simply generating function of K Now we are ready to state the following theorem, which helps us to find the generating function of a simplicial cone Theorem 31 [9, Theorem 35] Let K be an n-dimensional, rational, simplicial cone with generators, w 1, w 2,, w n Z n Then n σ K (z) = σ πk (z) (1 z w 1 )(1 z w 2) (1 z w n)

22 14 where π K is the half-open parrallelepiped π K := {λ 1 w 1 + λ 2 w λ n w n : 0 λ 1, λ 2,, λ n < 1} Example 31 Find S(z 1, z 2 ) = z m 1 1 z m 2 2m 1 m 2,2m 2 m 1 2, where the sum ranges over all pairs (m 1, m 2 ) of non-negative integers satisfying the indicated inequalities Figure 32: The cone K and its fundamental parallelogram Proof We see that all the integer points (m 1, m 2 ) satisfying the indicated inequalities lie on the intersection of the halfspaces m 1 2m 2 and m 2 2m 1, as shown in Figure 32 This intersection creates the two-dimensional simplicial cone K with

23 15 generators (2, 1) and (1, 2) Hence, K = {λ 1 (1, 2) + λ 2 (2, 1) : λ 1, λ 2 0} and π K = {λ 1 (1, 2) + λ 2 (2, 1) : 1 > λ 1, λ 2 0} Applying Theorem 31, we obtain σ K (z 1, z 2 ) = σ πk (z 1, z 2 ) (1 z 1 z 2 2)(1 z 2 1z 2 ) Figure 32 shows π K Z 2 = {(0, 0), (1, 1), (2, 2)} This implies π K = 1 + z 1 z 2 + z 2 1z 2 2 Therefore, S(z 1, z 2 ) = σ K (z 1, z 2 ) = 1 + z 1z 2 + z 2 1z 2 2 (1 z 2 1z 2 )(1 z 1 z 2 2) The following lemma is a well-known result in lattice-point enumeration This version was formulated in [10] for easy application to partition and composition enumeration problems Lemma 32 Let C = [c i,j ] be an n n matrix of integers such that C 1 = B = [b i,j ] exists and b i,j are all nonnegative integers Let e 1,, e n be nonnegative integers

24 16 For each 1 i n, let c i be the constraint c i,1 λ 1 + c i,2 λ c i,n λ n e i Let S C be the set of nonnegative integer sequences λ = (λ 1, λ 2, λ n ) satisfying the constraints c i for all i, 1 i n Then the generating function for S C is: F C (x 1, x 2,, x n ) = λ S C x λ 1 1 x λ 2 2 x λn n = n (x b 1,j j=1 1 x b 2,j n (1 x b 1,j j=1 2 x b n,j n 1 x b 2,j ) e j n ) 2 x b n,j

25 Chapter 4 Geometric Proofs of Ω Theorems This chapter contains proofs of the Ω theorems that are listed in Chapter 2, using lattice-point enumeration Before proving the Ω theorems, we start with an elementary problem 41 Integer-Sided Triangles of Perimeter n The following has been posed as a problem and solved using the Ω-operator in [7, Problem 1] We want to show that the problem can be solved using lattice-point enumeration Problem: Let t 3 (n) be the number of non-congruent triangles whose sides have integer length and whose perimeter is n For instance, t 3 (9) = 3, corresponding to , , Find t 3 (n)q n n 3 17

26 18 The corresponding generating function is T 3 (q) := n 3 t 3 (n)q n = q a 1 +a 2 +a 3, where is the restricted summation over all positive integer triples (a 1, a 2, a 3 ) satisfying a 1 a 2 a 3 1 and a 1 + a 2 > a 3 In other words, we want to find all partitions of n of the form a 1 + a 2 + a 3, where 1 a 1 a 2 a 3 and a 1 + a 2 > a 3 Proof Figure 41 shows all integer points (a 1, a 2, a 3 ) that satisfy the conditions above lie in the cone K = {(λ 1 (0, 1, 1) + λ 2 (1, 1, 1) + λ 3 (1, 1, 2) : λ 1, λ 3 0 and λ 2 > 0}, which is the intersection of three halfspaces: a 1 a 2, a 2 a 3 and a 1 + a 2 > a 3 This implies π K = {(λ 1 (0, 1, 1) + λ 2 (1, 1, 1) + λ 3 (1, 1, 2) : 1 > λ 1, λ 3 0 and 1 λ 2 > 0}, and π K Z 3 = {(1, 1, 1)} Using Theorem 31, we obtain σ K (x 1, x 2, x 3 ) = x 1 x 2 x 3 (1 x 1 x 2 )(1 x 1 x 2 x 2 )(1 x 2 1x 2 x 3 ) Note that σ K (x 1, x 2, x 3 ) = S 3 (x 1, x 2, x 3 ) in the language of Theorem 21

27 19 of K: Since n = a 1 + a 2 + a 3, we let q = x 1 = x 2 = x 3 Hence, the generating function σ K (q, q, q) = T 3 (q) = n 3 t 3 (n)q n = q x 1 +x 2 +x 3 = q 3 (1 q 2 )(1 q 3 )(1 q 4 ) (41) Figure 41: The cone K = {(λ 1 (0, 1, 1)+λ 2 (1, 1, 1)+λ 3 (1, 1, 2) : λ 1, λ 3 0 and λ 2 > 0}

28 20 Now we consider Theorem 21, the generalization of the triangle problem to k- gons, where k 3, which was proved using the Ω-operator in [7, Theorem 1] In the following section, with the lattice-point enumeration method in hand, we are able to reprove this main result for k-gon partitions 42 Geometric Proof of Theorem 21 In Section 41, we computed the generating functions T 3 (q) = n 3 t 3(n)q n and S 3 (x 1, x 2, x 3 ) Our goal in this section is to compute the generating function where S k (x 1, x 2,, x k ) = x a1 1 x a k (a 1,a 2,,a k ) τ k k, τ k = {(a 1, a 2,, a k ) Z k : a k a k 1 a 1 > 0 and a 1 +a 2 + +a k 1 > a k } Proof of Theorem 21 Let K := {(a 1, a 2,, a k ) Z k : a k a k 1 a 1 > 0} and P := {(a 1, a 2,, a k ) Z k : a k a k 1 a 1 > 0 and a 1 +a 2 + +a k 1 a k }

29 21 We see that τ k = K \ P The constraints of K are given by the system a 1 a 2 a k 1 a k (42) Now let C be the matrix on the left side of (42) Then det(c) = 1, which implies that C is invertible, and C 1 = We see that the integer entries of C 1 are all nonnegative, and Lemma 32 gives the generating function of K : σ K (x 1, x 2,, x k ) = X 1 (1 X 1 )(1 X 2 ) (1 X k )

30 22 Recall that X i = x i x k for 1 i k The constraints of P are given by the system a 1 a 2 a 3 a k 1 a k (43) Since x k 1 x k 2 x 1 1 and x k x 1 + x x k 1, this implies x k x k 1 Therefore, we do not need the condition x k x k 1 Now, let D be the matrix on the left side of (43) Then det(d) = 1, and D 1 = k k 1 k 2 k Again, we see that the integer entries of D 1 are all nonnegative, and Lemma 32

31 23 gives the generating function of P : σ P (x 1, x 2,, x k ) = X 1 X k 2 k (1 X k )(1 X k 1 )(1 X k 2 X k )(1 X k 3 X 2 k ) (1 X 1X k 2 k ) We have τ k = K \ P, and so S k (x 1, x 2,, x k ) = σ K (x 1, x 2,, x k ) σ P (x 1, x 2,, x k ) = X 1 (1 X 1 )(1 X 2 ) (1 X k ) X 1 X k 2 k (1 X k )(1 X k 1 )(1 X k 2 X k )(1 X k 3 Xk 2) (1 X 1X k 2 k ) To show that our method can be applied for other cases of partition identities, we will reprove Theorem 22 using our method in the following section 43 Geometric Proof of Theorem 22 In this section, our goal simplifies to computing the generating function of m (n): σ K (q) := m (n)q n = q a 1+a 2 + +a 2m+1, n 0 (a 1,a 2,,a 2m+1 ) K where K := {(a 1, a 2,, a 2m+1 ) Z 2m+1 : a 1 a 2 0 and

32 24 a 1 a 2 a 3 + a 4 0, a 3 a 4 a 5 + a 6 0, a 2m 3 a 2m 2 a 2m 1 + a 2m 0, a 2m 1 a 2m a 2m+1 0} Proof of Theorem 22 We see that a 1 a 2 a 3 + a 4 0 a 1 + a 4 a 2 + a 3 This implies a 3 a 4 because a 1 a 2 Similarly, a 2m 3 a 2m 2 a 2m 1 + a 2m 0 a 2m 3 +a 2m a 2m 2 +a 2m 1 This implies a 2m 1 a 2m, and a 2m 1 a 2m a 2m+1 0 a 2m 1 a 2m + a 2m+1 This condition guarantees that a 2m+1 0 because

33 25 a 2m 1 a 2m Therefore, the constraints of K are given by the system a 1 a 2 a 3 a 2m a 2m+1 0 (44)

34 26 Let A be the matrix on the left side of (44) Then det A = =

35 27 Adding all odd-numbered rows together we get det A =

36 28 Thus, det A = 1, and A 1 = m m m m m m m m m m m 1 m m m 1 m The integer entries of A 1 are all nonnegative, and Lemma 32 gives the generating function of K as σ K (x 1, x 2,, x 2m+1 ) = 1 (1 x m+1 1 x m 2 x 2m+1 )(1 x 1 x 2 ) (1 x 1 x 2 x 2m+1 ) Now let q = x 1 = x 2 = = x 2m+1, we get σ K (q, q,, q) as 1 (1 q 2 ) (1 q 2m )(1 q 2m+1 )(1 q 4m )(1 q 3(2m 1) ) (1 q m(m+2) )(1 q (m+1)(m+1) )

37 29 Hence, the generating function we set out to find is σ K (q) = m j=1 1 (1 q 2j ) m i=0 1 (1 q (i+1)(2m+1 i) ), and this is precisely the generating function for the partitions described in Theorem 22 Note that we actually derived the full generating function of Theorem 22 Theorem 41 For an integer m 0, let σ(x 1, x 2,, x 2m+1 ) := x a 1 1 x a 2 2 x a 2m+1 2m+1, where is the restricted summation over all nonnegative integers (a 1, a 2,, a 2m+1 ) satisfying a 1 a 2 0 and a 1 a 2 a 3 + a 4 0, a 3 a 4 a 5 + a 6 0, a 2m 3 a 2m 2 a 2m 1 + a 2m 0, a 2m 1 a 2m a 2m+1 0

38 30 Let X i = x 1 x i for 1 i 2m + 1 Then σ(x 1, x 2,, x 2m+1 ) = 1 (1 X 2 ) (1 X 2m )(1 X 1 X 3 X 5 X 2m+1 )(1 X 3 X 5 X 2m+1 ) (1 X 2m+1 ) Once we have seen the results of Theorems 22 and 41, it is natural to consider a variety of partition identities related to further difference conditions 44 Geometric Proofs of Theorem 23 In this section, we give a novel proof of Theorem 23 This theorem requires us to prove that p 2 (n) equals the number of partitions of n into triangular numbers Proof of Theorem 23 The generating function of p 2 (n) is p 2 (n)q n = q a 1+a 2 + +a s, n 0 (a 1,a 2,,a s) K where K := (a 1, a 2,, a s ) Z s : a 1 a 2 a s 0 and a i 2a i+1 + a i+2 0 for 1 i s 1 and a s+1 = 0

39 31 Claim: The conditions a i 2a i+1 + a i+2 0 for 1 i s 1 and a s+1 = 0 guarantee that a i a i+1 for 1 i s 1 Proof of Claim: We will use induction on s Base case: If s = 1 then a 1 0 If s = 2, we have a 1 2a 2 0, this implies a 1 2a 2 Thus a 1 a 2 Induction step: Assume the claim is true for s 1, ie, if a i 2a i+1 + a i+2 0 for 1 i s 2, then a 1 a 2 a s 1 We want to prove that it is also true for s The condition a i 2a i+1 + a i+2 0 implies a s 1 2a s Therefore, a s 1 a s Next, a s 2 2a s 1 + a s 0 a s 2 + a s 2a s 1 We have shown that a s 1 a s Thus a s 2 a s 1 Now we use the induction step for (a 1, a 2,, a s 1 ), and the claim is proven Thus, the conditions for K can be simplified: K := (a 1, a 2,, a s ) Z s : a s 0 and a i 2a i+1 + a i+2 0 for 1 i s 1 and a s+1 = 0

40 32 Therefore, the constraints of K are given by the system a 1 a 2 a 3 a s 2 a s 1 a s 0 (45) Let A be the matrix on the left side of (45) Then det(a) = 1, and A 1 = s 3 s 2 s 1 s s 4 s 3 s 2 s s 5 s 4 s 3 s The integer entries of A 1 are all nonnegative Thus, Lemma 32 gives the generating

41 33 function of K: σ K (x 1, x 2,, x s ) = = 1 (1 x 1 )(1 x 2 1x 2 ) (1 x s 1x s 1 2 x s ) s 1 (1 x i 1x2 i 1 x i ) i=1 Since n = a 1 +a 2 + +a s, we let q = x 1 = x 2 = = x s The generating function of p 2 (n) is σ K (q) = s i=1 1 s (1 q i q i 1 q) = i=1 1 1 q (i+1 2 ), which is the generating function for partitions into triangular numbers Thus, p 2 (n) equals the number of partitions of n into triangular numbers Note that we actually derived the full generating function of Theorem 23 Theorem 42 For an integer s 1, let σ(x 1, x 2,, x s ) := x a 1 1 x a 2 2 x a 3 3 x as s, where is the restricted summation over all nonnegative integers (a 1, a 2,, a s ) satisfying a s 0 and a i 2a i+1 + a i+2 0 for 1 i s 1 and a s+1 = 0 Then σ(x 1, x 2,, x s ) = s i=1 1 (1 x i 1x i 1 2 x i ) Once one becomes aware of the discoveries that the geometry of lattice-point

42 34 enumeration yields almost painlessly, it is possible to produce the next result 45 Geometric Proof Of Theorem 24 In this section, we give a geometric proof that p ± (m, n) equals the number of partitions of n of the form (m j)(m + j + 1)/2 where 0 j m Proof of Theorem 24 The generating function of p ± (m, n) is p ± (m, n)q n = q a 1+a 2 + +a m, n 0 (a 1,a 2,,a m) K where K := (a 1, a 2,, a m ) Z m : a 1 a 2 a m 0 and a i 2a i+1 + a i+2 0 for 1 i m 1 and a m+1 = 0 Claim: If a 1 a 2 and a i 2a i+1 + a i+2 0, then a i a i+1 and a m 0 for 1 i m 1 Proof of Claim: We will use induction on m Base case: If m = 2, we have a 1 a 2 and a 1 2a 2 0 a 1 a 2 and a 1 2a 2 Thus a 2 0 If m = 3, then we have a 1 a 2 and a 1 2a 2 +a 3 0 a 1 a 2 and a 1 +a 3 2a 2 Thus, a 2 must be greater or equal to a 3 Next, a 2 2a 3 0 a 2 2a 3 ; however, a 2 a 3 Therefore, a 3 0

43 35 Induction step: Assume the claim is true for m 1 We need to show it is also true for m From the induction step, we get a i a i+1 for 1 i m 2 Thus, a m 2 2a m 1 + a m 0 a m 2 + a m 2a m 1 implies that a m 1 a m because a m 2 a m 1 In addition, a m 1 2a m 0 a m 1 2a m and a m 1 a m imply that a m 0 Therefore, the claim is proven Thus, the conditions for K can be simplified: K := (a 1, a 2,, a s ) Z s : a 1 a 2 and a i 2a i+1 + a i+2 0 for 1 i m 1 and a m+1 = 0 Therefore, the constraints of K are given by the system a 1 a 2 a 3 a m 2 a m 1 a m 0 (46)

44 36 Now let C be the matrix on the left side of (46) Then det C = Adding the i th row to the (i + 1) th row for 1 i m, we obtain det C =

45 37 Thus det C = 1, and C 1 = m m 1 m 2 m m 1 m 1 m 2 m m 2 m 2 m 2 m The integer entries of C 1 are all nonnegative Thus, Lemma 32 gives the generating function of K: σ K (x 1, x 2,, x m ) = 1 (1 x m 1 x m 1 2 x 2 m 1x m )(1 x m 1 1 x m 1 2 x 2 m 1x m ) (1 x 1 x 2 x m 1 x m ) Since n = a 1 + a a m, we let q = x 1 = x 2 = = x m and obtain σ k (q) = 1 (1 q m+(m 1)+ +1 )(1 q (m 1)+(m 1)+ +1 ) (1 q m+(m 1) )(1 q m )

46 38 Thus, the generating function we wanted to find is σ K (q) = m j=1 1 (1 q j(2m j+1)/2 ), and this is precisely the generating function for the partitions described in Theoreom Note that in our proof we actually derived the full generating function of Theorem Theorem 43 For an integer m 2, let σ(x 1, x 2,, x m ) := x a 1 1 x a 2 2 x am m, where is the restricted summation over all nonnegative integers (a 1, a 2,, a m ) satisfying a 1 a 2 and a i 2a i+1 + a i+2 0 for 1 i m 1 and a m+1 = 0 Let X j = x 1 x j for 1 j m Then σ(x 1, x 2,, x m ) = 1 (1 X 1 X 2 X m )(1 X 2 X m ) (1 X m 1 X m )(1 X m )

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