On a conjecture of George Beck

Shane Chern

arXiv:1705.10700·math.NT·May 23, 2018

On a conjecture of George Beck

Shane Chern

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TL;DR

This paper proves a conjecture by George Beck related to special types of integer partitions, specifically gap-free partitions and partitions with distinct parts, advancing understanding in partition theory.

Contribution

The paper provides a proof for Beck's conjecture, establishing a new connection between gap-free partitions and partitions with distinct parts.

Findings

01

Confirmed the validity of Beck's conjecture.

02

Established a new relationship between two classes of partitions.

03

Enhanced theoretical understanding of partition structures.

Abstract

In this paper, we prove a conjecture proposed by George Beck, which involves gap-free partitions and partitions with distinct parts.

Equations52

n \geq 1 \sum a (n) q^{n} = m \geq 1 \sum \frac{q ^{m}}{1 - q ^{m}} (- q)_{m - 1},

n \geq 1 \sum a (n) q^{n} = m \geq 1 \sum \frac{q ^{m}}{1 - q ^{m}} (- q)_{m - 1},

(a)_{n} = (a; q)_{n} := k = 0 \prod n - 1 (1 - a q^{k}),

(a)_{n} = (a; q)_{n} := k = 0 \prod n - 1 (1 - a q^{k}),

(a)_{\infty} = (a; q)_{\infty} := k \geq 0 \prod (1 - a q^{k}) .

(a)_{\infty} = (a; q)_{\infty} := k \geq 0 \prod (1 - a q^{k}) .

n \geq 1 \sum b (n) q^{n} = \frac{1}{2} (π \in D \sum σ (π) q^{∣ π ∣} - π \in D \sum (- 1)^{♯ (π)} σ (π) q^{∣ π ∣}) .

n \geq 1 \sum b (n) q^{n} = \frac{1}{2} (π \in D \sum σ (π) q^{∣ π ∣} - π \in D \sum (- 1)^{♯ (π)} σ (π) q^{∣ π ∣}) .

π \in D \sum σ (π) q^{∣ π ∣}

π \in D \sum σ (π) q^{∣ π ∣}

= m \geq 1 \sum m q^{m} (- q^{m + 1})_{\infty},

π \in D \sum (- 1)^{♯ (π)} σ (π) q^{∣ π ∣}

π \in D \sum (- 1)^{♯ (π)} σ (π) q^{∣ π ∣}

= - m \geq 1 \sum m q^{m} (q^{m + 1})_{\infty} .

n \geq 1 \sum b (n) q^{n}

n \geq 1 \sum b (n) q^{n}

= \frac{1}{2} ((q)_{\infty} m \geq 1 \sum \frac{m q ^{m}}{( q ) _{m}} + (- q)_{\infty} m \geq 1 \sum \frac{m q ^{m}}{( - q ) _{m}}) .

m \geq 1 \sum \frac{m q ^{m}}{( q ) _{m}} = [z \frac{\partial}{\partial z} (m \geq 0 \sum \frac{z ^{m}}{( q ) _{m}})]_{z = q} .

m \geq 1 \sum \frac{m q ^{m}}{( q ) _{m}} = [z \frac{\partial}{\partial z} (m \geq 0 \sum \frac{z ^{m}}{( q ) _{m}})]_{z = q} .

m \geq 0 \sum \frac{z ^{m}}{( q ) _{m}} = \frac{1}{( z ) _{\infty}} .

m \geq 0 \sum \frac{z ^{m}}{( q ) _{m}} = \frac{1}{( z ) _{\infty}} .

z \frac{\partial}{\partial z} (m \geq 0 \sum \frac{z ^{m}}{( q ) _{m}})

z \frac{\partial}{\partial z} (m \geq 0 \sum \frac{z ^{m}}{( q ) _{m}})

= \frac{z}{( z ) _{\infty}} \frac{\partial}{\partial z} (lo g \frac{1}{( z ) _{\infty}})

= \frac{z}{( z ) _{\infty}} \frac{\partial}{\partial z} (m \geq 0 \sum lo g \frac{1}{1 - z q ^{m}})

= \frac{z}{( z ) _{\infty}} m \geq 0 \sum \frac{q ^{m}}{1 - z q ^{m}} .

(q)_{\infty} m \geq 1 \sum \frac{m q ^{m}}{( q ) _{m}} = (q)_{\infty} \frac{q}{( q ) _{\infty}} m \geq 0 \sum \frac{q ^{m}}{1 - q ^{m + 1}} = m \geq 1 \sum \frac{q ^{m}}{1 - q ^{m}} .

(q)_{\infty} m \geq 1 \sum \frac{m q ^{m}}{( q ) _{m}} = (q)_{\infty} \frac{q}{( q ) _{\infty}} m \geq 0 \sum \frac{q ^{m}}{1 - q ^{m + 1}} = m \geq 1 \sum \frac{q ^{m}}{1 - q ^{m}} .

m \geq 1 \sum \frac{m q ^{m}}{( - q ) _{m}} = [z \frac{\partial}{\partial z} (m \geq 0 \sum \frac{z ^{m}}{( - q ) _{m}})]_{z = q} .

m \geq 1 \sum \frac{m q ^{m}}{( - q ) _{m}} = [z \frac{\partial}{\partial z} (m \geq 0 \sum \frac{z ^{m}}{( - q ) _{m}})]_{z = q} .

_{r + 1} ϕ_{s} (a_{0}, a_{1}, a_{2} \dots, a_{r} b_{1}, b_{2}, \dots, b_{s}; q, z) := n \geq 0 \sum \frac{( a _{0} ; q ) _{n} ( a _{1} ; q ) _{n} \dots ( a _{r} ; q ) _{n}}{( q ; q ) _{n} ( b _{1} ; q ) _{n} \dots ( b _{s} ; q ) _{n}} ((- 1)^{n} q^{(2 n)})^{s - r} z^{n} .

_{r + 1} ϕ_{s} (a_{0}, a_{1}, a_{2} \dots, a_{r} b_{1}, b_{2}, \dots, b_{s}; q, z) := n \geq 0 \sum \frac{( a _{0} ; q ) _{n} ( a _{1} ; q ) _{n} \dots ( a _{r} ; q ) _{n}}{( q ; q ) _{n} ( b _{1} ; q ) _{n} \dots ( b _{s} ; q ) _{n}} ((- 1)^{n} q^{(2 n)})^{s - r} z^{n} .

_{2} ϕ_{1} (a, b c; q, z) = \frac{( b ) _{\infty} ( a z ) _{\infty}}{( c ) _{\infty} ( z ) _{\infty}}_{2} ϕ_{1} (c / b, z a z; q, b) .

_{2} ϕ_{1} (a, b c; q, z) = \frac{( b ) _{\infty} ( a z ) _{\infty}}{( c ) _{\infty} ( z ) _{\infty}}_{2} ϕ_{1} (c / b, z a z; q, b) .

m \geq 0 \sum \frac{z ^{m}}{( - q ) _{m}}

m \geq 0 \sum \frac{z ^{m}}{( - q ) _{m}}

= \frac{( q ) _{\infty}}{( - q ) _{\infty}} \frac{1}{( z ) _{\infty}} n \geq 0 \sum \frac{( - 1 ) _{n} ( z ) _{n}}{( q ) _{n}} q^{n}

= \frac{( q ) _{\infty}}{( - q ) _{\infty}} n \geq 0 \sum \frac{( - 1 ) _{n}}{( q ) _{n} ( z q ^{n} ) _{\infty}} q^{n} .

z \frac{\partial}{\partial z} (m \geq 0 \sum \frac{z ^{m}}{( - q ) _{m}})

z \frac{\partial}{\partial z} (m \geq 0 \sum \frac{z ^{m}}{( - q ) _{m}})

= \frac{z ( q ) _{\infty}}{( - q ) _{\infty}} n \geq 0 \sum \frac{( - 1 ) _{n} q ^{n}}{( q ) _{n}} \frac{1}{( z q ^{n} ) _{\infty}} m \geq 0 \sum \frac{q ^{m + n}}{1 - z q ^{m + n}} .

(- q)_{\infty} m \geq 1 \sum \frac{m q ^{m}}{( - q ) _{m}}

(- q)_{\infty} m \geq 1 \sum \frac{m q ^{m}}{( - q ) _{m}}

= n \geq 0 \sum (- 1)_{n} q^{n} m \geq 0 \sum \frac{q ^{m + n + 1}}{1 - q ^{m + n + 1}}

= m \geq 1 \sum \frac{q ^{m}}{1 - q ^{m}} n = 0 \sum m - 1 (- 1)_{n} q^{n}

= m \geq 1 \sum \frac{q ^{m}}{1 - q ^{m}} (1 + 2 n = 1 \sum m - 1 (- q)_{n - 1} q^{n})

= m \geq 1 \sum \frac{q ^{m}}{1 - q ^{m}} (2 (- q)_{m - 1} - 1) .

(- q)_{m - 1} = 1 + n = 1 \sum m - 1 (- q)_{n - 1} q^{n} .

(- q)_{m - 1} = 1 + n = 1 \sum m - 1 (- q)_{n - 1} q^{n} .

n \geq 1 \sum b (n) q^{n}

n \geq 1 \sum b (n) q^{n}

= m \geq 1 \sum \frac{q ^{m}}{1 - q ^{m}} (- q)_{m - 1},

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Full text

Int. J. Number Theory 14 (2018), no. 3, 647–651. MR3786639.

doi: 10.1142/S1793042118500392

On a conjecture of George Beck

Shane Chern

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

[email protected]; [email protected]

Abstract.

In this paper, we prove a conjecture proposed by George Beck, which involves gap-free partitions and partitions with distinct parts.

Keywords. Gap-free partition, distinct partition.

2010MSC. Primary 11P84; Secondary 05A17.

1. Introduction

A partition of a natural number $n$ is a nonincreasing sequence $\pi=(\pi_{1},\pi_{2},\ldots,\pi_{\ell})$ of positive integers whose sum equals $n$ , i.e. $\pi_{1}\geq\pi_{2}\geq\cdots\geq\pi_{\ell}$ and $\pi_{1}+\pi_{2}+\cdots+\pi_{\ell}=n$ . Furthermore, a partition is called gap-free if we restrict that the difference between each consecutive parts is at most $1$ . For example, there are five gap-free partitions of $5$ : $5$ , $3+2$ , $2+2+1$ , $2+1+1+1$ , and $1+1+1+1+1$ . Let $a(n)$ denote the number of gap-free partitions of $n$ . This sequence is listed as A034296 in the On-Line Encyclopedia of Integer Sequence [5]. To determine the generating function of $a(n)$ , we only need the following trivial observation (cf. [2]):

The conjugates of gap-free partitions are partitions where only the largest part may repeat.

Hence we have

[TABLE]

where and in the sequel, we employ the standard $q$ -series notations

[TABLE]

and

[TABLE]

Recently, George Beck proposed an interesting conjecture in [5, A034296]:

Conjecture 1.1.

$a(n)$ * is also the sum of the smallest parts in the distinct partitions (i.e. partitions with distinct parts) of $n$ with an odd number of parts.*

Let $b(n)$ denote the sum of the smallest parts in the distinct partitions of $n$ with an odd number of parts. In this paper, we aim to prove Beck’s conjecture.

Theorem 1.2.

For all $n\geq 1$ , $a(n)=b(n)$ .

2. Proof of Theorem 1.2

Let $\mathcal{D}$ be the set of distinct partitions. For any $\pi\in\mathcal{D}$ , we denote by $|\pi|$ the sum of the parts of $\pi$ , by $\sigma(\pi)$ the smallest part of $\pi$ , and by $\sharp(\pi)$ the number of parts of $\pi$ . Then

[TABLE]

One also readily sees that

[TABLE]

and

[TABLE]

Hence

[TABLE]

We first notice that

[TABLE]

Also the Euler’s first sum [1, Eq. (17.5.4)] tells that

[TABLE]

Hence

[TABLE]

This implies that

[TABLE]

On the other hand, we have

[TABLE]

Let

[TABLE]

We now need the Heine’s first transformation [1, Eq. (17.6.6)]

[TABLE]

Then

[TABLE]

It follows that

[TABLE]

Hence

[TABLE]

To see the last identity, we consider distinct partitions with largest part $\leq m-1$ . Obviously, its generating function is $(-q)_{m-1}$ . On the other hand, if we fix the largest part to be $n\geq 1$ , then it follows that

[TABLE]

Together with Eqs. (2.1)–(2.3), we conclude that

[TABLE]

which coincides with the generating function of $a(n)$ . Hence for all $n\geq 1$ , $a(n)=b(n)$ .

3. Conclusion

In a recent paper [3], Andrews proved two conjectures of the same flavor (that is, showing some (weighted) partition functions are identical), which are also proposed by George Beck. We also notice the power of the differentiation technique in both our proof of Theorem 1.2 and Andrews’ proofs in [3]. Shishuo Fu and Dazhao Tang [4] subsequently generalized Andrews’ results and provided a combinatorial proof. It would be interesting to know whether there also exists a combinatorial proof for our Theorem 1.2.

Bibliography5

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] G. E. Andrews, q 𝑞 q -Hypergeometric and related functions, NIST handbook of mathematical functions , 419–433, U.S. Dept. Commerce, Washington, DC, 2010.
2[2] G. E. Andrews, The Bhargava–Adiga summation and partitions, J. Indian Math. Soc. (N.S.) 84 (2017), no. 3-4, 151–160.
3[3] G. E. Andrews, Euler’s partition identity and two problems of George Beck, Math. Student 86 (2017), no. 1-2, 115–119.
4[4] S. Fu and D. Tang, Generalizing a partition theorem of Andrews, Preprint (2017), ar Xiv:1705.05046.
5[5] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, http://oeis.org .

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

On a conjecture of George Beck

Abstract.

1. Introduction

Conjecture 1.1**.**

Theorem 1.2**.**

2. Proof of Theorem 1.2

3. Conclusion

Conjecture 1.1.

Theorem 1.2.