1 Introduction

Special classes of -bracket operators

Tanay V. Wakhare

University of Maryland, College Park, MD 20742, USA

E-mail: twakhare@gmail.com

Received 23 December 2016 in final form ????; Published online ????

doi:10.3842/JOURNAL.201*.1

Abstract. We study the -bracket operator of Bloch and Okounkov when applied to and . We use these expansions to derive convolution identities for the functions and link both classes of -brackets through divisor sums. As a result, we generalize Euler’s classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley’s theorem as well as provide several new combinatorial results.

Key words: partitions, multiplicative number theory, additive number theory, q-series

2010 Mathematics Subject Classification: 05A17;11P81;11N99

1 Introduction

The -bracket operator was introduced by Bloch and Okounkov in 2000 [1] and is defined by [2, (Definition 1)] as

 ⟨f⟩q:=∑λ∈Pf(λ)q|λ|∑λ∈Pq|λ|=(q;q)∞∞∑n=0qn∑λ⊢nf(λ)∈C[[q]], (1.1)

where is a function and is the set of integer partitions. Throughout, we will refer to partitions by the variable and use the notation to show that is a partition of . Then, indicates that is a part of the partition .

We will denote by the sum of over all partitions of . We let denote the sum of over every part of , and the sum of over every distinct part of . For instance, . We also use to denote the sum of over all positive divisors of , including and .

We also use the convention throughout, where denotes the number of unrestricted partitions of .

2 Sums over all parts

Theorem 1.
 ⟨∑λi∈λf(λi)⟩q=∞∑n=1f(n)qn1−qn. (2.1)
Proof.

We first find a product that will generate , where for convenience. We begin with the usual[4, (27.14.3)]

 ∞∑n=0p(n)qn=1(q;q)∞=(1+q+q2+…)(1+q2+q4+…)⋯ (2.2)

We introduce a new variable and study

 ϕ(λ,q):=(1+λf(1)q+λ2f(1)q2+…)(1+λf(2)q2+λ2f(2)q4+…)⋯(1+λf(n)qn+λ2f(n)q2n+…)⋯ (2.3)

Now every coefficient of will be a polynomial in , with in equal to for that form a partition of . Therefore, by taking the partial derivative of with respect to and then setting we will sum the powers of each polynomial and make that the new coefficient of . We then have the relation

 ∂ϕ(λ,q)∂λ∣∣λ=1=∞∑n=0qn∑λ⊢ng(λ)=1(q;q)∞⟨g(λ)⟩q. (2.4)

Evaluating the partial by summing each term of as a geometric series and taking the logarithmic derivative of

 ϕ(λ,q)=∞∏n=1(11−λf(n)qn) (2.5)

while noting yields the desired theorem. ∎

We note that the case case was treated in [6, (Theorem 6.6)]. We also note that the expression on the right hand side of Theorem 1 is simply a Lambert series[4, (27.7.5)]. The case was considered by Zagier [9, (42)] as his ”moment function”. We find that the resulting Lambert series is equal to , which arises as part of the Fourier expansion of weight Eisenstein series. This shows the connection of certain -brackets and quasimodularity, and highlights the special nature of the moment function - it’s the only function additive over parts of partitions whose bracket will generate weighted Eisenstein series.

Theorem 2.

We have the convolution identity

 ∑λ⊢n∑λi∈λf(λi)=n∑k=1p(n−k)F(k), (2.6)

where

 F(n)=∑d|nf(d). (2.7)
Proof.

Recognizing the left hand side of the sum in 1 as a Lambert series and as the generating function for , we obtain

 ∞∑n=0∑λ⊢n∑λi∈λf(λi)qn=1(q;q)∞⟨∑λi∈λf(λi)⟩q=∞∑n=0p(n)qn∞∑n=1F(n)qn, (2.8)

where is as defined above. Taking a Cauchy product, reindexing, and comparing coefficients of yields the desired identity. ∎

3 Sums over distinct parts

Theorem 3.
 ⟨∑λi∈λλidistinctf(λi)⟩q=∞∑n=1f(n)qn. (3.1)
Proof.

Analogously to the proof of 1 we now study the product

 ϕ2(λ,q):=(1+λf(1)q+λf(1)q2+…)(1+λf(2)q2+λf(2)q4+…)⋯(1+λf(n)qn+λf(n)q2n+…)⋯ (3.2)

The coefficient of each will again be a polynomial in . However, in will now be equal to for distinct in a partition of since we do not give different weights based on how many times each appears in a partition. Therefore, as before we have

 ∂ϕ2(λ,q)∂λ∣∣λ=1=∞∑n=0qn∑λ⊢ng(λ)=1(q;q)∞⟨g(λ)⟩q, (3.3)

where for convenience. Evaluating the partial by summing each term in and taking the logarithmic derivative of

 ϕ2(λ,q)=∞∏n=1(1+λf(n)qn1−qn) (3.4)

while noting yields the desired theorem. ∎

The case for constant was treated with a similar method in [8, (Page 2)].

Theorem 4.

We have the convolution identity

 ∑λ⊢n∑λi∈λλidistinctf(λi)=n∑k=1p(n−k)f(n). (3.5)
Proof.

Rewriting Theorem 3 as

 ∞∑n=0∑λ⊢n∑λi∈λλidistinctf(λi)qn=∞∑n=0p(n)qn∞∑n=1f(n)qn, (3.6)

taking a Cauchy product, reindexing, and comparing coefficients of yields the desired identity. ∎

4 Connections to multiplicative number theory

Theorem 5.
 ∑λ⊢n∑λi∈λλidistinctF(λi)=∑λ⊢n∑λi∈λf(λi)=n∑k=1p(n−k)F(k), (4.1)

where

 F(n)=∑d|nf(d). (4.2)

In other words, the sum of over all parts in every partition of is equal to the divisor sum of , evaluated over all distinct parts in every partition of .

Proof.

Looking at the convolution identities 2 and 4, we see that a pair of functions that satisfies (4.2) both lead to expressions of the form

 ∑λ⊢n∑λi∈λλidistinctF(λi)=n∑k=1p(n−k)∑d|kf(d)=∑λ⊢n∑λi∈λf(λi). (4.3)

This completes the proof. ∎

This result generalizes Stanley’s Theorem [7, (Page 1)], which states that the total number of s in all partitions of is equal to the sum of the number of distinct parts in each partition. Theorem 5 reduces to Stanley’s Theorem if we let be the indicator function for , so that .

Theorem 5 is extremely interesting in that it links divisor sums, which are often found in multiplicative number theory, to sums over partitions of - a concept solely from additive number theory. Krishna Alladi has previously done extensive work at the confluence of multiplicative and additive number theory, for example [10]. By specializing in 5 we obtain several apparently new corollaries.

Corollary 6.

The number of partitions of containing is . The number of squarefree parts with an even number of prime factors minus the number of squarefree parts with an odd number of prime factors, summed over every partition of , is also given by .

Proof.

Take in Theorem 5, where is the Möbius function, so that is the indicator function for [3, (Page 25)]. The convolution on the right hand side then sums to while then counts the number of s in the distinct parts of each partition of . This is equivalent to simply counting the number of partitions of which contain a . We note that the this can be directly proved by removing a from every partition of that contains it, since these will then form partitions of [George Andrews, personal communication].

The second statement follows from applying the definition of the Möbius function to . ∎

Corollary 7.

We have the identity

 ∑λ⊢n∑λi∈λλαi=∑λ⊢n∑λi∈λλidistinctσα(λi)=n∑k=1σα(k)p(n−k). (4.4)
Proof.

Take in Theorem 5. ∎

This forms a natural generalization of the classical identity due to Euler[3, (Page 323)]

 np(n)=n∑k=1σ1(k)p(n−k), (4.5)

since . Both the parameter and the connection to sums over distinct parts are new.

Corollary 8.

We have the identity

 ∑λ⊢n∑λi∈λJα(λi)=∑λ⊢n∑λi∈λλidistinctλαi=n∑k=1Jα(k)p(n−k). (4.6)
Proof.

Take in Theorem 5, where is the Jordan totient function [3, (48)], so that . ∎

The forms a natural dual of Euler’s convolution, since the convolution arises from considering th moments of all parts of the partitions of , while the convolution arises from considering th moments of all distinct parts of the partitions of . We also note that taking provides an analog of (4.5) for the Euler totient function.

Corollary 9.

Let denote the number of squarefree parts in all partitions of . Then , where gives the number of distinct prime factors of . Therefore, .

Proof.

Take in Theorem 5, so that [3, (Page 45)], where is as given above. We have that is the indicator function for squarefree numbers, so gives the number of squarefree parts in all partitions of . The congruence arises from noting that unless . ∎

Corollary 10.

The sum of the number of distinct squares in every partition of is given by

 ⌊√n⌋∑k=1p(n−k2). (4.7)
Proof.

Take in Theorem 5, where is the Liouville function, so that is the indicator function for the squares[3, (Page 38)]. This implies is the sum of the number of distinct squares in every partition of . Rewriting the summation completes the proof. ∎

Corollary 11.

We have the identity

 ∑λ⊢n∑λi∈λΛ(λi)=∑λ⊢n∑λi∈λλidistinctlog(λi)=n∑k=1p(n−k)logk. (4.8)

Furthermore, equals the product of all distinct parts in every partition of .

Proof.

Take in Theorem 5, where is the von Mangoldt function, which is if is an integer power of the prime and otherwise. Then, [3, (Page 32)], which yields the corollary. By exponentiating the sum over distinct and the right hand side, we obtain the second part of the corollary. ∎

An open question for future research is now to provide combinatorial or alternate proofs for these formulae, or apply Theorem 5 to any other divisor sum and find new consequences. For instance, studying the prime factorization of or the role of the von Mangoldt function in Corollary 11 should yield information about the multiplicities of primes in the parts of the partitions of .

5 Acknowledgments

I would like to thank Christophe Vignat for guiding me through my first forays into mathematics, and Armin Straub for first introducing me to Stanley’s Theorem and the current literature on partitions. I would also like to thank George Andrews and Robert Schneider for helpful comments.

References

• [1] S. Bloch and A. Okounkov, The character of the infinite wedge representation, Advances in Mathematics 149 (2000), 1-60
• [2] R. Schneider, Arithmetic of partitions and the -bracket operator To appear in Proceedings of the AMS (2016), http://arxiv.org/abs/1601.07466
• [3] T. M. Apostol, Introduction to Analytic Number Theory. Springer-Verlag, New York, NY, 1976.
• [4] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov, Release 1.0.9 of 2014-08-29. Online companion to [5].
• [5] F. W. J. Olver and D. W. Lozier and R. F. Boisvert and C. W. Clark, editor. NIST Handbook of Mathematical Functions. Cambridge University Press, New York, NY, 2010. Print companion to [4].
• [6] G.-N. Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, preprint, 2008, http://arxiv.org/abs/0804.1849.
• [7] M. G. Dastidar and S. S. Gupta, Extension of Stanley’s Theorem for Partitions, preprint, 2010, http://arxiv.org/abs/1007.3459.
• [8] M. D. Hirschhorn, The number of different parts in the partitions of , Fibonacci Quart. 52 (2014), 10-15
• [9] D. Zagier, Partitions, quasimodular forms, and the Bloch-Okounkov theorem, The Ramanujan Journal (2015), 1-24
• [10] K. Alladi and P. ErdÅs, On an additive arithmetic function, Pacific J. Math. 2 (1977), 275–294
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