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Abstract

A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of partitions of n that are 7-cores with given BG-rank can be written as certain sum of multi-theta functions. We give explicit representations for these generating functions in terms of sums of positive eta-quotients and derive inequalities for the their coefficients. New identities for the generating function of unrestricted 7-cores and inequalities for their coefficients are also obtained. Our proofs utilize Ramanujan's theory of modular equations.

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... Recently, Ramanujan's modular equations have been applied by Baruah and Berndt [1] to obtain a linear relation for 5core partitions and by Berkovich and Yesilyurt [2] to get inequalities for 7-core partitions. In particular, in [1,Theorem 4.2], Baruah and Berndt proved that a 5 (4n + 3) = a 5 (2n + 1) + 2a 5 (n), (2) and in [2, Theorem 1.1], Berkovich and Yesilyurt proved the inequalities a 7 (2n + 2) ≥ 2a 7 (n) and a 7 (4n + 6) ≥ 10a 7 (n), for all n ≥ 0. In the same paper, Berkovich and Yesilyurt conjectured the stronger inequalities: ...
... Recently, Ramanujan's modular equations have been applied by Baruah and Berndt [1] to obtain a linear relation for 5core partitions and by Berkovich and Yesilyurt [2] to get inequalities for 7-core partitions. In particular, in [1,Theorem 4.2], Baruah and Berndt proved that a 5 (4n + 3) = a 5 (2n + 1) + 2a 5 (n), (2) and in [2, Theorem 1.1], Berkovich and Yesilyurt proved the inequalities a 7 (2n + 2) ≥ 2a 7 (n) and a 7 (4n + 6) ≥ 10a 7 (n), for all n ≥ 0. In the same paper, Berkovich and Yesilyurt conjectured the stronger inequalities: ...
... The next goal of this paper is to obtain linear relations for 7-core partitions, which are analogous to (2). In Section 4, we will note that (2) can be seen from the fact that the generating function for 5-core partitions is essentially a Hecke eigenform, which implies the following generalization of (2). ...
Article
Recently, Ramanujan's modular equations have been applied by N.D. Baruah and B.C. Berndt to obtain a linear relation for 5-core partitions and by A. Berkovich and H. Yesilyurt to obtain inequalities for 7-core partitions. In this paper, we generalize their results by using the theory of modular forms. In particular, we prove conjectures of Berkovich and Yesilyurt.
... Moreover, the generating function of c t (n) can be represented as the product of the Dedekind eta function. Therefore, many researches on core partitions are being made through various ways, such as representation theory and analytic methods-see, for example, [7,8,10,12,15,16]. Stanton [17] conjectured the monotonicity of c t (n), that is, c t+1 (n) ≥ c t (n) for t ≥ 4 and n ≥ t + 1. Motivated by this conjecture, Anderson [4], Kim and Rouse [14] found the asymptotics of c t (n) and proved Stanton's conjecture partially. ...
... • Since D 1 (λ) = {5} and D 3 (λ) = {31, 19, 11},λ ∈ SC (4) and ((1), (8,5,3)) is the diagonal sequence pair ofλ. If we letμ be the partition φ (4) 14 (λ), then µ 1 = 1 + 1 + 2 = 4 and (µ 2 , µ 3 , . . . ) is the conjugate of the partition (8 − 3, 5 − 3 + 1, 3 − 3 + 2). ...
Preprint
We give a bijection between the set of self-conjugate partitions and that of ordinary partitions. Also, we show the relation between hook lengths of self conjugate partition and corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.
... Following [2], we say that a q-series is positive if its power series coefficients are all non-negative. We define P [q] to be the set of all such series. ...
... However, it is not at all obvious that (1.13) ψ(q)(φ(q) 2 − φ(q 7 ) 2 ) ∈ P [q]. Motivated by their studies of 7-core partitions, H. Yesilyurt and the first author conjectured (1.13) in ((6.2), [2]). The reader should be cautioned that other similar conjectures there: (6.1), (6.3) and (6.4) are false. ...
Article
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We show that many of Ramanujan’s modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n ∈ N, {(x,y,z)Z3:x(x+1)2+y2+z2=n}{(x,y,z)Z3:x(x+1)2+3y2+3z2=n},\begin{array}{rcl} & & \left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + {y}^{2} + {z}^{2} = n\right \}\right \vert \geq \\ & &\left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + 3{y}^{2} + 3{z}^{2} = n\right \}\right \vert, \\ \end{array} just to name one among many similar “positivity” results of this type. In particular, we prove the recent conjecture of H. Yesilyurt and the first author, stating that for any n ∈ N, {(x,y,z)Z3:x(x+1)2+y2+z2=n}{(x,y,z)Z3:x(x+1)2+7y2+7z2=n}.\begin{array}{rcl} & & \left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + {y}^{2} + {z}^{2} = n\right \}\right \vert \geq \\ & &\left \vert \left \{(x,y,z) \in {\mathbf{Z}}^{3} : \frac{x(x + 1)} {2} + 7{y}^{2} + 7{z}^{2} = n\right \}\right \vert.\end{array} We prove a number of identities for certain ternary forms with discriminants 144, 400, 784, or 3, 600 by converting every ternary identity into an identity for the appropriate η-quotients. In the process, we discover and prove a few new modular equations of degree 5 and 7. For any square free odd integer S with prime factorization p 1…p r , we define the S-genus as a union of 2r specially selected genera of ternary quadratic forms, all with discriminant 16S 2. This notion of S-genus arises naturally in the course of our investigation. It entails an interesting injection from genera of binary quadratic forms with discriminant − 8S to genera of ternary quadratic forms with discriminant 16S 2.
... (1. 2) It has been studied extensively in [2] and [5]. Let L, m, n be non-negative integers. ...
Article
Full-text available
In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo t which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and the fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new elegant generating function formulas for unrestricted partitions, self-conjugate partitions, and partitions whose parts repeat a finite number of times.
... It has been studied extensively in [2] and [5]. ...
Preprint
Full-text available
In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo t which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new elegant generating function formulas for unrestricted partitions, self-conjugate partitions, and partitions whose parts repeat finite number of times.
... In 2010, B.Kim [7] established some inequalities and linear relations for 7-core partitions. Radu and Sellers [8] showed that a 7 (14n + 7, 9, 13) ≡ 0 (mod 8), by using the theory of modular forms, ( see [3,4] for further results). ...
Conference Paper
A partition λ is said to be a t-core if and only if it has no hook numbers that are multiples of t. In this paper, we find several new and interesting congruences for 7-core partitions modulo 3 by making use of Ramanujan’s theta function identities. We obtain several infinite families of congruences modulo 3 for 7-core partitions. For example, if p ≥ 5 is a prime with ( − 7 p ) = − 1 and r ∈ {3, 4, 6}, then for all non-negative integers n and k, a 7 (147 · p 2k n + 7 · p 2k (3r + 1) − 2) ≡ a 7 (21 · p 2k n + p 2k (3r + 1) − 2) (mod 3).
... Next, we find the congruence subgroup corresponding to the function field C(q α g5,s g5,−1 ) for (α, s) = (0, 1), (1,2) and (3,4). Denote h ...
Article
We apply modular function theory to find the relation among t-core partitions. By using the generators of function field corresponding to a certain modular group, we reprove the identities in [1] because their relations are linear for t = 3 or 5.
... As a second application, we will prove an inequality involving c p (n). Recently, many interesting inequalities for the number of p-core partitions have been investigated using either modular equations or modular forms (see [5], [6], and [23] ...
Article
In this article, we derive explicit bounds on c t ( n ) c_{t} (n) , the number of t t -core partitions of n n . In the case when t = p t = p is prime, we express the generating function f ( z ) f(z) as the sum f(z)=epE(z)+irigi(z)f(z)=epE(z)+irigi(z) f ( z ) = e p E ( z ) + ∑ i r i g i ( z ) f(z) = e_{p} E(z) + \sum _{i} r_{i} g_{i}(z) of an Eisenstein series and a sum of normalized Hecke eigenforms. We combine the Hardy-Littlewood circle method with properties of the adjoint square lifting from automorphic forms on G L ( 2 ) \rm {GL}(2) to G L ( 3 ) \rm {GL}(3) to bound R ( p ) := ∑ i | r i | R(p) := \sum _{i} |r_{i}| , solving a problem raised by Granville and Ono. In the case of general t t , we use a combination of techniques to bound c t ( n ) c_{t}(n) and as an application prove that for all n ≥ 0 n \geq 0 , n ≠ t + 1 n \ne t+1 , ct+1(n)ct(n)ct+1(n)ct(n) c t + 1 ( n ) ≥ c t ( n ) c_{t+1}(n) \geq c_{t}(n) provided 4 ≤ t ≤ 198 4 \leq t \leq 198 , as conjectured by Stanton.
... For a partition π , BG-rank(π ) is defined as an alternating sum of parities of parts of π [2,3]. In [4] , the authors found positive etaquotient representations for the 7-core generating functions n0 a 7, j (n)q n , where a 7, j (n) denotes the number of 7-cores of n with BG-rank = j and established a number of inequalities for a 7, j (n) with j = −1, 0, 1, 2 and a 7 (n). In this paper, we prove lower and upper bounds for a 7 (n), namely ...
Article
In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x(2) + y(2) + z(2) + 7s(2) + 7t(2) + 7u(2). We establish the following intriguing inequalities 2 omega(n + 2) >= a(7) (n) >= omega(n + 2) for n not equal 0, 2, 6, 16. Here a(7)(n) is the number of partitions of n that are 7-cores and omega(n) is the number of representations of n by the sextenary form (x(2) + y(2) + z(2) + 7s(2) + 7t(2) + 7u(2))/8 with x, y, z, s, t and u being odd positive integers.
... Clearly, one wants to find " positive " etaquotient representations for other admissible values of BG-rank. (See [3] for a fascinating discussion of the t = 7 case). Finally, we observe that (1.2) is the s = 2 case of the following more general definition ...
Article
Let π denote a partition into parts . In a 2006 paper we defined BG-rank(π) as This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let pj(n) denote the number of partitions of n with BG-rank=j. Here, we provide a combinatorial proof that by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by pj(5n+4) into five equal classes. This proof uses the orbit construction from our previous paper and a new identity for the BG-rank. Let at,j(n) denote the number of t-cores of n with BG-rank=j. We find eta-quotient representations for when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a5,j(n), j=0,±1.
... For a partition π , BG-rank(π ) is defined as an alternating sum of parities of parts of π [2,3]. In [4], the authors found positive etaquotient representations for the 7-core generating functions n 0 a 7, j (n)q n , where a 7, j (n) denotes the number of 7-cores of n with BG-rank = j and established a number of inequalities for a 7, j (n) with j = −1, 0, 1, 2 and a 7 (n). In this paper, we prove lower and upper bounds for a 7 (n), namely Theorem 1.1. ...
Article
Full-text available
In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2. We establish the following intriguing inequalities 2b(n)>=a_7(n)>=b(n) for n not equal to 0,2,6,16. Here a_7(n) is the number of partitions of n that are 7-cores and b(n) is the number of representations of n+2 by the sextenary form (x ^2+ y ^2+z ^2+ 7s ^2 + 7t ^2+ 7u^2)/8 with x,y,z,s,t and u being odd.
... We refer to this statistic as the GBG-rank of π mod s. The special case s = 2 was studied in great detail in [2] and [3]. In particular, we have shown in [2] ...
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Let r_j(\pi,s) denote the number of cells, colored j, in the s-residue diagram of partition \pi. The GBG-rank of \pi mod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*\Pi*I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-core may assume. The above inequality becomes an equality when s is prime or when s is composite and t<=2p_s, where p_s is a smallest prime divisor of s. We will show that the generating functions for 4-cores with the prescribed values of GBG-rank mod 3 are all eta-products.
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. We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero pGammablocks remained unclassified were the alternating groups An . Here we show that these all have a p-block with defect 0 for every prime p 5. This follows from proving the same result for every symmetric group Sn , which in turn follows as a consequence of the t-core partition conjecture, that every non-negative integer possesses at least one t-core partition, for any t 4. For t 17, we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with t ! 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for t-core partitions in terms of modular forms, and then controlling the size of the coefficients usin...
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Let π denote a partition into parts . In a 2006 paper we defined BG-rank(π) as This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let pj(n) denote the number of partitions of n with BG-rank=j. Here, we provide a combinatorial proof that by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by pj(5n+4) into five equal classes. This proof uses the orbit construction from our previous paper and a new identity for the BG-rank. Let at,j(n) denote the number of t-cores of n with BG-rank=j. We find eta-quotient representations for when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a5,j(n), j=0,±1.
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B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991.