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Singular Overpartitions
Abstract
The objective in this paper is to present a general theorem for overpartitions analogous to Rogers–Ramanujan type theorems for ordinary partitions with restricted successive ranks.
... (4), (4), (3,1), (3,1), (3,1), (3,1), (2,2), (2,2), (2, 1, 1), (2, 1, 1), (2, 1, 1), (2, 1, 1), (1, 1, 1, 1), (1, 1, 1, 1). ...
... (4), (4), (3,1), (3,1), (3,1), (3,1), (2,2), (2,2), (2, 1, 1), (2, 1, 1), (2, 1, 1), (2, 1, 1), (1, 1, 1, 1), (1, 1, 1, 1). ...
... (4), (4), (3,1), (3,1), (3,1), (3,1), (2,2), (2,2), (2, 1, 1), (2, 1, 1), (2, 1, 1), (2, 1, 1), (1, 1, 1, 1), (1, 1, 1, 1). ...
In this paper, we consider the 2-adic valuation of integers and provide an alternative representation for the generating function of the number of overpartitions of an integer. As a consequence of this result, we obtain a new formula and a new combinatorial interpretation for the number of overpartitions of an integer. This formula implies a certain type of partitions with restrictions for which we provide two Ramanujan-type congruences and present as open problems two infinite families of linear inequalities. Connections between overpartitions and the game of m-Modular Nim with two heaps are presented in this context.
... Andrews [4] extended the idea of -regular overpartitions by considering the enumeration C k,i (n) of so-called singular overpartitions of n that correspond to -regular overpartitions of n in which only parts ≡ ±i (mod k) may be overlined. Clearly, ...
... Therefore, it follows that the above sums of partition numbers are divisible by 3 k for almost all n. In the following corollary, we present selected congruences for the sums in nondecreasing order of the moduli that arise from the congruences for A 3 (n) or C 3,1 (n), which either appeared in [2,4,6,8,9,11,14,16,22,23,26] or are easily deduced from these results. For any nonnegative integers k and n, S(3n + 2) ≡ 0 (mod 2), S(4n + 2) ≡ 0 (mod 2), ...
... Most of the congruences follow easily from the corresponding congruences and generating function representations of A 3 (n) or C 3,1 (n) in [2,4,6,8,9,11,14,16,22,23,26] and Theorem 2.1. Therefore, we only prove the last three congruences in Corollary 2.2, that is, But, by the binomial theorem, (q j ; q j ) 2 ∞ ≡ (q 2j ; q 2j ) ∞ (mod 2) for any integer j ≥ 1. ...
We show that certain sums of partition numbers are divisible by multiples of 2 and 3. For example, if p(n) denotes the number of unrestricted partitions of a positive integer n (and p(0)=1 , p(n)=0 for ), then for all nonnegative integers m , \begin{align*}\sum_{k=0}^\infty p(24m+23-\omega(-2k))+\sum_{k=1}^\infty p(24m+23-\omega(2k))\equiv 0~ (\text{mod}~144),\end{align*}
where .
... For any δ|N , since γ 3 ∈ Γ 1 (N ), we see that a 3 ≡ 1 (mod δ), δ|c 3 and gcd(δ, d 3 ) = 1. Using (2.27), it can be verified that ...
... While many partition functions a(n) are of the form (1.1), there are partition functions that do not seem to fall into this framework, such as Andrews' (k, i)singular overpartition function Q k,i (n). Andrews [3] derived the generating function: ...
... Note that Theorem 10.2 is needed to find a generalized eta-quotient h(τ ), such that hF has a pole only at infinity. For example, we can derive Ramanujan-type identities on the singular overpartition function introduced by Andrews [3]. The number of (k, i)-singular overpartitions of n is denoted by Q k,i (n) (1 ≤ i < k 2 ). ...
This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu’s algorithms, we present an algorithm to find Ramanujan-type identities for a(mn+t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n+6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions and and Andrews–Paule’s broken 2-diamond partition functions and . It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews’ singular overpartition functions and due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.
... In [5], Andrews introduced singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row. For integers K, i with 1 ≤ i < K/2, Andrews defined a subclass of singular overpartitions with some restrictions subject to K and i, namely (K, i)-singular overpartitions. ...
... He then showed interesting combinatorial and arithmetic properties of (K, i)singular overpartitions. As seen in [5], (K, i)-singular overpartitions are closely related to partitions counted by partition sieves, which were first employed by Andrews [1,2] to discover Rogers-Ramanujan type partitions and later generalized further by Bressoud [7]. ...
... where Q K,i,α,α (0) = 1 and (a; q) ∞ = lim n→∞ n j=0 (1 − aq j ). We note that Q K,i,1,1 (n) becomes the number of (K, i)-singular overpartitions of n given by Andrews in [5]. ...
Singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row, were first introduced by Andrews in 2015. In his paper, Andrews investigated an interesting subclass of singular overpartitions, namely, (K, i)-singular overpartitions for integers K, i with . The definition of such singular overpartitions requires successive ranks, parity blocks and anchors. The concept of successive ranks was extensively generalized to hook differences by Andrews, Baxter, Bressoud, Burge, Forrester and Viennot in 1987. In this paper, employing hook differences, we generalize parity blocks. Using this combinatorial concept, we define -singular overpartitions for positive integers with , and then we show some connections between such singular overpartitions and ordinary partitions.
... After Ramanujan died, H. Gupta extended MacMahon's table up to n = 300. Chowla [19] after observing the Gupta's table, found that p(243) is not divisible by 7 3 , despite the fact that 24 · 243 ≡ 1 (mod 7 3 ). To correct Ramanujan's conjecture, de ne ζ = Watson [73] published a proof of (1.10) for a = c = 0 and noticed a more detailed version of Ramanujan's proof of (1.10) in case b = c = 0. Finally, Atkin [6] proved (1.10) for arbitrary c and a = b = 0. ...
... After Ramanujan died, H. Gupta extended MacMahon's table up to n = 300. Chowla [19] after observing the Gupta's table, found that p(243) is not divisible by 7 3 , despite the fact that 24 · 243 ≡ 1 (mod 7 3 ). To correct Ramanujan's conjecture, de ne ζ = Watson [73] published a proof of (1.10) for a = c = 0 and noticed a more detailed version of Ramanujan's proof of (1.10) in case b = c = 0. Finally, Atkin [6] proved (1.10) for arbitrary c and a = b = 0. ...
... The total number of partitions of n with designated summands is denoted by P D(n). The authors [5] have derived the following generating function of P D(n): ∞ n=0 P D(n)q n = (q 6 ; q 6 ) ∞ (q; q) ∞ (q 2 ; q 2 ) ∞ (q 3 ; q 3 ...
We study several congruence properties of restricted partition functions such as:
k-color overpartition functions, Andrews' singular overpartitions, Designated summands, -regular cubic partition pairs, -regular bipartition triples and Partition quadruple with t-cores.
... Recently, George Andrews introduced a certain subclass of overpartitions, namely singular overpartitions which are Frobenius symbols with at most one overlined entry in each row [3]. For integers k, i with k ≥ 3 and 1 ≤ i < k, Andrews found interesting combinatorial and arithmetic properties of (k, i)-singular overpartitions, which are singular overpartitions with some restrictions subject to k and i. ...
... Because of the complexity of the restrictions, we defer the exact definition to Section 2.4. One of the main results of Andrews in [3] is the following. Theorem 1.1 (Andrews, [3]). ...
... One of the main results of Andrews in [3] is the following. Theorem 1.1 (Andrews, [3]). The number of (k, i)-singular overpartitions of n equals the number of overpartitions of n in which no part is divisible by k and only parts congruent to ±i mod k may be overlined. ...
Singular overpartitions, which are defined by George Andrews, are overpartitions whose Frobenius symbols have at most one overlined entry in each row. In his paper, Andrews obtained interesting combinatorial results on singular overpartitions, one of which relates a certain type of singular overpartitions with a subclass of overpartitions. In this paper, we provide a combinatorial proof of Andrews' result, which answers to one of his open questions.
... Overpartitions, first described by MacMahon [8], are important objects of study in partition theory [2,3,4,6]. Corteel and Lovejoy write, "the theory of basic hypergeometric series contains a wealth of information about overpartitions, [and] many theorems and techniques for ordinary partitions have analogues for overpartitions." [6] In this note, I prove the set of overpartitions (under the same partition multiplication operation) and the positive rational numbers (under multiplication in Q + ) are isomorphic as groups. ...
... Cuthbertson and Waldron both independently showed me that negative multiplicities can be interpreted as overlines. 2 One can directly verify the bijection between overlined parts and negative multiplicities. If λ ∈ P is the partition one forms from the parts of α ∈ O whose multiplicities are positive, and γ ∈ P the partition one forms from the parts with negative multiplicities, we also define a rational partition form for the overpartition by writing α = λ/γ ∈ O. ...
In a 2022 paper, Dawsey, Just and the present author prove that the set of integer partitions, taken as a monoid under a partition multiplication operation I defined in my Ph.D. work, is isomorphic to the positive integers as a monoid under integer multiplication. In this note, I extend partition multiplication to the set of overpartitions, which are of much interest in partition theory. I prove the overpartitions form an Abelian group under partition multiplication. Moreover, the overpartitions and the positive rational numbers are isomorphic as multiplicative groups. I then prove further overpartition isomorphisms and discuss approaches to a ring theory of overpartitions.
... As usual, we denote by Q(n) the number of integer partitions of n into distinct parts. For example, Q(7) = 5 because the five partitions of 7 into distinct parts are (7), (6,1), (5,2), (4,3), (4, 2, 1). ...
... Overpartitions were introduced by S. Corteel and J. Lovejoy in [8] and have been the subject of many recent studies including G. E. Andrews [2], K. Bringmann and J. Lovejoy [5], W. Y. C. Chen and J. J. Y. Zhao [7], S. Corteel and P. Hitczenko [9], S. Corteel, W. M. Y. Goh and P. Hitczenko [10], S. Corteel and O. Mallet [11], A. M. Fu and A. Lascoux [12], M. D. Hirschhorn and J. A. Sellers [24,25], Kim [27], J. Lovejoy [29][30][31][32][33][34], K. Mahlburg [36], M. Merca [40,42] and A. V. Sills [46]. ...
In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function Q(n) which counts partitions of n into distinct parts. Computing Q(n) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of n into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function Q(n). One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of Q(n). This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.
... As usual, we denote by Q(n) the number of integer partitions of n into distinct parts. For example, Q(7) = 5 because the five partitions of 7 into distinct parts are (7), (6,1), (5,2), (4,3), (4, 2, 1). ...
... Overpartitions were introduced by S. Corteel and J. Lovejoy in [8] and have been the subject of many recent studies including G. E. Andrews [2], K. Bringmann and J. Lovejoy [5], W. Y. C. Chen and J. J. Y. Zhao [7], S. Corteel and P. Hitczenko [9], S. Corteel, W. M. Y. Goh and P. Hitczenko [10], S. Corteel and O. Mallet [11], A. M. Fu and A. Lascoux [12], M. D. Hirschhorn and J. A. Sellers [24,25], Kim [27], J. Lovejoy [29][30][31][32][33][34], K. Mahlburg [36], M. Merca [40,42] and A. V. Sills [46]. ...
... The function A (n) was investigated by J. Lovejoy [10,11,12] and E. Y. Y. Shen [15]. Recently, G. E. Andrews [4] introduced a class of interesting overpartitions and referred to them as singular overpartitions. Singular overpartitions depend on two parameters and are enumerated by the function C k,i (n) which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
... In [4], Andrews remarks that the generating function for C k,i (n) can be expressed as ...
Let be the number of -regular overpartitions of n, i.e., overpartitions of n into parts not divisible by . Let be the number of almost -regular overpartitions of n, i.e., overpartitions of n in which none of its overlined parts is divisible by . In this paper, we study the connections between , respectively , and the singular overpartition functions and which count the number of overpartitions into parts not divisible by 12 and in which only parts congruent to , respectively , may be overlined. We give a combinatorial proof for the the surprising identity . We also provide a linear homogeneous recurrence relation for and give an alternate combinatorial interpretation for .
... A 4 (4p 2α n + (4j + 3p)p 2α−1 ) ≡ 0 (mod 16) where p ≥ 5 is a prime such that the Legendre symbol ( −2 p ) = −1, with n, α are positive integers. Andrews [3] defined the singular overpartition function C k,i (n), which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. For k ≥ 3 and 1 ≤ i ≤ k 2 , the generating function for C k,i (n) is given by ...
... It is noteworthy that C 3,1 (n) has attracted the attention of many mathematicians. Andrews [3] proved the following Ramanujan-type congruences: for n ≥ 0, ...
Let be the number of overpartitions of n into parts not divisible by . In this paper, we prove that is almost always divisible by if , where j is a fixed positive integer and with primes . We obtain a Ramanujan-type congruence for modulo 7. We also exhibit infinite families of congruences and multiplicative identities for .
... While many partition functions a(n) are of the form (1.1), there are partition functions that do not seem to fall into this framework, such as Andrews' (k, i)-singular overpartition function Q k,i (n). Andrews [3] derived the generating function ...
... For example, we can derive Ramanujan-type identities on the singular overpartition function introduced by Andrews [3]. The number of (k, i)-singular overpartitions of n is denoted by Q k,i (n) (1 ≤ i < k 2 ). ...
This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an algorithm to find Ramanujan-type identities for a(mn+t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n+6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions and and Andrews--Paule's broken 2-diamond partition functions and . It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews' singular overpartition functions and due to Shen, the 2-dissection formulas of Ramanujan and the 8-dissection formulas due to Hirschhorn.
... In a recent work, Andrews [1] de ned combinatorial objects that he called singular overpartitions. Moreover, Andrews proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function C k,i (n) which denotes the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
... Furthermore, by elementary generating function manipulations, Andrews [1] proved that for all n ≥ , ...
Singular overpartition functions were defined by Andrews. Let Ck , i ( n ) denote the number of ( k , i )-singular overpartitions of n , which counts the number of overpartitions of n in which no part is divisible by k and only parts ± i (mod k ) may be overlined. A number of congruences modulo 3, 9 and congruences modulo powers of 2 for Ck , i ( n ) were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pairs ( k , i ). In this paper, we prove some congruences modulo powers of 2 for C48, 6 ( n ) and C48, 18 ( n ).
... (1.1) Setting ℓ = 3 in (1.1), Shen [18] observed that A ! (n) = C !, (n), where C k,j (n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±j (mod k) may be overlined. This function was introduced and investigated by Andrews in [3]. As noted in [3], the generating function for C k,j (n) is given by ...
... This function was introduced and investigated by Andrews in [3]. As noted in [3], the generating function for C k,j (n) is given by ...
... In [1] and [9], the concept of oscillations of the successive ranks were introduced and used to study Rogers-Ramanujan type partition identities, and further study was made on variations of successive ranks in [5]. Recently, in the study of singular overpartitions [4], Andrews revisited successive ranks and parity blocks. ...
Successive ranks of a partition, which were introduced by Atkin, are the difference of the ith row and the ith column in the Ferrers graph. Recently, in the study of singular overpartitions, Andrews revisited successive ranks and parity blocks. Motivated by his work, we investigate partitions with prescribed successive rank parity blocks. The main result of this paper is the generating function of partitions with exactly d successive ranks and m parity blocks.
... An overpartition of n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a part may be overlined. Andrews [2] defined a new type of overpartitions, called singular overpartition to give the overpartition analogoue to Rogers-Ramanujan type theorems for ordinary partitions with restricted successive ranks. The singular overpartition function, C k,i (n), counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
Andrews introduced the partition function , called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts may be overlined. We study the parity and distribution results for where and . More particularly, we prove that for each integer depending on k and i, the interval resp. contains an integer n such that is even (resp. odd). Finally we study the distribution for where is a prime number.
... An overpartition of n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. Andrews [2] defined a new type of overpartitions, called singular overpartition to give the overpartition analogous to Rogers-Ramanujan type theorems for ordinary partitions with restricted successive ranks. The singular overpartition, C k,i (n), counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
Andrews introduced the partition function , called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts may be overlined. We study the parity and distribution results for where and . More particularly, we prove that for each integer depending on k and i, the interval resp.\ contains an integer n such that is even (resp.\ odd). Finally we study the distribution for where be a prime number.
... The properties of the overpartitions have been the subject of many recent studies of Andrews [1], Corteel and Lovejoy [4], Hirschhorn [6], Kim [9], Lovejoy [11,12,13], Mahlburg [14], Merca [15,16,17], and Merca, Wang and Yee [18]. ...
Let denote the number of overpartitions of n into odd parts. In this paper, we provide a complete characterization of Ramanujan-type congruences modulo 8 for the overpartition function considering the number of odd positive divisors of and the relations of the form
with and
... An overpartition of n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function with restricted successive ranks, Andrews [2] defined the so-called singular overpartitions. Andrews' singular overpartition function C k,i (n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely p t,t (n) and p 2t,t (n). We establish identities connecting the ordinary partition function p(n) to p t,t (n) and p 2t,t (n) for all t ≥ 1. Using these identities, we prove that Ramanujan's famous congruences for p(n) are also satisfied by p t,t (n) and p 2t,t (n) for infinitely many values of t. Very recently, da Silva and Sellers provided complete parity characterizations of p 1,1 (n) and p 3,3 (n). We prove that p t,t (n) ≡ C 4t,t (n) (mod 2) for all n ≥ 0 and t ≥ 1, where C 4t,t (n) is Andrews' singular overpartition function. Using this congruence, the parity characterization of p 1,1 (n) given by da Silva and Sellers follows from that of C 4,1 (n). We also give elementary proofs of certain congruences already proved by da Silva and Sellers.
... Overpartitions were introduced by Corteel and Lovejoy in [7] and have been the subject of many recent studies including Andrews [2], Bringmann and Lovejoy [4], Chen and Zhao [6], Corteel and Hitczenko [8], Corteel, Goh and Hitczenko [9], Corteel and Mallet [10], Fu and Lascoux [12], Hirschhorn and Sellers [14,15], Kim [17], Lovejoy [19,20,21,22,23,24], Mahlburg [25], Merca [27] and Sills [30]. ...
In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function p ( n ). Computing p ( n ) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of p ( n ) that requires only the values of p ( k ) with k ≤ n /2. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p ( n ) to obtain a simple and fast computation of the value of p ( n ). This new method uses only (large) integer arithmetic and it is simpler to program.
... Thus the 14 overpartitions of 4 are 4, 4, 3 + 1, 3 + 1, 3 + 1, 3 + 1, 2 + 2, 2 + 2, 2 + 1 + 1, 2 + 1 + 1, 2 + 1 + 1, 2 + 1 + 1, 1 + 1 + 1 + 1, 1 + 1 + 1 + 1. In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function with restricted successive ranks, Andrews [2] defined the so-called singular overpartitions. Andrews' singular overpartition function C k,i (n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
Andrews introduced the singular overpartition function C‾k,i(n) which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. In this article, we study the divisibility properties of C‾4k,k(n) and C‾6k,k(n) by arbitrary powers of 2 and 3 for infinite families of k. For an infinite family of k, we prove that C‾4k,k(n) is almost always divisible by arbitrary powers of 2. We also prove that C‾6k,k(n) is almost always divisible by arbitrary powers of 3 for an infinite family of k. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of 2 satisfied by C‾4⋅2α,2α(n) and C‾4⋅3⋅2α,3⋅2α(n).
... Thus the eight overpartitions of 3 are 3, 3, 2 + 1, 2 + 1, 2 + 1, 2 + 1, 1 + 1 + 1, 1 + 1 + 1. In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function with restricted successive ranks, Andrews [1] defined the so-called singular overpartitions. Andrews' singular overpartition function C k,i (n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
Andrews' -singular overpartition function counts the number of overpartitions of n in which no part is divisible by k and only parts may be overlined. In recent times, divisibility of , and by 2 and 3 are studied for certain values of . In this article, we study divisibility of , and by primes . For all positive integer and prime divisors of , we prove that , and are almost always divisible by arbitrary powers of p. For , we next show that the set of those n for which is infinite, where k is a positive integer satisfying . We further improve a result of Gordon and Ono on divisibility of -regular partitions by powers of certain primes. We also improve a result of Ray and Chakraborty on divisibility of -regular overpartitions by powers of certain primes.
... Thus the eight overpartitions of 3 are 3, 3, 2 + 1, 2 + 1, 2 + 1, 2 + 1, 1 + 1 + 1, 1 + 1 + 1. In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function with restricted successive ranks, Andrews [2] defined the so-called singular overpartitions. Andrews' singular overpartition function C k,i (n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function, Andrews defined the so-called singular overpartitions. Singular overpartition function C‾k,i(n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. Andrews also proved two beautiful Ramanujan type congruences modulo 3 satisfied by C‾3,1(n). Later on, Aricheta proved that for an infinite family of ℓ, C‾3ℓ,ℓ(n) is almost always divisible by 2. In this article, for an infinite subfamily of ℓ considered by Aricheta, we prove that C‾3ℓ,ℓ(n) is almost always divisible by arbitrary powers of 2. We also prove that C‾3ℓ,ℓ(n) is almost always divisible by arbitrary powers of 3 when ℓ=3,6,12,24. Proofs of our density results rely on the modularity of certain eta-quotients which arise naturally as generating functions for the Andrews' singular overpartition functions.
... An overpartition of n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function with restricted successive ranks, Andrews [2] defined the so-called singular overpartitions. Andrews' singular overpartition function C k,i (n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. ...
In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely and . We establish identities connecting the ordinary partition function p(n) to and for all . Using these identities, we prove that the Ramanujan's famous congruences for p(n) are also satisfied by and for infinitely many values of t. Very recently, da Silva and Sellers provide complete parity characterizations of and . We prove that for all and , where is the Andrews' singular overpartition function. Using this congruence, the parity characterization of given by da Silva and Sellers follows from that of . We also give elementary proofs of certain congruences already proved by da Silva and Sellers.
... Overpartitions were introduced by Corteel and Lovejoy in [7] and have been the subject of many recent studies including Andrews [2], Bringmann and Lovejoy [4], Chen and Zhao [6], Corteel and Hitczenko [8], Corteel, Goh and Hitczenko [9], Corteel and Mallet [10], Fu and Lascoux [12], Hirschhorn and Sellers [14,15], Kim [17], Lovejoy [19][20][21][22][23][24], Mahlburg [25], Merca [27] and Sills [30]. ...
In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function . Computing by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of that requires only the values of with . This formula is combined with a known linear homogeneous recurrence relation for the overpartition function to obtain a simple and fast computation of the value of . This new method uses only (large) integer arithmetic and it is simpler to program.
... The properties of the overpartitions have been the subject of many recent studies of Andrews [2], Corteel and Lovejoy [5], Hirschhorn [7], Kim [9], Lovejoy [12,13,14], Mahlburg [15] and Merca [19]. From our example, we see that p o (3) = 4. ...
In this paper, we invoke the bisectional pentagonal number theorem to prove that the number of overpartitions of the positive integer n into odd parts is equal to twice the number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20 or . This result allows us to experimentally discover new infinite families of linear partition inequalities involving Euler's partition function p(n). In this context, we conjecture that for , the theta series
has non-negative coefficients.
... An overpartition of n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. In [2], Andrews defined the partition function C k,i (n), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. For example, C 3,1 (4) = 10 with the relevant partitions being 4, 4, 2 + 2, 2 + 2, 2 + 1 + 1, 2 + 1 + 1, 2 + 1 + 1, 2 + 1 + 1, 1 + 1 + 1 + 1, 1 + 1 + 1 + 1. ...
Andrews introduced the partition function , called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts may be overlined. He also proved that and are divisible by 3 for . Recently Aricheta proved that for an infinite family of k, is almost always even. In this paper, we prove that for any positive integer k, is almost always divisible by and $3^k.
... An overpartition of n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. In Reference [2], Andrews defined the partition function C k,i (n), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. For example, C 3,1 (4) = 10 with the relevant partitions being 4, 4, 2 + 2, 2 + 2, 2 + 1 + 1, 2 + 1 + 1, 2 + 1 + 1, 2 + 1 + 1, 1 + 1 + 1 + 1, 1 + 1 + 1 + 1. ...
Andrews introduced the partition function C¯k,i(n), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. He also proved that C¯3,1(9n+3) and C¯3,1(9n+6) are divisible by 3 for n≥0. Recently Aricheta proved that for an infinite family of k, C¯3k,k(n) is almost always even. In this paper, we prove that for any positive integer k, C¯3,1(n) is almost always divisible by 2k and 3k.
... We see thatp(3) = 8. The properties of the overpartitions functionp(n) have been the subject of many recent studies [3,8,13,15,[17][18][19][21][22][23][24][25]. The partition functions p(n),p(n) and S e−o (n) are connected to r 4 (n) by the following result. ...
We consider the function which gives the number of ways to write n as the sum of s squares. Since the generating functions for and are Lambert series, we use Merca’s factorization theorem for Lambert series to establish relationships between these functions and partitions into distinct parts. We also obtain convolutions involving overpartition functions as well as pentagonal recurrence formulas for and . These results lead to new connections between divisors and partitions.
... and its properties have been the subject of many recent studies [4,6,8,9,10,11,12,13]. It is clear that, the partition function R 2 (n) can be expressed in terms of the overpartition functionp(n) as combinatorial interpretations of Theorems 6.1 and 6.2. ...
Let denote the number of partitions of n into parts that are odd or congruent to �. In 2007, Andrews considered partitions with some negative parts and provided a second combinatorial
interpretation for . In this paper, we give a collection of linear recurrence relations for the partition function . As a corollary, we obtain a simple criterion for deciding whether is odd or even.
Some identities involving overpartitions and partitions into distinct parts are derived in this context.
... In [1] and [9], the concept of oscillations of the successive ranks were introduced and used to study Rogers-Ramanujan type partition identities, and further study was made on variations of successive ranks in [5]. Recently, in the study of singular overpartitions [4], Andrews revisited successive ranks and parity blocks. ...
Successive ranks of a partition, which were introduced by Atkin, are the difference of the ith row and the ith column in the Ferrers graph. Recently, in the study of singular overpartitions, Andrews revisited successive ranks and parity blocks. Motivated by his work, we investigate partitions with prescribed successive rank parity blocks. The main result of this paper is the generating function of partitions with exactly d successive ranks and m parity blocks.
... We see thatp(3) = 8. Properties of the overpartitions functionp(n) have been the subject of many recent studies [4,6,10,12,13,14,15,16]. The partition function ped(n) can be expressed in terms of the overpartitions functionp(n) as follows. ...
The number of partitions of n wherein even parts are distinct and odd parts are unrestricted, often denoted by ped(n)ped(n), has been the subject of many recent studies. In this paper, the author provides an efficient linear recurrence relation for ped(n)ped(n). A simple criterion for deciding whether ped(n)ped(n) is odd or even is given as a corollary of this result. Some connections with partitions into parts not congruent to 2(mod4), overpartitions and partitions into distinct parts are presented in this context.
The primary purpose of this article is to provide an explicit interpretation of \textit{Ramanujan}'s Partition Congruences, namely, p(5n+4)\equiv 0\mbox{ }(mod\mbox{ }5), p(7n+5)\equiv 0\mbox{ }(mod\mbox{ }7) and, p(11n+6)\equiv 0\mbox{ }(mod\mbox{ }11), p(n) being the \textit{partition function} corresponding to any positive integer n, in terms of their corresponding \textit{Cranks} using various combinatorial arguments implemented by \textit{Dyson}, \textit{Atkin}, \textit{Swinnerton-Dyer} and later by \textit{Andrews} and, \textit{Garvan} to justify all the necessary concepts pertaining to this topic.
2020 MSC: Primary 11-02, 11P81, 11P82, 11P83, 11P84. Secondary 11B75, 05A16, 05A17, 05A30.
For two integers , we define (k, j)-colored partitions to be those partitions in which parts may appear in k different types and at most j types can appear for a given part size. Let be the number of (k, j)-colored partitions of n. Recently, Keith studied (k, j)-colored partitions and proved the following results: For , we have for all . For , we have for all . In this paper, we determine all a, b, c, j with (a,b)=1 and such that for all nonnegative integers n.
Lovejoy introduced the partition function as the number of 𝑙-regular overpartitions of 𝑛. Andrews defined (𝑘, 𝑖)-singular overpartitions counted by the partition function , and pointed out that . Meanwhile, Andrews derived an interesting divisibility property that (mod 3). Recently, we constructed the partition statistic 𝑟 𝑙 -crank of 𝑙-regular overpartitions and give combinatorial interpretations for some congruences of as well as the congruences of Andrews. In this paper, we aim to prove some equalities for the 𝑟 3 -crank of 3-regular overpartitions.
The primary purpose of this article is to provide an explicit interpretation of Ramanujan’s Partition Congruences, namely, p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7) and, p(11n + 6) ≡ 0 (mod 11), p(n) being the partition function corresponding to any positive integer n, in terms of their corresponding Cranks using various combinatorial arguments implemented by Dyson, Atkin, Swinnerton-Dyer and later by Andrews and, Garvan to justify all the necessary concepts pertaining to this topic.
In addition to introducing readers to the Theory of Partitions, as well as defining formally the notion of q-Series and Ramanujan’s Theta Function, we shall further deduce a rough estimate for p(n), a priori applying the so called Jacobi Triple Product Identity and Euler Pentagonal Theorem, and subsequently, providing a thorough explanation for the special case, when n ≡ 4 (mod 5).
Important to mention that, a whole section in this article has been dedicated towards gaining an detailed understanding of Rank of a partition and Crank of a Vector Partition, both introduced by Dyson, due to its significance and relevance to our study. Furthermore, in the later half of the text, we shall indeed prove our main result with the help of an important property of the partition function M(m, j, n) corresponding to any positive integer n with crank congruent to m modulo j. Exclusively, in the final section, rigorous derivations have been provided individually for each case to facilitate the motivated readers of this paper with a better understanding of this subject. Moreover, adequate references have been included, considering the broader aspect of research in this area.
2020 MSC: Primary 11-02, 11P81, 11P82, 11P83, 11P84. Secondary 11B75, 05A16, 05A17, 05A30.
Andrews introduced the partition function , called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts may be overlined. We prove that is almost always divisible by for any positive integer k . We also prove that and are almost always divisible by . Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of 2 satisfied by .
Let [Formula: see text] be the number of [Formula: see text]-colored partitions of [Formula: see text]. Recently, Keith proved that for [Formula: see text], if [Formula: see text] for all [Formula: see text], then [Formula: see text] is large. We prove that such [Formula: see text] do not exist. Furthermore, for any positive integers [Formula: see text] with [Formula: see text], there exist infinitely many positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text].
Purpose
In this paper, the author defines the function B ¯ i , j δ , k ( n ) , the number of singular overpartition pairs of n without multiples of k in which no part is divisible by δ and only parts congruent to ± i , ± j modulo δ may be overlined.
Design/methodology/approach
Andrews introduced to combinatorial objects, which he called singular overpartitions and proved that these singular overpartitions depend on two parameters δ and i can be enumerated by the function C ¯ δ , i ( n ) , which gives the number of overpartitions of n in which no part divisible by δ and parts ≡ ± i (Mod δ ) may be overlined.
Findings
Using classical spirit of q -series techniques, the author obtains congruences modulo 4 for B ¯ 2,4 8,3 ( n ) , B ¯ 2,4 8,5 and B ¯ 2,4 12,3 .
Originality/value
The results established in this work are extension to those proved in Andrews’ singular overpatition pairs of n .
An [Formula: see text]-regular overpartition of [Formula: see text] is an overpartition of [Formula: see text] into parts not divisible by [Formula: see text]. Let [Formula: see text] be the number of [Formula: see text]-regular overpartitions of [Formula: see text]. Andrews defined singular overpartitions counted by the partition function [Formula: see text]. It denotes the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts[Formula: see text] may be overlined. He proved that [Formula: see text] and [Formula: see text] are divisible by [Formula: see text]. In this paper, we aim to introduce a crank of [Formula: see text]-regular overpartitions for [Formula: see text] to investigate the partition function [Formula: see text]. We give combinatorial interpretations for some congruences of [Formula: see text] including infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text] as well as the congruences of Andrews for [Formula: see text] and [Formula: see text].
Let be the number of overpartitions of n into parts not divisible by . In this paper, we prove that is almost always divisible by if , where j is a fixed positive integer and with primes . We obtain a Ramanujan-type congruence for modulo 7. We also exhibit infinite families of congruences and multiplicative identities for .
Let denote the number of Andrews singular overpartition pairs of n in which no parts are divisible by δ and only parts congruent to ± i, ± j modulo δ may be overlined. In this paper, we establish some new infinite families of congruences modulo powers of 2 and 3 for , the number of Andrews singular overpartition pairs of n with odd parts in which no parts are divisible by δ and only parts congruent to ± i, ± j modulo δ may be overlined. For example, for all n ≥ 0 and β ≥ 0,
where a1 ∈{22,34,46,58}.
In a recent work, Andrews defined the singular overpartition functions, denoted by , which count the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. Moreover, many congruences modulo 3, 9 and congruences modulo powers of 2 for were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pair (k, i). In this paper, we proved new infinite families of congruences modulo 27 for and infinite families of congruences modulo 4 and 8 for , , .
Singular overpartitions, which were defined by George Andrews, are overpartitions whose Frobenius symbols have at most one overlined entry in each row. In his paper, Andrews obtained interesting combinatorial results on singular overpartitions, one of which relates a certain type of singular overpartition with a subclass of overpartitions. In this paper, we provide a combinatorial proof of Andrews’s result, which answers one of his open questions.KeywordsPartitionsOverpartitionsFrobenius symbolsSingular overpartitionsDyson’s mapWright’s map2010 Mathematics Subject Classification05A1711P81
In a recent work, Andrews defined the singular overpartitions with the goal of presenting an overpartition analogue to the theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo 3 for the number of singular overpartitions prescribed by parameters k=3 and i=1. It should be noticed that this number equals the number of the Rogers--Ramanujan--Gordon type overpartitions with k=i=3 which come from the overpartition analogue of Gordon's Rogers--Ramanujan partition theorem introduced by Chen, Sang and Shi. In this paper, we derive numbers of congruence identities modulo 4 for the number of Rogers--Ramanujan--Gordon type overpartitions.
Let A3(n) and A9(n) denote the number of 3-and 9-regular overpartitions of n. For each α > 0, we obtain the generating functions for A3(32α n), A3(32α−1 n) and A9(3α n). We show that A3(n) and A9(n) satisfy certain internal congruences.
A partition of n is called a t-core partition of n if none of its hook numbers are multiples of t. Let the number of t-core partitions of n be denoted by . Recently, G. E. Andrews defined combinatorial objects which he called singular overpartitions, overpartitions of n in which no part is divisible by k and only parts may be overlined. Let the number of singular overpartitions of n be denoted by . The object of this paper is to obtain new congruences modulo 2 for and . We also obtain congruences modulo 2 for and .
We explore new combinatorial properties of overpartitions which are natural generalizations of integer partitions. Building on recent work we state general combinatorial identities between standard partition, overpartition and l-regular partition, functions. We provide both generating function and bijective proofs. We also prove the congruences for certain overpartition functions combinatorially.
Let be the number of overpartitions of n into parts not divisible by . In a recent paper, Shen calls the overpartitions enumerated by the function as -regular overpartitions. In this paper, we find certain congruences for , when , and 9. Recently, Andrews introduced the partition function , called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts may be over-lined. He also proved that and are divisible by 3. In this paper, we prove that is divisible by 144 which was conjectured to be true by Naika and Gireesh.
In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result
of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem
of Watson on basic hypergeometric series, generating functions and miscellaneous methods.
Polynomial generalizations of all 130 of the identities in Slater's list of identities of the Rogers-Ramanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial identities were previously known but many are new. The author has implemented much of the nitization process in a Maple package which is available for free download from the author's website.
This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17.
Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook.
We give one-parameter overpartition-theoretic analogues of two classical families of partition identities: Andrews' combinatorial generalization of the Gollnitz-Gordon identities and a theorem of Andrews and Santos on partitions with attached odd parts. We also discuss geometric counterparts arising from multiple q-series identities. These involve representations of overpartitions in terms of generalized Frobenius partitions.
We demonstrate the correspondence which lies behind certain partition identities used by Andrews in his partition sieve. This leads to an extension of his methods and a generalization of his results.
A proof of the Rogers-Ramanujan identities is presented which is brief, elementary, and well motivated; the “easy” proof of whose existence Hardy and Wright had despaired. A multisum generalization of the Rogers-Ramanujan identities is shown to be a simple consequence of this proof.
We discuss a generalization of partitions, called overpartitions, which have proven useful in several combinatorial studies of basic hyper- geometric series. After showing how a number of nite products occurring in q-series have natural interpretations in terms of overpartitions, we present an introduction to their rich structure as revealed by q-series identities. 4; 4; 3+1 ; 3+1 ; 3+ 1; 3+ 1; 2+2 ; 2+2 ; 2+1+1 ; 2+1+1 ; 2+ 1+1 ; 2+ 1+1 ; 1+1+1+1 ; 1+1+1+1 : These objects are natural combinatorial structures associated with the q-binomial theorem, Heine's transformation, and Lebesgue's identity (see (20) for a summary with references). In (18), they formed the basis for an algorithmic approach to the combinatorics of basic hypergeometric series. More recently, overpartitions have been found at the heart of bijective proofs of Ramanujan's 1 1 summation and the q-Gauss summation (14), (15). It should come as no surprise, then, that the theory of basic hypergeometric series contains a wealth of information about overpartitions
We discuss conjugation and Dyson's rank for overpartitions from the perspective of the Frobenius representation. More specifically, we translate the classical definition of Dyson's rank to the Frobenius representation of an overpartition and define a new kind of conjugation in terms of this representation. We then use q-series identities to study overpartitions that are self-conjugate with respect to this conjugation.
Generalized Frobeniuspartitionsor F-partitions
have recently playedan important role in severalcombinatorial investigations of basic
hypergeometric series identities. The goal of this paper isto use the framework ofthese investigationsto interpret families of infinite productsas generating functions for F-partitions.We employ q-series identities and bijective
combinatorics.
We give overpartition-theoretic analogues of certain combinatorial generalizations of the Rogers–Ramanujan identities.
It is shown how each of the classical identities of Rogers-Ramanujan type can be embedded in an infinite family of multiple series identities. The method of construction is applied to four of L. J. Rogers’ elegant series related to the quintuple product identity. Other applications are also presented.
Abstracts of Short Communications (I.C.M
- B Gordon