Ramanujan's Partition Congruence Research Articles

A survey on t-core partitions

$t$-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on $t$-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on $t$-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of $t$-cores in number theoretic aspects of partition theory. This includes the modularity of $t$-core partition generating functions, the existence of $t$-core partitions, asymptotic formulas and arithmetic properties of $t$-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of $t$-core partitions, which include relations between core partitions and self-conjugate core partitions, a $t$-core crank explaining Ramanujan's partition congruences, and relations with class numbers.

Ramanujan's circular summation, t-cores and twisted partition identities

In this paper, we give new evaluations for Ramanujan's circular summation function. We also provide simpler proofs for known evaluations and give some generalizations. We discover modular relations among circular summation function partition function and give uniform proof of Ramanujan's partition congruences for the moduli 5, 7 and 11. We also prove several interesting congruence relations.

Dancing samba with Ramanujan partition congruences

The article presents an algorithm to compute a C[t]-module basis G for a given subalgebra A over a polynomial ring R=C[x] with a Euclidean domain C as the domain of coefficients and t a given element of A. The reduction modulo G allows a subalgebra membership test. The algorithm also works for more general rings R, in particular for a ring R⊂C((q)) with the property that f∈R is zero if and only if the order of f is positive. As an application, we algorithmically derive an explicit identity (in terms of quotients of Dedekind η-functions and Klein's j-invariant) that shows that p(11n+6) is divisible by 11 for every natural number n where p(n) denotes the number of partitions of n.

A short and simple proof of Ramanujan's mod 11 partition congruence

We present a short and simple proof of Ramanujan's partition congruence p(11n+6)≡0(mod11), and dedicate it to him on the occasion of his 125th birthday.

Congruences for Andrews’ spt-function modulo powers of $5$, $7$ and $13$

Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo p for all primes p>3 which were conjectured earlier by the author. We extend Ono's method to handle the powers of 5, 7 and 13 congruences. We need the theory of weak Maass forms as well as certain classical modular equations for the Dedekind eta-function.

New identities for 7-cores with prescribed BG-rank

Let π be a partition. BG-rank ( π ) is defined as an alternating sum of parities of parts of π [A. Berkovich, F.G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc. 358 (2006) 703–726. [1]]. Berkovich and Garvan [The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear in 〈 http://arxiv.org/abs/math/0602362 〉 ] found theta series representations for the t-core generating functions ∑ n ⩾ 0 a t , j ( n ) q n , where a t , j ( n ) denotes the number of t-cores of n with BG-rank = j . In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme values of BG-rank. In this paper we discuss representations of this type for all 7-cores with prescribed BG-rank. We make an essential use of the Ramanujan modular equations of degree seven [B.C. Berndt, Ramanujan's Notebooks, Part III, Springer, New York, 1991] to prove a variety of new formulas for the 7-core generating function ∏ j ⩾ 1 ( 1 - q 7 j ) 7 ( 1 - q j ) . These formulas enable us to establish a number of striking inequalities for a 7 , j ( n ) with j = - 1 , 0 , 1 , 2 and a 7 ( n ) , such as a 7 ( 2 n + 2 ) ⩾ 2 a 7 ( n ) , a 7 ( 4 n + 6 ) ⩾ 10 a 7 ( n ) . Here a 7 ( n ) denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only.

Quintuple products and Ramanujan's partition congruence p(11n+6)≡0 (mod 11)

Quintuple products and Ramanujan's partition congruence p(11n+6)≡0 (mod 11)

A New Identity for (q;q) 10 with an Application to Ramanujan's Partition Congruence Modulo 11

Our principal second goal is to show that this identity leads to a short proof of Ramanujan’s famous congruence p(11n+6) ≡ 0 (mod 11), where p(n) denotes the number of unrestricted partitions of the positive integer n. Our proof of Theorem 2.2 is short but depends upon some results of Ramanujan from his notebooks [18]. In Section 4, we give a more elementary proof, based upon Ramanujan’s 1ψ1 summation formula, of Ramanujan’s principal result, Lemma 2.1, which is employed in our proof of Theorem 2.2. In Section 5, we give an entirely different and direct proof of Theorem 2.2 based on several elementary identities for theta functions due to Ramanujan. In fact, during our first proof of Theorem 2.2, we establish a special case of a general result on Eisenstein series found in Ramanujan’s lost notebook [19, p. 369]. More precisely, Ramanujan asserts that every member of a certain class of infinite series can be expressed in terms of Ramanujan’s Eisenstein series P , Q, and R (to be defined in Section 6). A less precise version of this claim appears in Ramanujan’s notebooks [18], [2, p. 65, Entry 35(i)], but we prove the better version in Section 6. D. Stanton empirically discovered an analogue of Theorem 2.2, and in Section 7 we give a proof of Stanton’s identity. In Section 8, we prove that (q; q) ∞ is lacunary; in this connection, see a result of J.–P. Serre [20].

A New Identity for (q;q) 10 with an Application to Ramanujan's Partition Congruence Modulo 11

A New Identity for (q;q) 10 with an Application to Ramanujan's Partition Congruence Modulo 11

More cranks andt-cores

Dedicated to George Szekeres on the occasion of his 90th BirthdayIn 1990, new statistics on partitions (calledcransk) were found which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan's partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.

Powers of Euler's Product and Related Identities

Ramanujan's partition congruences can be proved by first showing that the coefficients in the expansions of (q; q)r∞ satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r, the coefficients in the expansions of (q; q)r∞ satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to above. We also obtain arithmetic relations for the coefficients in the expansions of (q; q)r∞(qt; qt)s∞, for t = 2, 3, 4 and various values of r and s. Our proofs are explicit and elementary, and make use of the Macdonald identities of ranks 1 and 2 (which include the Jacobi triple product, quintuple product and Winquist's identities). The paper concludes with a list of conjectures.

A reformulation of Ramanujan's partition congruences

The Ramanujan partition congruences concern p ( nδ + λ ), where δ = 5 α 7 β 11 γ , 24 λ ≡ 1 (mod δ ) and 0 < λ < δ . We give an explicit expression for λ in terms of δ and a new formulation of the partition congruences.

A Reformulation of Ramanujan's Partition Congruences

The Ramanujan partition congruences concern p ( nδ + λ ), where δ = 5 α 7 β 11 γ , 24 λ ≡ 1 (mod δ ) and 0 < λ < δ . We give an explicit expression for λ in terms of δ and a new formulation of the partition congruences.

On the Equivalence of Ramanujan's Partition Identities and a Connection with the Rogers–Ramanujan Continued Fraction

A famous identity of Ramanujan connected with partitions modulo 5 is shown to be equivalent to another identity of Ramanujan. The latter identity is used to establish a differential equation for the Rogers–Ramanujan continued fraction found in Ramanujan's lost notebook. We also prove that two other identities of Ramanujan are equivalent, one of which is associated with Ramanujan's partition congruence modulo 7. Last, we give a new proof of the transformation formula for the Dedekind eta-function, which is used in our proofs of equivalence.

Ramanujan's partition congruences

Simple, uniform proofs are given of Ramanujan's partition congruences p(5n+4)≡0 mod 5, p(7n+5)≡0 mod 7, p(11n+6)≡0 mod 11 . Also, generating functions are obtained for the partitions involved in Ramanujan's more general congruences.

On the rank and the crank modulo $4$ and $8$

In this paper we prove some identities, conjectured by Lewis, for the rank and crank of partitions concerning the modulo 4 and 8. These identities are similar to Dyson's identities for the rank modulo 5 and 7 which give a combinatorial interpretation to Ramanujan's partition congruences. For this, we use multisection of series and some of the results that Watson established for the third order mock theta functions.

On some relations for the rank moduli 9 and 12

In this paper we prove some identities, conjectured by Lewis, concerning the rank moduli 9 and 12, which are similar to Dyson's identities for the rank moduli 5 and 7 which give a combinatorial explanation to Ramanujan's partition congruences. For this we use multisection of series and some identities for the third and sixth order Mock theta functions, in such a way that all the identities for a given modulus reduce to a single theta identity.

Proof of Ramanujan's Partition Congruence for the Modulus 11 3

Proof of Ramanujan's Partition Congruence for the Modulus 11 3