Partition Congruences Research Articles

Congruences modulo powers of 2 for a certain partition function

We study the divisibility properties of the coefficients c(n) defined by $$\prod_{n=1}^\infty\frac{1}{(1-q^n)^2(1-q^{3n})^2}=\sum _{n=0}^\infty c(n)q^n.$$ An analogue of Ramanujan’s partition congruences is obtained for certain coefficients c(n) modulo powers of 2. Furthermore, an analogue of the identity that Hardy regarded as Ramanujan’s most beautiful is proved.

Partitions weighted by the parity of the crank

The ‘crank’ is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank M e ( n ) minus the number of partitions of n with odd crank M o ( n ) . We also discuss the combinatorial implications of q-series identities involving M e ( n ) − M o ( n ) . Finally, we determine the exact values of M e ( n ) − M o ( n ) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.

Convolution Sums of Some Functions on Divisors

One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining $r_s(n)$ and $\delta_s(n)$, $s=4,$ $8$, in terms of $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $\delta_s(n)$ denotes the number of representations of $n$ as a sum of $s$ triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions.

On the work of Basil Gordon

A review is given of several aspects of the work of Basil Gordon. These include: Rogers–Ramanujan identities, plane partitions, the method of weighted words, modular forms and partition congruences, and the asymptotics of partitions and related q-series.

Partition congruences and the Andrews-Garvan-Dyson crank

In 1944, Freeman Dyson conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

On the Andrews-Stanley refinement of Ramanujan’s partition congruence modulo 5 and generalizations

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.

Partition congruences by involutions

We present a general construction of involutions on integer partitions which enables us to prove a number of modulo 2 partition congruences.

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.

Partition identities involving gaps and weights

We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers–Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.

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.

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.

Some Congruences for Partitions that are p -Cores

A number of linear congruences modulo r are proved for the number of partitions that are p-cores where p is prime, 5 ⩽ p ⩽ 23, and r is any prime divisor of ½(p − 1). Analogous results are derived for the number of irreducible p-modular representations of the symmetric group Sn. The congruences are proved using the theory of modular forms.

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.

The crank of partitions mod 8,9 and 10

Recently new combinatorial interpretations of Ramanujan’s partition congruences modulo 5 5 , 7 7 and 11 11 were found. These were in terms of the crank. A refinement of the congruence modulo 5 5 is proved. The number of partitions of 5 n + 4 5n + 4 with even crank is congruent to 0 0 modulo 5 5 . The residue of the even crank modulo 10 10 divides these partitions into five equal classes. Other relations for the crank modulo 8 8 , 9 9 and 10 10 are also proved. The dissections of certain generating functions associated with these results are calculated. All of the results are proved by elementary methods.

On Ramanujan’s modular identities

For Ramanujan’s modular identities connected with his well-known partition congruences for the moduli 5 or 7, we had given, in an earlier paper, natural and uniform proofs through the medium of modular forms. Analogous (modular) identities corresponding to the (more difficult) case of the modulus 11 are provided here, with the consequent partition congruences; the relationship with relevant results of N J Fine is also sketched.

A simple proof of Watson's partition congruences for powers of 7

AbstractRamanujan conjectured that ifnis of a specific form thenp(n), the number of unrestricted partitions ofn, is divisible by a high power of 7. A modified version of Ramanujan's conjecture was proved by G. N. Watson.In this paper we establish appropriate generating formulae, from which Watson's results follow easily.Our proofs are more straightforward than those of Watson. They are elementary, depending only on classical identities of Euler and Jacobi. Watson's proofs rely on the modular equation of seventh order. We also need the modular equation but we derive it using the elementary techniques of O. Kolberg.