On the number of partitions with designated summands

Abstract

Andrews, Lewis and Lovejoy introduced the partition function P D(n) as the number of partitions of n with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that P D(3n + 2) is divisible by 3. We obtain a Ramanujan type identity for the generating function of P D(3n + 2) which implies the congruence of Andrews, Lewis and Lovejoy. For P D(3n), Andrews, Lewis and Lovejoy showed that the generating function can be expressed as an infinite product of powers of (1 − q 2n+1 ) times a function F (q 2 ). We find an explicit formula for F (q 2 ), which leads to a formula for the generating function of P D(3n). We also obtain a formula for the generating function of P D(3n + 1). Our proofs rely on Chan's identity on Ramanujan's cubic continued fraction and some identities on cubic theta functions. By introducing a rank for the partitions with designed summands, we give a combinatorial interpretation of the congruence of Andrews, Lewis and Lovejoy.

Congruences modulo 9 for bipartitions with designated summands

“…In this section, we first introduce a generalized crank which called pd-crank for bipartitions with designated summands. The definition of the pd-crank relies on the construction of the following bijection ∆ which Chen, Ji, Jin and the second author established in [7]. There is a bijection ∆ between the set of partitions of n with designated summands and the set of vector partitions (α, β) with |α| + |β| = n, where α is an ordinary partition and β is a partition into parts ≡ ±1 (mod 6).…”

“…In particular, they obtained a 2 -dissection formula for the generating function of P D(n) and a Ramanujan-type congruence as given by [22] also investigated the arithmetic properties of the partition function P D(n) . He proved several infinite families of congruences modulo 9 and 27 for P D(n) by utilizing the generating function of P D(3n) and P D(3n + 1) derived in [8]. Xia [22] also found some congruences modulo 27 for P D(n) by employing some results due to Newman [19].…”

“…([7, Definition 3.2]) Let λ be a partition with designated summands and let (α, β) = ∆(λ). Then the pd-rank of λ, denoted r d (λ), is defined by…”

Arithmetic properties of bipartitions with designated summands

“…He considered partitions with designated summands wherein exactly k different magnitudes occur among all the parts; see also Andrews and Rose [2]. Recently, Chen et al [6] established a Ramanujan-type identity for the partition function P D(3n + 2) which implies the congruence of Andrews et al [1] and also gave a combinatorial interpretation of the congruence for P D(3n +2) by introducing a rank for partitions with designated summands. More recently, Xia [12] studied various arithmetic properties of P D(n) by employing the generating functions to P D(3n) and P D(3n + 2) due to Chen et al [6].…”

“…= −1, then the congruence relation (4.51) holds if and only if both k = m = ± p−16 . Substitute (2.10) into (4.49) and extract the terms in which the powers of q are congruent to p 2 −1 12 modulo p and then divide q…”