A partition of a positive integer n is a nonincreasing sequence of positive integers λ1≥λ2≥…≥λk such that λ1+λ2…+λk=n. Let p(n) be the number of partitions of n. For example p(5)=7. The seven partitions of 5 are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. The generating function for p(n) is given by

where throughout this paper, for any complex numbers a and q<1 we define

Almost a century back Ramanujan established the following identity [7],

which in fact implies Ramanujan's congruences for p(n) modul 5,

Recently, Andrews [2] introduced the partition function EO(n) which counts the number of partitions of n in which every even part is less than each odd part. For example, EO(6)=7. The seven partitions of 6 it enumerates are 6, 5+1, 4+2, 3+3, 3+1+1+1, 2+2+2, 1+1+1+1+1+1. In [2], Andrews shows that the generating function for EO(n) is

Andrews [2], also defined the partition function EO¯(n) which counts the number of partitions enumerated by EO(n) in which only the largest even part appears an odd number of times. For example, EO¯(6)=4. The four partitions of 6 it enumerates are 6, 3+3, 2+2+2, 1+1+1+1+1+1. In [2], Andrews shows that the generating function for EO¯(n) is

In Section 3 of this paper, we prove some congruences modulo 2 and 4 for the partition function EO¯(n). In Section 4, we give a simple proof of Andrews' congruences

and we prove some interesting congruences modulo 10 and 20. In the Section 5, we consider

where the function EOe(n) counts the elements in the set of partitions which are enumerated by EO¯(n) together with the partitions enumerated by EO(n) where all parts are odd and number of parts is even, i.e, EOe(n) denotes the number of partitions enumerated by EO(n) in which only the largest even part appears an odd number of times except when parts are odd and number of parts is even. For example, EOe(6)=6. The six partitions of 6 it enumerates are 6, 3+3, 2+2+2, 1+1+1+1+1+1 (which are counted by EO¯(n)) and 5+1 and 3+1+1+1 (only counted by EO(n) in which all parts are odd and the number of parts is even). We prove some arithmetic properties modulo 2 satisfied by EOe(n). All of the proofs will follow from elementary generating function considerations and q-series manipulations. The paper concludes with a conjecture on EO¯(n).

We require the following definitions and lemmas to prove the main results in the next three sections. For ∣ab∣<1, Ramanujan's general theta function f(a,b) is defined as

Using Jacobi's triple product identity [1, Theorem 2.8], (2.1) takes the shape

The special cases of f(a,b) are

Lemma 2.1.

(Hirschhorn [6, p. 14, Eqn. 1.9.4]) We have the following 2-dissection of ϕ(q),

Lemma 2.2.

(Hirschhorn [5] or Hirschhorn [6, p. 36, Eqn. 3.6.4]) we have

Lemma 2.3.

(Hirschhorn [6, p. 105, Eqn. 10.7.6]) We have the following beautiful identity due to Ramanujan,

From the Binomial Theorem, for any positive integer, k,

In this section we prove some congruences modulo 2 and 4 satisfied by EO¯(n).

We require the following generating functions to prove congruences for EO¯(n).

Theorem 3.1.

We have,

Proof. From (1.4), we have

since there are no terms on the right in which the power of q is odd, we have

thus by using (2.6), we obtain

It follows that

and

which is our (3.1) and (3.2). We have

It follows that

and

which is our (3.3) and (3.5). We have

It follows that

and

which is our (3.4) and (3.6).

We have the following congruences.

Corollary 3.2.

For all n≥0,

Remark 3.3.

The congruences (3.11)-(3.13) were obtained earlier by Andrews et al. [4]. Andrews et al. [3] introduced a partition function pν(n) which counts the number of partitions of n in which the parts are distinct and all odd parts are less than twice the smallest part.

where ν(q) is a mock theta function. Andrews [2, Corollary 5.2] noted that

He proved the congruences using the properties of mock theta function, whereas we use the q-series identities.

In this section we prove some congruences modulo 5, 10 and 20 for EO¯(n). In the next theorem, we give a simple proof of the Andrews' result [2, Eqn. 1.6], which can be tracked back to [3, Thrm. 6.7]. He used the properties of mock theta functions to prove the congruence, whereas we manipulate the q-series identities to get the result.

Theorem 4.1.

For all n≥0,

Proof. Applying (2.9) in (1.4), we obtain

From (2.7), we have

where J_i contains terms in which the power of q is congruent to i modulo 5, then

where J^*_i contains terms in which the power of q is congruent to i modulo 5. Substituting (4.3) and (4.4) in (4.2), we have

There are no terms on the right in which the power of q is 4 modulo 5, so

from which we deduce (4.1).

In the next theorem, we derive two congruences modulo 10 from the generating Łinebreak functions (3.2) and (3.5).

Theorem 4.2.

For all n≥0,

Proof. Using (2.9) in (3.2), we have

Replacing q by q² in (2.8), we have

where R^*_i contains terms in which the power of q is congruent to i modulo 5.

Substituting (4.3) and (4.9) in (4.8), we obtain

There are no terms on the right in which the power of q is 4 modulo 5, so

from which we deduce (4.6). Using (2.9) in (3.5), we have

From (2.7), we have

where Ji** contains terms in which the power of q is congruent to i modulo 5. Substituting (4.9) and (4.12) in (4.11), we obtain

There are no terms on the right in which the power of q is 3 modulo 5, so

from which we deduce (4.7).

In the next theorem, we derive a congruences modulo 20 from the generating Łinebreak function (3.6).

Theorem 4.3.

For all n≥0,

Proof. Using (2.9) in (3.6), we have

From (2.8), we have

where R_i contains terms in which the power of q is congruent to i modulo 5. Substituting (4.9) and (4.16) in (4.15), we obtain

There are no terms on the right in which the power of q is 4 modulo 5, so

from which we deduce (4.14).

In this section we prove some congruences modulo 2 for EOe(n).

Theorem 5.1.

Proof. From (1.5), we have

since there are no terms on the right in which the power of q is odd, we have

by using (2.6), we obtain

It follows that

and

which is our (5.1) and (5.2).

We have the following congruences.

Corollary 5.2.

For all n≥0,

Andrews [2, Problem 4], proposed to further investigate the properties of EO¯n. We conclude the paper with the following conjecture. Using maple, we found the following congruences hold up to n = 2000.

Conjecture 6.1.

Andrews [2, Problem 4], proposed to further investigate the properties of EO¯n. We conclude the paper with the following conjecture. Using maple, we found the following congruences hold up to n = 2000.

For all n≥0,

The authors would like to thank Professor Michael Hirschhorn for his valuable comments and helpful suggestions.