1 Introduction
Let n be a positive integer. A partition of n is a nonincreasing sequence
$(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{k})$
satisfying
$\lambda _{1} + \lambda _{2} + \cdots + \lambda _{k} = n$
[Reference Andrews1]. Each
$\lambda _{i}$
is called a part of the partition. The number of all partitions of n is denoted by
$p(n)$
. A partition is said to be two-coloured if each part is assigned one of two colours, for instance, green or blue. In 2004, Corteel and Lovejoy [Reference Corteel and Lovejoy5] introduced the notion of an overpartition of n, denoted by
$\overline {p}(n)$
. An overpartition of n is a partition of n in which the first occurrence of each distinct part may be overlined. The number of overpartitions of n into odd parts is labelled by
$\overline{p}_{o}(n)$
. Recently, Andrews and El Bachraoui [Reference Andrews and El Bachraoui2] introduced
$E(n)$
with the following definition.
Definition 1.1 [Reference Andrews and El Bachraoui2, Definition 1].
Let n be a nonnegative integer and let
$E(n)$
be the number of two-colour partitions of n in which parts are distinct and the even parts may occur only in the blue colour. Let
$E_{0}(n)$
(respectively
$E_{1}(n)$
) denote the number of partitions counted by
$E(n)$
in which the number of even parts is even (respectively odd). Further, let
$E_{2}(n)$
(respectively
$E_{3}(n)$
) be the number of partitions counted by
$E(n)$
in which the number of parts is even (respectively odd).
Andrews and El Bachraoui gave analytical proofs of the following relations between two-coloured partitions and overpartitions.
Theorem 1.2 [Reference Andrews and El Bachraoui2, Theorem 1].
For any nonnegative integer n:
$$ \begin{align*} (a) \quad\ E(n) & = \overline{p}_{o}(n); \\ (b) \quad E_0(n) & =\begin{cases} \frac{1}{2}\overline{p}_{o}(n) +1 & \text{if } n \text{ is a square}, \\ \frac{1}{2}\overline{p}_{o}(n) & \text{otherwise}; \end{cases}\\ (c) \quad E_1(n) & =\begin{cases} \frac{1}{2}\overline{p}_{o}(n)-1 & \text{if } n \text{ is a square}, \\ \frac{1}{2}\overline{p}_{o}(n) & \text{otherwise}; \end{cases} \\ (d) \quad E_2(n) & =\begin{cases} \frac{1}{2}\overline{p}_{o}(n) +(-1)^n & \text{if } n \text{ is a square} ,\\ \frac{1}{2}\overline{p}_{o}(n) & \text{otherwise}; \end{cases} \\ (e) \quad E_3(n) & =\begin{cases} \frac{1}{2}\overline{p}_{o}(n)-(-1)^n & \text{if } n \text{ is a square} ,\\ \frac{1}{2}\overline{p}_{o}(n) & \text{otherwise.} \end{cases} \end{align*} $$
They also proved that
$\overline {p}(n) = F(n),$
where
$F(n)$
counts the number of two-colour partitions of n in which the even parts occur only in the blue colour and defined the functions in the following definition.
Definition 1.3 [Reference Andrews and El Bachraoui2, Definition 8].
Let n be a nonnegative integer and let
$F_{0}(n)$
(respectively
$F_{1}(n)$
) be the number of partitions counted by
$F(n)$
in which the number of even parts is even (respectively odd). Furthermore, let
$F_{2}(n)$
(respectively
$F_{3}(n)$
) be the number of partitions counted by
$F(n)$
in which the number of parts is even (respectively odd).
They posed some questions about these functions.
Conjecture 1.4 [Reference Andrews and El Bachraoui2, Section 9].
-
(i) Are there formulae for
$F_{j}(n)$
for
$j = 0, 1, 2, 3$
? -
(ii) Is it possible to give combinatorial proofs for the identities in Theorem 1.2?
Recently, Chen and Zou [Reference Chen and Zou4] established such identities for
$F_{j}(n) $
for
$j = 0, 1, 2, 3$
.
Theorem 1.5 [Reference Chen and Zou4, Theorem 1.3].
For any nonnegative integer
$n $
:
$$ \begin{align*} (a) \,\ \quad F(n) & = \overline{p}(n); \\ (b) \quad F_{0}(n) & = \tfrac{1}{2}(\overline{p}(n) + \overline{p}_{o}(n));\\ (c) \quad F_{1}(n) & = \tfrac{1}{2}(\overline{p}(n) - \overline{p}_{o}(n));\\ (d) \quad F_{2}(n) & = \tfrac{1}{2}(\overline{p}(n) + (-1)^{n} \overline{p}_{o}(n));\\ (e) \quad F_{3}(n) & = \tfrac{1}{2}(\overline{p}(n) - (-1)^{n} \overline{p}_{o}(n)). \end{align*} $$
Chen and Zou employed q-series identities and generating functions to establish Theorem 1.5(b)–(e), and provided combinatorial proofs for Theorems 1.2(a) and 1.5(a). They proposed the following conjecture.
Conjecture 1.6 [Reference Chen and Zou4, Section 4].
It is natural to ask if there are combinatorial proofs for parts (b)–(e) of Theorems 1.2 and 1.5.
Very recently, Bugleev [Reference Bugleev3] provided a combinatorial proof for parts (b)–(e) of Theorem 1.2. Our primary objective in this paper is to give a combinatorial proof for parts (b)–(e) of Theorem 1.5.
2 Combinatorial proof for parts (b)–(e) of Theorem 1.5
To prove our results, we first recall two maps given by Chen and Zou [Reference Chen and Zou4].
Definition 2.1 [Reference Chen and Zou4, Section 3].
We denote by
$\pi _{F(n)}$
and
$\pi _{\overline {p}(n)}$
the sets of partitions in
$F(n)$
and
$\overline {p}(n)$
, respectively. We first define a map
$\sigma $
on
$\pi _{F(n)}$
. For
$\alpha \in \pi _{F(n)}$
, we define
$\beta = \sigma (\alpha )$
as follows.
-
(1) Map the green parts in
$\alpha $
to the nonoverlined part in
$\beta $
and keep the part value unchanged. -
(2) Map the blue parts in
$\alpha $
that appear only once to the overlined parts in
$\beta $
and keep the part value unchanged. -
(3) For the blue parts that appear at least twice:
-
(a) if the part occurs an odd number of times, the map retains one of the parts which is overlined, and adds the remaining parts two by two and makes them nonoverlined;
-
(b) if the part occurs an even number of times, the map adds the parts two by two and makes them nonoverlined.
-
Define the map
$\delta : \pi _{\overline {p}(n)} \to \pi _{F(n)}$
.
-
(1) Map the overlined parts of a partition
$\beta \in \pi _{\overline {p}(n)}$
to blue parts of
$\alpha \in \pi _{F(n)}$
. -
(2) Map the nonoverlined odd parts of
$\beta $
to the green parts of
$\alpha $
and divide the nonoverlined even parts of
$\beta $
between the same two blue parts of
$\alpha $
.
Let
$S_{\overline {p}(n)}$
denote the set of overpartitions of n and let
$S_{\overline {p}_{o}(n)}$
denote the set of overpartitions of n into odd parts. We define
$ S: = S_{\overline {p}(n)} \setminus S_{\overline {p}_{o}(n)}$
as the set of overpartitions of n containing at least one even part. Since
$S_{\overline {p}_{o}(n)} \subseteq S_{\overline {p}(n)}$
, we obtain
$$ \begin{align*} |S| = |S_{\overline{p}(n)} \setminus S_{\overline{p}_{o}(n)}| = |S_{\overline{p}(n)}| - |S_{\overline{p}_{o}(n)}| = \overline{p}(n) - \overline{p}_{o}(n). \end{align*} $$
In addition, we define
$$ \begin{align*} \tilde{S} := \{\lambda \in S : \text{the number of overlined even parts in } \lambda \text{ is odd}\}. \end{align*} $$
Denote by
$S_{F_{1}(n)}$
the set of two-colour partitions (blue, green) of n, in which there are odd numbers of even parts and the even parts can occur only in the blue colour. We use the maps
$$ \begin{align*} \sigma : S_{F_{1}(n)} \to \tilde{S} \quad \text{and} \quad \delta : \tilde{S} \to S_{F_{1}(n)}, \end{align*} $$
defined in the same way as
$\sigma $
and
$\delta $
in Definition 2.1, except that here, the domain and codomain sets are different. The map
$\delta $
will serve as the inverse of the map
$\sigma $
.
Example 2.2. Let
$n = 6$
. We denote by
$\lambda ^{\prime }_{B}$
and
$\lambda ^{\prime }_{G}$
the part
$\lambda '$
occurring in blue and green, respectively. Then, we obtain the following sets, where the corresponding positions of two elements from the two sets are images with respect to the corresponding functions
$$ \begin{align*} &S_{F_{1}(6)}\\ &\quad = \left\{\!\! \begin{array}{ccc} 4_B + 1_B + 1_B, & 4_B + 1_B + 1_G, & 4_B + 1_G + 1_G, \\ 3_B + 2_B + 1_B, & 3_B + 2_B + 1_G, & 3_G + 2_B + 1_B, \\ 3_G + 2_B + 1_G, & 2_B + 2_B + 2_B, & 2_B + 1_B + 1_B + 1_B + 1_B, \\ 2_B + 1_G + 1_G + 1_G + 1_G, & 2_B + 1_B + 1_B + 1_B + 1_G, & 2_B + 1_B + 1_B + 1_G + 1_G, \\ 2_B + 1_B + 1_G + 1_G + 1_G, & 6_B \end{array} \!\!\!\right\} \end{align*} $$
$$ \begin{align*} \tilde{S} = \left\{\!\! \begin{array}{ccc} \overline{4} + 2, & \overline{4}+\overline{1}+1, & \overline{4}+1+1,\\ \overline{3}+\overline{2}+\overline{1}, &\overline{3} + \overline{2} + 1, & 3 + \overline{2} + \overline{1}, \\ 3 + \overline{2} + 1, & 4 + \overline{2}, & \overline{2} + 2 + 2, \\ \overline{2} + 1 + 1 +1+1, & \overline{2} + 2 + \overline{1} + 1, & \overline{2} + 2 + 1 + 1,\\ \overline{2} + \overline{1} + 1 + 1 + 1, & \overline{6}. \end{array} \!\!\right\}\end{align*} $$
Lemma 2.3.
$|\tilde {S}| = \tfrac 12 (\overline {p}(n) - \overline {p}_{o}(n))$
.
Proof. Assume
$\lambda := (\lambda _{1}^{m_1}, \ldots , \lambda _{k}^{m_{k}})$
is a partition of n, where each
$\lambda _i$
denotes the corresponding distinct parts of the partition and each
$m_i$
specifies the respective multiplicity of each part. Assume that of the k distinct parts in
$\lambda $
, there is a total of h distinct even parts. First, by fixing the
$k - h$
distinct odd parts of the partition
$\lambda $
, we get
$2^{k - h}$
overpartitions, since each of those distinct
$k - h$
odd parts can either be overlined or kept nonoverlined. To count the total number of possible
$\lambda \in \tilde {S}$
that can be generated from each
$\lambda $
consisting of k distinct parts, it is evident that an odd number of those h distinct even parts in
$\lambda $
should be overlined in each of those
$2^{k - h}$
overpartitions. This argument leads us to the equation
$$ \begin{align} |\tilde{S}| = \sum_{h = 1}^{\max h}\sum_{s \text{ is odd}} 2^{k - h} {h \choose s}. \end{align} $$
Next, define a set
$S' := \{\lambda \in S : \text {the number of overlined even parts in } \lambda \text { is even}\}$
. In the same way, for
$\lambda \in S'$
, we derive the equation
$$ \begin{align} |S'| = \sum_{h = 1}^{\max h}\sum_{s \text{ is even}} 2^{k - h} {h \choose s}. \end{align} $$
Recall the following binomial identity:
$$ \begin{align} \sum_{s \text{ is odd}} {h \choose s} = \sum_{s \text{ is even}} {h \choose s}. \end{align} $$
Using (2.1), (2.2) and (2.3), we arrive at the conclusion
$$ \begin{align} |\tilde{S}| = |S'|. \end{align} $$
Since
$|\tilde {S}| + |S'| = |S| = \overline {p}(n) - \overline {p}_{o}(n)$
, from (2.4), we reach our desired identity
$|\tilde {S}| = \tfrac 12(\overline {p}(n) - \overline {p}_{o}(n))$
.
Lemma 2.4. Assume that
$\lambda \in S_{F_{1}(n)}$
and denote
$\sigma (\lambda ) = \lambda '$
. Then,
$\lambda ' \in \tilde {S}$
.
Proof. The proof of this lemma requires us to show that there are an odd number of overlined even parts in
$\lambda '$
.
Let
$\lambda := (\lambda _{1}^{m_1}, \ldots , \lambda _{k}^{m_{k}})$
be a partition of n, where
$\lambda _i$
denote the distinct parts of the partition and each
$m_i$
specifies the multiplicity of the corresponding part. From the definition of the set
$S_{F_{1}(n)}$
, it follows that
$\lambda $
contains an odd number of even parts. For each
$\lambda $
, we define a new set
$R_{\lambda } := \{m_i : i \in \{1, 2, \ldots , k\}\}$
, which collects the multiplicity of each part of a particular coloured partition
$\lambda $
. We define a subset of
$R_{\lambda }$
by
$$ \begin{align*} R_{\lambda, \text{ even }} := \{m_i : i \in \{1, 2, \ldots, k\} \text{ and } \lambda_i \equiv 0 \pmod{2}\}. \end{align*} $$
It is clear from the definition of the set
$S_{F_{1}(n)}$
that if
$ \lambda \in S_{F_{1}(n)} \text { and } m_i \in R_{\lambda , \text { even}}$
, then
$\sum _{m_i \in R_{\lambda , \text { even }}} m_i \equiv 1 \pmod {2} $
. This further implies that there exist an odd number of
$m_i$
terms in
$R_{\lambda , \text { even}}$
which are odd. According to the definition of
$\sigma $
, all these
$\lambda _i \in \lambda $
with odd multiplicities
$m_i$
generate a single overlined even part in
$\lambda '$
. The rest of the even parts will always be nonoverlined, based on the definition of
$\sigma $
. Since there are an odd number of even
$\lambda _i$
terms and each of them generates a single overlined even part in
$\lambda '$
, it follows that
$\lambda '$
always has an odd number of overlined even parts.
Lemma 2.5. Assume that
$\lambda ' \in \tilde {S}$
and denote
$\delta (\lambda ') = \lambda $
. Then,
$\lambda \in S_{F_{1}(n)}$
.
Proof. We need to show that if
$\lambda $
has an odd number of even parts, then the even parts are blue follows directly from the definition of
$\delta $
.
Let
$\lambda ' = ({\lambda _{1}'}^{m_1}, \ldots , {\lambda _{k}'}^{m_{k}})$
be as stated in Lemma 2.4. Since
$\lambda ' \in \tilde {S}$
, there is at least an odd number of distinct even parts in
$\lambda '$
. From the definition of
$\delta $
, it is clear that there is at least an odd number of blue even parts from all these overlined even parts in
$\lambda '$
. Now, from the definition of
$\delta $
, nonoverlined even parts get broken into two equal blue parts. This also means that all the nonoverlined parts can either generate a pair of equal odd numbers or a pair of equal even numbers. There will always be an even number of blue even parts generated by using
$\delta $
on the nonoverlined even parts of
$\lambda '$
. Hence, we have an odd number of blue even parts arising from overlined even parts plus an even number of blue even parts arising from nonoverlined even parts. This concludes our proof, as the number of even parts in
$\lambda = \delta (\lambda ')$
will always contain an odd number of even numbers.
Lemma 2.6. The map
$\sigma $
is a bijection from
$S_{F_{1}(n)}$
to
$\tilde {S}$
.
Proof. From the definitions of
$\sigma $
and
$\delta $
, it is easy to verify that
$\delta (\sigma (\lambda )) = \lambda $
for all
$\lambda \in S_{F_{1}(n)}$
and also
$\sigma (\delta (\lambda ')) = \lambda '$
for all
$\lambda ' \in \tilde {S}$
. The lemma then follows immediately from Lemmas 2.4 and 2.5.
Proof of Theorem 1.5(b) and (c).
Clearly, Lemma 2.6 gives
$$ \begin{align*} \tfrac{1}{2} (\overline{p}(n) - \overline{p}_{o}(n))=|\tilde{S}| = F_{1}(n). \end{align*} $$
Also,
$ F_{0}(n) + F_{1}(n) = F(n)=\overline {p}(n),$
which completes the proof.
Proof of Theorem 1.5(d) and (e).
We consider two separate cases, one when n is odd and the other when n is even. For the case in which n is odd, we rewrite equation (d) of Theorem 1.5 as
$$ \begin{align*} F_{2}(n) = \tfrac{1}{2}(\overline{p}(n) - \overline{p}_{o}(n)). \end{align*} $$
It is clear that if we can show there is a bijection between the sets
$S_{F_{1}(n)}$
and
$S_{F_{2}(n)}$
, when n is odd, then we are done. We define the simple identity mapping
$\psi : S_{F_{1}(n)} \rightarrow S_{F_{2}(n)}$
when n is odd. Take a partition
$\lambda \in S_{F_{1}(n)}$
and apply
$\psi $
. Since
$\psi $
is an identity map, we need to show that
$\lambda \in S_{F_{2}(n)}$
. As
$\lambda \in S_{F_{1}(n)}$
, it has an odd number of even parts. Since n is an odd number, the total number of odd parts in
$\lambda $
must be odd. From the observation that
$\lambda $
has an odd number of even and odd parts, we conclude that the total number of parts in
$\lambda $
is even, implying
$\lambda \in S_{F_{2}(n)}$
. Now, we try to show that if we apply the inverse map (which is also the identity map) to an element
$\lambda ' \in S_{F_{2}(n)}$
, we get an element of the set
$S_{F_{1}(n)}$
. Again, since n is an odd number,
$\lambda '$
must contain an odd number of odd parts. In addition,
$\lambda '$
must have an even number of parts in total by the definition of
$S_{F_{2}(n)}$
. This is sufficient to conclude that
$\lambda '$
must have an odd number of even parts.
For the case where n is even, again, take an identity map
$\chi : S_{F_{0}(n)} \rightarrow S_{F_{2}(n)}$
. It is possible to show that
$\chi $
will be a bijection by similar arguments. This completes the proof of our theorem for part (d). Part (e) follows immediately from the fact that
$F_{2}(n)+F_{3}(n)=F(n)= \overline {p}(n).$
