1 Introduction and statement of results

We recall that a partition of an integer n is a non-increasing sequence $\lambda = \lambda _1+ \dots + \lambda _\ell $ of positive integers whose parts, denoted by $\lambda _j$ ( $1 \leq j \leq \ell $ ), sum up to n. Restrictions on partitions concerned with the parity of the parts in the partition have played a major role in the history of partition theory; see, e.g., Andrews’ article [Reference Andrews3] for an excellent overview of many ways parity has appeared in partition theory. In Andrews’ recent works in this direction [Reference Andrews4, Reference Andrews5], he introduced the notion of “partitions with parts separated by parity” in connection with mock theta functions. A partition $\lambda $ has parts separated by parity Footnote ¹ if each even part $\lambda _j$ is larger than each odd part $\lambda _k$ , or if each odd part $\lambda _j$ is larger than each even part $\lambda _k$ . Andrews used the notation $p_{\mathrm {xy}}^{\mathrm {zw}}(n)$ to count various kinds of partitions of n with parts separated by parity. In his notation, we let $\{\mathrm {x},\mathrm {z}\}=\{\mathrm {e},\mathrm {o}\}$ , where the case $\mathrm {z} = \mathrm {e}, \mathrm {x} = \mathrm {o}$ (resp., $\mathrm {z} = \mathrm {o}, \mathrm {x} = \mathrm {e}$ ) signifies that even parts are larger than odd parts (resp., that odd parts are larger than even parts). Andrews then allowed $\mathrm {y}$ and $\mathrm {w}$ to denote either $\mathrm {u}$ or $\mathrm {d}$ , which represent unrestricted and distinct, respectively, and signify whether parts of parity $\mathrm {x}$ and $\mathrm {z}$ are unrestricted or must be distinct, respectively. Andrews studied the eight possible functions built this way; e.g., $p_{\mathrm{od}}^{\mathrm{eu}}(n)$ counts the number of partitions of n with even parts larger than odd parts which have all odd parts distinct.

Andrews’ papers [Reference Andrews4, Reference Andrews5] and the follow-up work of the first author and Jennings–Shaffer [Reference Bringmann and Jennings-Shaffer12] focused on the generating functions

$$ \begin{align*} F_{\mathrm{xy}}^{\mathrm{zw}}(q) := \sum_{n \geq 0} p_{\mathrm{xy}}^{\mathrm{zw}}(n) q^n. \end{align*} $$

These generating functions have connections to a wide variety of modular-type objects. To state some of these results, we define the q-Pochhammer symbol

$$ \begin{align*} \left( a \right)_n = \left( a;q \right)_n := \prod_{k=0}^{n-1} \left( 1 - aq^k \right), \ \ \ \ \ \left( a_1, a_2, \dots, a_m; q \right)_n := \prod_{k=1}^m \left( a_k; q \right)_n, \end{align*} $$

for $n \in \mathbb {N}_0 \cup \{\infty \},\,m\in \mathbb {N}$ . As a first example, Andrews proved [Reference Andrews4, Equation (2.1)] that

$$ \begin{align*} F_{\mathrm{eu}}^{\mathrm{ou}}(q) = \dfrac{1}{(1-q)\left( q^2;q^2 \right)_\infty}, \end{align*} $$

which is essentially a modular form of weight $-\frac {1}{2}$ . The papers [Reference Andrews5, Reference Bringmann and Jennings-Shaffer12] both proved (see also notation and comments from [Reference Bringmann, Craig and Nazaroglu10, Reference Bringmann, Craig and Nazaroglu11]) that

$$ \begin{align*} F_{\mathrm{od}}^{\mathrm{eu}}(q) = \dfrac{1}{\left( q^2;q^2 \right)_\infty}\left( 1 - \dfrac{\sigma(-q)}{2} + \dfrac{\left( -q;-q \right)_\infty}{2} \right), \end{align*} $$

where

$$ \begin{align*} \sigma(q) := \sum_{n \geq 0} \dfrac{q^{\frac{n(n+1)}{2}}}{\left( -q;q \right)_n} \end{align*} $$

is the famous Ramanujan $\sigma $ -function from his lost notebook [Reference Ramanujan22]. It was shown by Andrews–Dyson–Hickerson [Reference Andrews, Dyson and Hickerson6] and Cohen [Reference Cohen13] that $\sigma (q)$ is related to Maass forms. The variety of generating functions in [Reference Andrews5] also connects to modular forms and mock modular forms. Andrews’ work has given rise to many other examples, looking at further generating function identities [Reference Bringmann and Jennings-Shaffer12], asymptotic formulas [Reference Bringmann, Craig and Nazaroglu10, Reference Bringmann, Craig and Nazaroglu11], and connections with mock theta functions [Reference Fu and Tang17]. Although the definitions of each $p_{\mathrm {xy}}^{\mathrm {zw}}(n)$ are quite similar, the properties that arise from their generating functions are very different (including modular forms, mock modular forms, false modular forms, and mock Maass theta functions like $\sigma (q)$ ); this suggests a rich theory yet to be fully understood.

In this article, we extend this framework to overpartitions. An overpartition [Reference Corteel and Lovejoy14] is a partition where the first instance of each part may possibly be overlined. For instance, there are 8 overpartitions of 3, given by

$$ \begin{align*} 3,\ \overline 3,\ 2+1,\ \overline 2 + 1,\ 2 + \overline 1,\ \overline 2 + \overline 1,\ 1 + 1 + 1,\ \overline 1 + 1 + 1. \end{align*} $$

Overpartitions have played numerous roles in q-series and combinatorics (see, e.g., [Reference Bessenrodt and Pak8, Reference Lovejoy20] and the references therein), mathematical physics (see, e.g., [Reference Fortis, Jacob and Mathieu16]), symmetric functions (see, e.g., [Reference Brenti9]), representation theory (see, e.g., [Reference Kang and Kwon19]), and algebraic number theory (see, e.g., [Reference Lovejoy21]). The goal of this article is to pursue this theme in the context of partitions with parts separated by parity.

We define two variations of overpartitions with parts separated by parity. In both cases, we say that an overpartition $\lambda $ has parts separated by parity Footnote ² if each even $\lambda _j$ is larger than each odd $\lambda _k$ , or if each odd $\lambda _j$ is larger than each even $\lambda _k$ . Because allowing the distinct parts to be overlined introduces powers of two that are both complicated and unenlightening, we introduce an additional constraint in order to create an alternative family of overpartitions. If we impose the restriction that any parts required to be distinct cannot be overlined, then we call these modified. We let $\overline {p}_{\mathrm {xy}}^{\mathrm {zw}}\left ( n\right )$ and $\underline {p}_{\mathrm {xy}}^{\mathrm {zw}}\left ( n\right )$ denote the functions which count the number of overpartitions with parts separated by parity and the modified variation, respectively. For instance, we have that $\overline {p}_{\mathrm{od}}^{\mathrm{eu}}(3) = 6$ and $\underline {p}_{\mathrm{od}}^{\mathrm{eu}}(3) = 3$ , respectively, since the corresponding overpartitions are

$$ \begin{align*} 3 ,\ 2+1,\ \overline 3,\ \overline 2 + 1,\ 2 + \overline 1,\ \overline 2 + \overline 1 \end{align*} $$

and

$$ \begin{align*} 3,\ 2+1,\ \overline 2 + 1. \end{align*} $$

The difference between the two cases is that the overpartitions counted by $\overline {p}_{\mathrm{od}}^{\mathrm{eu}}(3)$ are permitted to have overlines on both the even and odd parts, whereas the overpartitions counted by $\underline {p}_{\mathrm{od}}^{\mathrm{eu}}(3)$ are only permitted to have overlines on the even parts.

In this article, we consider the 16 generating functions

$$ \begin{align*} \overline{F}_{\mathrm{xy}}^{\mathrm{zw}}(q) := \sum_{n \geq 0} \overline{p}_{\mathrm{xy}}^{\mathrm{zw}}(n) q^n, \ \ \ \ \ \underline{F}_{\mathrm{xy}}^{\mathrm{zw}}(q) := \sum_{n \geq 0} \underline{p}_{\mathrm{xy}}^{\mathrm{zw}} (n) q^n. \end{align*} $$

We also need the Jacobi theta function

$$ \begin{align*} \Theta\left( \tau\right) := \sum_{n \in \mathrm{z}Z} q^{n^2} \ \ \ \ \ (q:=e^{2\pi i\tau}). \end{align*} $$

This is a modular form of weight $\frac {1}{2}$ . Our first results are formulas for the generating functions $\overline {F}_{\mathrm {xy}}^{\mathrm {zw}}(q)$ .

Theorem 1.1 The following identities hold:

(1.1) $$ \begin{align} \overline{F}_{\mathrm{eu}}^{\mathrm{ou}}(q) &= \frac{\left( -q^2; q^2\right) _\infty}{2\left( q^2; q^2\right) _\infty} \left( \Theta^2(\tau) + 1 \right), \end{align} $$

(1.2) $$ \begin{align} \overline{F}_{\mathrm{ou}}^{\mathrm{eu}}(q) &= \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} + \frac{2q}{1-q} \frac{\left( -q^2; q^2\right) _\infty}{\left( q^2; q^2\right) _\infty} \sum_{n \geq 0} \frac{\left( -q, q^2; q^2\right) _{n}q^{2n}}{\left(q^3,-q^2; q^2\right)_{n}} , \end{align} $$

(1.3) $$ \begin{align} \overline{F}_{\mathrm{ed}}^{\mathrm{od}}(q) &= \left( -2q;q^2 \right)_\infty - \dfrac{q}{1-q} \left( -2q^2;q^2 \right)_\infty + \dfrac{q}{1-q} \left( -2q;q^2 \right)_\infty, \end{align} $$

(1.4) $$ \begin{align} \overline{F}_{\mathrm{od}}^{\mathrm{ed}}(q) &= \left( -2q^2;q^2 \right)_\infty + \dfrac{q}{1-q} \left( 3 \left( -2q^2;q^2 \right)_\infty - \left( -2q;q^2 \right)_\infty \right), \end{align} $$

(1.5) $$ \begin{align} \overline{F}_{\mathrm{eu}}^{\mathrm{od}}(q) &= \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{2^n q^{n^2}}{\left( -q^2;q^2 \right)_n}, \end{align} $$

(1.6) $$ \begin{align} \overline{F}_{\mathrm{od}}^{\mathrm{eu}}(q) &= - \dfrac{\left( -2q, q^2; q^2 \right)_\infty}{\left( -q^2;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{(-1)^n q^{2n}}{\left( 2q; q^2 \right)_{n+1}} + 2 \sum_{n \geq 0} \dfrac{(-1)^n 2^n q^{n^2+2n}}{\left( 1 + q^{2n+2} \right) \left( 2q; q^2 \right)_{n+1}}, \end{align} $$

(1.7) $$ \begin{align}\overline{F}_{\mathrm{ed}}^{\mathrm{ou}}(q) &= \frac{\left(-q^2;q^2\right)_\infty}{\left(q^2;q^2\right)_\infty} + 2q\left(-2q^2;q^2\right)_\infty \sum_{n\ge0} \frac{(-1)^n q^{2n}}{\left(2q;q^2\right)_{n+1}} \nonumber\\&\quad + \frac{2q\left(-q;q^2\right)_\infty}{\left(q^3;q^2\right)_\infty} \sum_{n\ge0} \frac{(-1)^n 2^n\left(-q;q^2\right)_n q^{n^2+3n}}{\left(2q,-q^2;q^2\right)_{n+1}}, \end{align} $$

(1.8) $$ \begin{align} \overline{F}_{\mathrm{ou}}^{\mathrm{ed}}(q) &= \left( -2q^2;q^2 \right)_\infty + \sum_{n \geq 0} \dfrac{\left( -q;q^2 \right)_n \left( -2q^{2n+2};q^2 \right)_\infty}{\left( q;q^2 \right)_{n+1}} q^{2n+1}. \end{align} $$

We next turn to $\underline {F}^{\mathrm {zw}}_{\mathrm {xy}}(q)$ . We note that in cases where odd and even parts are either both unrestricted or both distinct, the generating functions boil down to either $\overline {F}$ or F, respectively; that is,

$$ \begin{align*} \underline{F}_{\mathrm{eu}}^{\mathrm{ou}}(q) = \overline{F}_{\mathrm{eu}}^{\mathrm{ou}}(q), \ \ \ \underline{F}^{\mathrm{eu}}_{\mathrm{ou}}(q) = \overline{F}^{\mathrm{eu}}_{\mathrm{ou}}(q), \ \ \ \underline{F}_{\mathrm{ed}}^{\mathrm{od}}(q) = F_{\mathrm{ed}}^{\mathrm{od}}(q), \ \ \ \underline{F}_{\mathrm{od}}^{\mathrm{ed}}(q) = F_{\mathrm{od}}^{\mathrm{ed}}(q). \end{align*} $$

These are treated either in Theorem 1.1 or in previous works [Reference Andrews5, Reference Bringmann and Jennings-Shaffer12]. Thus, we focus on the remaining four examples of interest, in which one parity is restricted and one is unrestricted. In order to state this theorem, we recall the third order mock theta function

$$ \begin{align*} \phi(q) := \sum_{n \geq 0} \dfrac{q^{n^2}}{\left( -q^2;q^2 \right)_n}. \end{align*} $$

We prove the following result on the remaining modified generating functions.

Theorem 1.2 The following identities hold:

(1.9) $$ \begin{align} \,\underline{F}_{\mathrm{eu}}^{\mathrm{od}}(q) &= \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} \phi(q), \end{align} $$

(1.10) $$ \begin{align} \underline{F}_{\mathrm{od}}^{\mathrm{eu}}(q) &= \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} - 2q \left( -q; q^2 \right)_\infty \sum_{n\ge0}\frac{q^n}{\left(-q^2;q^2\right)_{n+1}}\nonumber\\ &\quad + 4q \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{(-1)^n q^{n^2 + 2n}}{\left( 1 + q^{2n+2} \right) \left( q;q^2 \right)_{n+1}}, \end{align} $$

(1.11) $$ \begin{align} \underline{F}_{\mathrm{ed}}^{\mathrm{ou}}(q) &= - 2q \left( -q^2; q^2 \right)_\infty \sum_{n \geq 0} \dfrac{q^{n}}{\left( -q^2;q^2 \right)_{n+1}} + \dfrac{\left( -q;q^2 \right)_\infty}{\left( q;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{ (-1)^n \left( -q;q^2 \right)_{n+1} q^{n^2+n}}{\left( -q^2, q; q^2 \right)_{n+1}}, \end{align} $$

(1.12) $$ \begin{align} {\underline{F}}_{\mathrm{ou}}^{\mathrm{ed}}(q) &= \left( -q^2;q^2 \right)_\infty \left( 1 + 2q \sum_{n \geq 0} \dfrac{\left( -q;q^2 \right)_n q^{2n} }{\left( q, -q^2;q^2 \right)_n}\right). \end{align} $$

The article is organized as follows. In Section 2, we provide several known q-series identities which we use in the proof of our main results. In Sections 3 and 4, we prove the results regarding standard and modified overpartitions with parts separated by parity, respectively. In Section 5, we discuss future directions for work.