# On the parity of the number of $(a,b,m)$-copartitions of $n$

Hannah E. Burson and Dennis Eichhorn

###### Abstract

We continue the study of the $(a,b,m)$-copartition function
$\mathrm{cp}_{a,b,m}(n)$, which arose as a combinatorial generalization of
Andrews’ partitions with even parts below odd parts. The generating function
of $\mathrm{cp}_{a,b,m}(n)$ has a nice representation as an infinite product.
In this paper, we focus on the parity of $\mathrm{cp}_{a,b,m}(n)$. As with the
ordinary partition function, it is difficult to show positive density of
either even or odd values of $\mathrm{cp}_{a,b,m}(n)$ for arbitrary $a,b$, and
$m$. However, we find specific cases of $a,b,m$ such that
$\mathrm{cp}_{a,b,m}(n)$ is even with density 1. Additionally, we show that
the sequence $\\{\mathrm{cp}_{a,m-a,m}(n)\\}_{n=0}^{\infty}$ takes both even
and odd values infinitely often.

##  1 Introduction

In [4], the authors introduce and develop the theory of copartitions, which
have connections to mock theta functions, Rogers-Ramanujan partitions, the
capsids of Garvan and Schlosser [6], and many other classical partition-
theoretic objects. Each $(a,b,m)$-copartition is comprised of three
partitions: a partition into parts $\equiv a\pmod{m}$, a partition into parts
$\equiv b\pmod{m}$, and a rectangular partition that unites them.
Additionally, the $(a,b,m)$-copartition generating function can be written as
an infinite product, further motivating interest in the copartition counting
functions. As with any type of partitions, one of the first questions one may
ask is about how frequently these counting functions take even and odd values.

For example, Kolberg showed that $p(n)$, the ordinary partition function,
takes both even and odd values infinitely often [7]. Parkin and Shanks studied
the parity of $p(n)$ computationally, and their evidence strongly suggests
that $p(n)$ is even about half of the time [8]. Sadly, the best known results
on the parity of $p(n)$ are spectacularly far from proving anything of the
sort [2]. In fact, it is still an open problem to even show that $p(n)$ is
even or odd with positive density.

In this paper, we study the parity of the copartition counting functions.
Although we conjecture that some copartition functions are equally often even
and odd, we show that this is not the case for all such functions.
Furthermore, in some special cases, we are able to demonstrate explicit sets
with positive density on which $\mathrm{cp}_{a,b,m}(n)$, the number of
$(a,b,m)$-copartitions of $n$, is always even. These explicit sets allow us to
give infinitely many arithmetic progressions on which certain copartition
functions are always even. For example, we show that for $r=3,17,24,31,38,45$,
we have

$\mathrm{cp}_{3,1,4}(49k+r)\equiv 0\pmod{2},$

and for $s=9,14,19,24$, we have

$\mathrm{cp}_{5,1,6}(25k+s)\equiv 0\pmod{2}$

for every nonnegative integer $k$.

In Section 2, we recall the definition and generating function for
copartitions. In Section 3, we recall conjugation, discuss self-conjugate
copartitions, and give lower bounds on the number of even values of
$\mathrm{cp}_{a,a,m}(n)$. In Section 4, we focus on
$\mathrm{cp}_{a,m-a,m}(n)$; we show that $\mathrm{cp}_{a,m-a,m}(n)$ takes both
odd and even values infinitely often, give explicit sets with density one on
which $\mathrm{cp}_{3,1,4}(n)$ and $\mathrm{cp}_{5,1,6}(n)$ are even, and
provide infinitely many congruences modulo two in arithmetic progressions for
each of these functions. Sections 3 and 4 also include several open problems.
In Appendix A, we provide computational data surrounding our open questions
and conjectures.

##  2 Background on Copartitions

In this section, we review $(a,b,m)$-copartitions and the function
$\mathrm{cp}_{a,b,m}(n),$ which were first introduced in [4].

######  Definition.

An $(a,b,m)$-copartition is a triple of partitions $(\gamma,\rho,\sigma)$,
where each of the parts of $\gamma$ is at least $a$ and congruent to
$a\pmod{m}$, each of the parts of $\sigma$ is at least $b$ and congruent to
$b\pmod{m}$, and $\rho$ has the same number of parts as $\sigma$, each of
which have size equal to $m$ times the number of parts of $\gamma$.
When $a,b,m\geq 1$, we let $\mathrm{cp}_{a,b,m}(n)$ denote the number of
$(a,b,m)$-copartitions of size $n$.

Although $\rho$ is completely determined by $\gamma$ and $\sigma$, the
graphical representation we use suggests that the natural way to write the
triple is $(\gamma,\rho,\sigma)$. We refer to $\gamma$ as the _ground_ of a
copartition and $\sigma$ as the _sky_.

######  Example 2.1.

The $(2,1,3)$-copartitions of size 9 are

$\left(\\{5,2^{2}\\},\emptyset,\emptyset\right),\;\left(\\{5\\},\\{3\\},\\{1\\}\right),\;\left(\\{2\\},\\{3\\},\\{4\\}\right),\;\left(\emptyset,\emptyset,\\{7+1^{2}\\}\right),\;$
$\left(\emptyset,\emptyset,\\{4^{2}+1\\}\right),\left(\emptyset,\emptyset,\\{4+1^{5}\\}\right),\text{
and }\left(\emptyset,\emptyset,\\{1^{9}\\}\right).$

Thus, $\mathrm{cp}_{2,1,3}(9)=7$.

To represent the $(a,b,m)$ copartition $(\sigma,\rho,\gamma)$ graphically, we
append the $m$-modular diagram for $\sigma$ to the right of the $m$-modular
diagram for $\rho$. Then, we append the conjugate of the $m$-modular diagram
for $\gamma$ below $\rho$.

######  Example 2.2.

The following diagram represents the $(a,b,m)$-copartition
$(\\{3m+a,2m+a,2m+a,a\\},\\{4m,4m\\},\\{3m+b,2m+b\\})$.

m&mmmbmmm mmmmbmm aaaa mmm mmm m

In [4], we also show that

$\displaystyle{\mathbf{cp}}_{a,b,m}(q)$
$\displaystyle:=\sum_{n=0}^{\infty}\mathrm{cp}_{a,b,m}(n)q^{n}=\frac{(q^{a+b};q^{m})_{\infty}}{(q^{b};q^{m})_{\infty}(q^{a};q^{m})_{\infty}}.$
(1)

##  3 Conjugation, Self-Conjugate Copartitions, and the Parity of
$\mathrm{cp}_{a,a,m}(n)$

In this section, we explore the parity of $\mathrm{cp}_{a,a,m}(n)$. Many of
our results follow from considering the fixed points of the conjugation
involution on $\mathrm{cp}_{a,a,m}(n)$. As in [4], we define the conjugate of
a copartition $(\gamma,\rho,\sigma)$ as the copartition obtained by reflecting
the graphical representation about the line $y=-x$. Defining $\nu(\lambda)$ to
be the number of parts of a partition $\lambda$, the conjugate copartition is
precisely $(\sigma,\rho^{\prime},\gamma),$ where $\rho^{\prime}$ consists of
exactly $\nu(\gamma)$ parts of size $m\times\nu(\sigma)=m\times\nu(\rho)$.
Equivalently, $\rho^{\prime}$ is the partition obtained by conjugating the
$m$-modular diagram representing $\rho$.

m&mmbmmm

mmmbm

aaa

mm

mm

m

$\xrightarrow{\hskip 21.68121pt}\quad$ m&mammm mmamm mma bb mm m m

Figure 1: Conjugation of an $(a,b,m)$-copartition.

######  Remark.

Conjugation is a size-preserving bijection from $\mathcal{CP}_{a,b,m}$ to
$\mathcal{CP}_{b,a,m}$.

######  Remark.

When $a\neq b$, there are no self-conjugate $(a,b,m)$-copartitions.

######  Theorem 3.1.

Let $\mathrm{scp}_{a,m}(n)$ denote the number of self-conjugate
$(a,a,m)$-copartitions of size $n$. Then, we have the following generating
function:

$\sum_{n=0}^{\infty}\mathrm{scp}_{a,m}(n)=(-q^{m+2a};q^{2m})_{\infty}.$

###### Proof.

We prove this theorem combinatorially by adapting Sylvester’s proof that self-
conjugate partitions of $n$ are equinumerous with partitions of $n$ into
distinct odd parts [9, p. 275]. Consider the graphical representation of a
self-conjugate $(a,a,m)$-copartition. For each cell on the line $y=-x$, there
is a corresponding hook that consists of that cell, all the cells directly
below it, and all the cells directly to the right of it. Note that, because
the diagram is self conjugate, each hook has an odd number of $m$’s and
exactly two $a$’s. Thus, if we consider the hooks as the parts of a new
partition, this new partition will have distinct parts that are congruent to
$m+2a\pmod{2m}$. Similarly, given a partition into distinct parts congruent to
$m+2a\pmod{2m}$, we can create a self-conjugate $(a,a,m)$ copartition. Since
$(-q^{m+2a};q^{2m})_{\infty}$ generates partitions into distinct parts that
are congruent to $m+2a\pmod{2m}$, our result follows. ∎

Since conjugation is an involution on $(a,a,b)$-copartitions,
$\mathrm{cp}_{a,a,m}(n)$ is odd exactly when there are an odd number of self-
conjugate $(a,a,m)$-copartitions of size $n$. Thus, by Theorem 3.1,

$\sum_{n=0}^{\infty}\mathrm{cp}_{a,a,m}(n)q^{n}\equiv(-q^{m+2a};q^{2m})_{\infty}\pmod{2}.$
(2)

This congruence immediately implies the following facts about the parity of
$\mathrm{cp}_{a,a,m}(n)$.

######  Corollary 3.2.

For even $m$, $\mathrm{cp}_{a,a,m}(2n+1)\equiv 0\pmod{2}$.

######  Corollary 3.3.

For even $m$,

$\liminf_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{a,a,m}(k)\text{
is even}\\}}{n}\geq\frac{1}{2}.$

######  Corollary 3.4.

For $m\equiv 2\pmod{4}$ and odd $a$,

$\liminf_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{a,a,m}(k)\text{
is even}\\}}{n}\geq\frac{3}{4}.$

###### Proof.

Note that, because $m\equiv 2\pmod{4}$ and $a$ is odd, $m+2a\equiv 0\pmod{4}$
and $2m\equiv 0\pmod{4}$. Thus, all self-conjugate copartitions must be of
size $0\pmod{4}$. ∎

######  Corollary 3.5.

For $a$ odd,

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{a,a,2a}(k)\text{
is even}\\}}{n}=1.$

Moreover, $\mathrm{cp}_{a,a,2a}(k)$ is odd if and only if $k$ is $4a$ times a
pentagonal number; that is, if and only if $k=2an(3n-1)$ for some integer $n$.

###### Proof.

From (2), we see that

$\displaystyle\sum_{n=0}^{\infty}\mathrm{cp}_{a,a,2a}(n)q^{n}$
$\displaystyle\equiv(-q^{4a};q^{4a})_{\infty}\pmod{2}$
$\displaystyle\equiv(q^{4a};q^{4a})_{\infty}$
$\displaystyle=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{2an(3n-1)}.$

The final equality follows from Euler’s pentagonal number theorem:

$(q;q)_{\infty}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}.$

Then, because the final series is lacunary, we have proved that
$\mathrm{cp}_{a,a,2a}(n)$ is even with density $1$. ∎

Corollaries 3.3 and 3.4 come from explicit sets upon which
$\mathrm{cp}_{a,a,m}(n)$ is always even. Outside of those sets, empirical
evidence suggests that the parity is equally balanced, which leads us to the
following conjecture.

######  Conjecture 3.6.

For even $m$ and odd $a$,

$\displaystyle\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq
n\mid\mathrm{cp}_{a,a,m}(k)\text{ is even}\\}}{n}=$ $\displaystyle 1$ if
$m=2a$ (3) $\displaystyle\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq
n\mid\mathrm{cp}_{a,a,m}(k)\text{ is even}\\}}{n}=$ $\displaystyle\frac{3}{4}$
if $m\equiv 0\pmod{4}$ (4) $\displaystyle\lim_{n\to\infty}\frac{\\#\\{1\leq
k\leq n\mid\mathrm{cp}_{a,a,m}(k)\text{ is even}\\}}{n}=$
$\displaystyle\frac{7}{8}$ otherwise. (5)

Note that (3) is Corollary 3.5.

On the other hand, for $m$ odd, as is the case with $p(n)$, we do not know
explicit sets upon which $\mathrm{cp}_{a,a,m}(n)$ is always even, and we make
the following conjecture.

######  Conjecture 3.7.

For odd $m$ and $\gcd(a,m)=1$, $\mathrm{cp}_{a,a,m}(n)$ is even (odd) with
density $\frac{1}{2}$. That is,

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{a,a,m}(k)\text{ is
even (odd)}\\}}{n}=\frac{1}{2}.$

Table 3.1 provides some computational evidence for Conjecture 3.6.

$n$ | $\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{3,3,4}(k)\text{ is even}\\}}{n}$ | $\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{1,1,6}(k)\text{ is even}\\}}{n}$
---|---|---
$1000$ | $0.765$ | $0.871$
$3000$ | $0.752$ | $0.875$
$5000$ | $0.753$ | $0.874$
$7000$ | $0.749$ | $0.875$
$9000$ | $0.748$ | $0.873$
$11000$ | $0.749$ | $0.874$
$13000$ | $0.750$ | $0.875$
$15000$ | $0.749$ | $0.875$
Table 3.1: The proportion of even values of $\mathrm{cp}_{a,a,m}(k)$ for
$1\leq k\leq n$ when $(a,m)$ is $(3,4)$ and $(1,6)$.

##  4 The Parity of $\mathrm{cp}_{a,m-a,m}(n)$

In this section, we explore the parity of the values of another special family
of copartition functions: $\mathrm{cp}_{a,m-a,m}(n).$ Note that

$\mathbf{cp}_{a,m-a,m}(q)=\frac{(q^{m};q^{m})_{\infty}}{(q^{a};q^{m})_{\infty}(q^{m-a};q^{m})_{\infty}}=\frac{(q^{m};q^{m})^{2}_{\infty}}{f(q^{a},q^{m-a})},$

where $f(x,y)=\sum_{n=-\infty}^{\infty}x^{n(n+1)/2}y^{n(n-1)/2}$ is
Ramanujan’s theta function. This form for $\mathbf{cp}_{a,m-a,m}(q)$ suggests
that there is a wide range of analytic tools that we can use to study this
family of functions.

A very basic question one may ask is simply, for which $a,m$ does the sequence
$\\{\mathrm{cp}_{a,m-a,m}(n)\\}_{n=0}^{\infty}$ take both even and odd values
infinitely often?

######  Theorem 4.1.

For all $a,m$, the sequence $\\{\mathrm{cp}_{a,m-a,m}(n)\\}_{n=0}^{\infty}$
takes both even and odd values infinitely often.

###### Proof.

Since $\mathrm{cp}_{a,m-a,m}(n)=\mathrm{cp}_{m-a,a,m}(n)$, without loss of
generality, we assume $0<a\leq m/2$.

If $a=m/2$, by (1) we have

$\displaystyle{\mathbf{cp}}_{m/2,m/2,m}(q)$
$\displaystyle=\frac{(q^{m};q^{m})_{\infty}}{(q^{m/2};q^{m})_{\infty}(q^{m/2};q^{m})_{\infty}}\equiv\frac{(q^{m};q^{m})_{\infty}}{(q^{m};q^{2m})_{\infty}}\equiv(q^{2m};q^{2m})_{\infty}\pmod{2}.$
(6)

By Euler’s Pentagonal Number Theorem, we see that the right-hand side of (6)
takes both even and odd values infinitely often.

For $a\neq m/2$, by (1) we have

$\displaystyle{\mathbf{cp}}_{a,m-a,m}(q)$
$\displaystyle=\frac{(q^{m};q^{m})^{2}_{\infty}}{(q^{m-a};q^{m})_{\infty}(q^{a};q^{m})_{\infty}(q^{m};q^{m})_{\infty}}\equiv\frac{(q^{2m};q^{2m})_{\infty}}{(q^{m-a};q^{m})_{\infty}(q^{a};q^{m})_{\infty}(q^{m};q^{m})_{\infty}}\pmod{2}.$
(7)

By applying Jacobi’s triple product identity to the denominator of (7),
multiplying both sides by that denominator, and applying Euler’s Pentagonal
Number Theorem to the remaining right-hand side, we have

$\displaystyle{\mathbf{cp}}_{a,m-a,m}(q)\sum_{n=-\infty}^{\infty}(-1)^{n}q^{an+mn(n-1)/2}\equiv\sum_{k=-\infty}^{\infty}q^{mk(3k-1)}\pmod{2}.$
(8)

Now suppose $\mathrm{cp}_{a,m-a,m}(n)$ has finitely many odd values, and let
$d$ be the largest integer such that $\mathrm{cp}_{a,m-a,m}(d)$ is odd. Notice
that $\mathrm{cp}_{a,m-a,m}(0)=1$ is also odd. If $d=0$, then we have from (8)
that $\\{an+mn(n-1)/2\mid n\in\mathbb{Z}\\}=\\{mn(3n-1)\mid
n\in\mathbb{Z}\\}$, which is not possible. For $d>0$, we derive a
contradiction by showing that if we go out far enough, the left-hand side of
(8) has two close odd values, but the right-hand side does not. A short
computation shows that since $a\neq m/2$,
$an_{1}+mn_{1}(n_{1}-1)/2=an_{2}+mn_{2}(n_{2}-1)/2$ if and only if
$n_{1}=n_{2}$. Let $N_{0}$ be so large that for $|n|\geq N_{0}$, consecutive
values of each set $\\{an+mn(n-1)/2\mid n\in\mathbb{Z}\\}$ and
$\\{mn(3n-1)\mid n\in\mathbb{Z}\\}$ are more than $d$ apart. Let $N_{1}>N_{0}$
be so large that $aN_{1}+mN_{1}(N_{1}-1)/2>mN_{0}(3N_{0}-1)$. Then, since
$\mathrm{cp}_{a,m-a,m}(0)$ and $\mathrm{cp}_{a,m-a,m}(d)$ are both odd and
$N_{1}>N_{0}$, the coefficients of $q^{aN_{1}+mN_{1}(N_{1}-1)/2}$ and
$q^{aN_{1}+mN_{1}(N_{1}-1)/2+d}$ on the left-hand side of (8) are both odd.
However, for exponents in that range, the terms of the right-hand side of (8)
with odd coefficients are more than $d$ terms apart, a contradiction. Thus
$\mathrm{cp}_{a,m-a,m}(n)$ has infinitely many odd values.

To show $\mathrm{cp}_{a,m-a,m}(n)$ is even infinitely often, define $E_{a,m}$
to be the set of nonnegative integers $n$ such that $\mathrm{cp}_{a,m-a,m}(n)$
is even, and define $G_{a,m}(q)=\sum_{n\in E_{a,m}}q^{n}$. Notice

$\frac{1}{1-q}\equiv{\mathbf{cp}}_{a,m-a,m}(q)+G_{a,m}(q)\pmod{2}.$

Multiplying both sides by the denominator in (7) after applying Jacobi’s
triple product identity, we have

$\displaystyle\frac{\sum_{n=-\infty}^{\infty}(-1)^{n}q^{an+mn(n-1)/2}}{1-q}$
$\displaystyle\equiv(q^{m};q^{m})^{2}_{\infty}+G_{a,m}(q)\sum_{n=-\infty}^{\infty}(-1)^{n}q^{an+mn(n-1)/2}\pmod{2}$
(9)
$\displaystyle\equiv(q^{2m};q^{2m})_{\infty}+G_{a,m}(q)\sum_{n=-\infty}^{\infty}q^{an+mn(n-1)/2}\pmod{2}.$
(10)

We can rewrite the left-hand side of (9) by pairing summands as

$\displaystyle\frac{1}{1-q}$
$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}[q^{-an+mn(n+1)/2}-q^{a(n+1)+mn(n+1)/2}]=\sum_{n=0}^{\infty}(-1)^{n}q^{-an+mn(n+1)/2}\frac{1-q^{a(2n+1)}}{1-q}$
$\displaystyle\equiv\sum_{n=0}^{\infty}q^{-an+mn(n+1)/2}+q^{-an+mn(n+1)/2+1}+\cdots+q^{a(n+1)+mn(n+1)/2-1}\pmod{2},$
(11)

where each term of the sum in (11) has been expanded into a finite geometric
series, the sum of $a(2n+1)$ consecutive powers of $q$. These finite geometric
series do not overlap, and so a short computation shows that (11) has $aN^{2}$
odd terms up through the $q^{\lfloor mN^{2}/2\rfloor}$ term, so that the terms
with odd coefficients in (11) and the left-hand side of (9) have density
$2a/m$.

By Euler’s Pentagonal Number Theorem, the odd values of
$(q^{2m};q^{2m})_{\infty}$ have density zero, thus the nonzero values of the
product $G_{a,m}(q)\sum_{n=-\infty}^{\infty}(-1)^{n}q^{an+mn(n-1)/2}$ must
have density $2a/m$. Since the nonzero values of
$\sum_{n=-\infty}^{\infty}(-1)^{n}q^{an+mn(n-1)/2}$ have density zero, there
must be infinitely many nonzero terms of $G_{a,m}(q)$. ∎

######  Remark.

Some of the analysis above is very similar to that of Berndt, Yee, and
Zaharescu [3] in treating the parity of $p(r,s;n)$, the number of partitions
of $n$ into parts congruent to $r$, $s$, or $r+s$ modulo $r+s$. In the second
half of the proof above, tracking the number of odd terms as they did, one can
arrive at the quantitative result that
$\\#\\{n<N\mid\mathrm{cp}_{a,m-a,m}(n)\text{ is
even}\\}>(a/\sqrt{2m}-o(1))\sqrt{N}$ for all $a\leq m/2$. In the special case
$a=m/2$, we have $\\#\\{n<N\mid\mathrm{cp}_{a,m-a,m}(n)\text{ is
even}\\}>(1-o(1))N$.

Empirical evidence suggests that the parity of $\mathrm{cp}_{a,m-a,m}(n)$ may
be balanced for many but not all $a,m$ with $\gcd(a,m)=1$. This leads us to
the following conjecture and question.

######  Conjecture 4.2.

For odd $m$ and $\gcd(a,m)=1$, the sequence
$\\{\mathrm{cp}_{a,m-a,m}(n)\\}_{n=0}^{\infty}$ takes both even and odd values
with density $\frac{1}{2}$. That is,

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{a,m-a,m}(k)\text{
is even (odd)}\\}}{n}=\frac{1}{2}.$

######  Question 4.3.

For which $a,m$ with $\gcd(a,m)=1$ is

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{a,m-a,m}(k)\text{
is even}\\}}{n}=\frac{1}{2}?$

Furthermore, when the limit is not $1/2$, what is

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{a,m-a,m}(k)\text{
is even}\\}}{n}?$

When the limit is not $1/2$, is it always $1$?

In Appendix A, we provide computational data surrounding Conjecture 4.2 and
Question 4.3.

###  4.1 The parity of $\mathrm{cp}_{3,1,4}(n)$ and $\mathrm{cp}_{5,1,6}(n)$

In two special cases, we can answer Question 4.3 and give the exact asymptotic
density of $n$ for which $\mathrm{cp}_{a,b,m}(n)$ is even. The generating
functions for $\mathrm{cp}_{3,1,4}(n)$ and $\mathrm{cp}_{5,1,6}(n)$ have nice
properties that allow us to apply the classical theory of binary quadratic
forms to show that $\mathrm{cp}_{3,1,4}(n)$ and $\mathrm{cp}_{5,1,6}(n)$ are
even on sets with arithmetic density one.

We now provide an explicit set with arithmetic density one on which
$\mathrm{cp}_{3,1,4}(n)$ is even.

######  Theorem 4.4.

When $a=3,b=1,$ and $m=4$,

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{3,1,4}(k)\text{ is
even}\\}}{n}=1.$

In particular, $\mathrm{cp}_{3,1,4}(n)$ is even if the prime factorization of
$24n+5$ has a prime $\equiv 3\pmod{4}$ occurring with an odd exponent.

######  Remark.

Note that $24n+5$ having a prime $\equiv 3\pmod{4}$ occurring with an odd
exponent is a sufficient condition, but is not necessary.

###### Proof.

We can rewrite the generating function for $\mathrm{cp}_{3,1,4}$ in a very
useful form. Consider

$\displaystyle\sum_{n=0}^{\infty}\mathrm{cp}_{3,1,4}(n)q^{n}$
$\displaystyle=\frac{(q^{4};q^{4})_{\infty}}{(q^{3};q^{4})_{\infty}(q;q^{4})_{\infty}}=\frac{(q^{4};q^{4})_{\infty}}{(q;q^{2})_{\infty}}$
$\displaystyle=\frac{(q^{4};q^{4})_{\infty}(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}}\equiv(q;q)_{\infty}(q^{4};q^{4})_{\infty}\pmod{2}.$
(12)

Applying Euler’s Pentagonal Number Theorem, we then have that

$\sum_{n=0}^{\infty}\mathrm{cp}_{3,1,4}(n)q^{n}\equiv\left(\sum_{j=-\infty}^{\infty}q^{j(3j+1)/2}\right)\left(\sum_{k=-\infty}^{\infty}q^{2k(3k+1)}\right)\pmod{2}.$

Thus, $\mathrm{cp}_{3,1,4}(n)$ must be even unless $n=j(3j+1)/2+2k(3k+1)$ for
some integers $j$ and $k$, or equivalently, unless
$24n+5=(6j+1)^{2}+4(6k+1)^{2}$. Analyzing this modulo $24$, this occurs if and
only if $24n+5$ is represented by the form $A^{2}+B^{2}$. From the classical
theory of binary quadratic forms, we know that the integers $24n+5$
representable by the form $A^{2}+B^{2}$ are precisely those with prime
factorizations having all powers of primes $\equiv 3\pmod{4}$ occurring with
an even exponent. Since that set of representable $(24n+5)$s has density zero,
we have

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{3,1,4}(k)\text{ is
even}\\}}{n}=1$

as desired. ∎

Considering the set of even values of $\mathrm{cp}_{3,1,4}$ guaranteed by
Theorem 4.4, it is straightforward to write down arithmetic progressions on
which $\mathrm{cp}_{3,1,4}$ is always even.

######  Corollary 4.5.

For any prime $p>3,p\equiv 3\pmod{4}$, let $24\delta\equiv 1\pmod{p^{2}}$.
Then

$\mathrm{cp}_{3,1,4}(p^{2}k+pt-5\delta)\equiv 0\pmod{2}$

for $t=1,2,\dots,p-1$ and every nonnegative integer $k$.

Below we give two specific examples.

######  Corollary 4.6.

For $r=3,17,24,31,38,45$, we have

$\mathrm{cp}_{3,1,4}(49k+r)\equiv 0\pmod{2}$

for every nonnegative integer $k$.

######  Corollary 4.7.

For $r=3,14,36,47,58,69,80,91,102,113$, we have

$\mathrm{cp}_{3,1,4}(121k+r)\equiv 0\pmod{2}$

for every nonnegative integer $k$.

In order to treat $\mathrm{cp}_{5,1,6}(n)$, we use the same techniques, but
the argument is slightly more complicated. We first require a lemma showing
the relationship between two binary quadratic forms.

######  Lemma 4.8.

An integer $N\equiv 1\pmod{6}$ is representable by the binary quadratic form
$A^{2}+3B^{2}$ if and only if $4N$ is representable by the binary quadratic
form $(6J+1)^{2}+3(6K+1)^{2}$.

###### Proof.

Suppose $N\equiv 1\pmod{6}$ is representable by the form $A^{2}+3B^{2}$. Then
there exist $a,b\in\mathbb{Z}$ such that

$N=a^{2}+3b^{2}.$ (13)

Specifically, because $a^{2}=(-a)^{2}$ and $a\not\equiv 0\pmod{3}$, if
$a\equiv b\pmod{3}$, let us instead choose $a$ to be $-a$ in (13) to obtain a
representation where $a\not\equiv b\pmod{3}$. Now notice

$\displaystyle(a+3b)^{2}+3(a-b)^{2}=4a^{2}+12b^{2}$ $\displaystyle=4N$
$\displaystyle(a+3b)^{2}+3(b-a)^{2}=4a^{2}+12b^{2}$ $\displaystyle=4N$
$\displaystyle(-a-3b)^{2}+3(a-b)^{2}=4a^{2}+12b^{2}$ $\displaystyle=4N,\text{\
and}$ $\displaystyle(-a-3b)^{2}+3(b-a)^{2}=4a^{2}+12b^{2}$ $\displaystyle=4N.$
(14)

Since $a$ and $b$ must be of opposite parity by (13), we have one
representation of $4N$ in (4.1) of the form $(6J+1)^{2}+3(6K+1)^{2}$.

Now instead suppose $N$ is any positive integer such that
$4N=(6j+1)^{2}+3(6k+1)^{2}$. If $j\equiv k\pmod{2}$, let $a=(3j+9k)/2+1$ and
$b=(3j-3k)/2$. Then

$a^{2}+3b^{2}=\frac{36j^{2}+12j+1}{4}+3\frac{36k^{2}+12k+1}{4}=\frac{(6j+1)^{2}}{4}+3\frac{(6k+1)^{2}}{4}=N.$
(15)

Otherwise, if $j\not\equiv k\pmod{2}$, let $a=(3j-9k-1)/2$ and
$b=(3j+3k+1)/2$. Then

$a^{2}+3b^{2}=\frac{36j^{2}+12j+1}{4}+3\frac{36k^{2}+12k+1}{4}=\frac{(6j+1)^{2}}{4}+3\frac{(6k+1)^{2}}{4}=N.$
(16)

In either case, we see that $N$ is of the form $A^{2}+3B^{2}$, and the lemma
follows.

∎

We now provide an explicit set with arithmetic density one on which
$\mathrm{cp}_{5,1,6}(n)$ is even.

######  Theorem 4.9.

When $a=5,b=1,$ and $m=6$,

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{5,1,6}(k)\text{ is
even}\\}}{n}=1.$

In particular, $\mathrm{cp}_{5,1,6}(n)$ is even if the prime factorization of
$6n+1$ has a prime $\equiv 2\pmod{3}$ occurring with an odd exponent.

######  Remark.

Note that $6n+1$ having a prime $\equiv 2\pmod{3}$ occurring with an odd
exponent is a sufficient condition, but is not necessary. For example,
$\mathrm{cp}_{5,1,6}(5)=2$ is even, even though $6(5)+1=31$.

###### Proof.

We can rewrite the generating function for $\mathrm{cp}_{5,1,6}$ in a very
useful form. Consider

$\displaystyle\frac{(q^{2};q^{2})_{\infty}(q^{3};q^{3})_{\infty}}{(q;q)_{\infty}(q^{6};q^{6})_{\infty}}$
$\displaystyle=\frac{(q^{2};q^{2})_{\infty}(q^{3};q^{6})_{\infty}(q^{6};q^{6})_{\infty}}{(q;q^{2})_{\infty}(q^{2};q^{2})_{\infty}(q^{6};q^{6})_{\infty}}$
$\displaystyle=\frac{(q^{3};q^{6})_{\infty}}{(q;q^{2})_{\infty}}$
$\displaystyle=\frac{1}{(q;q^{6})_{\infty}(q^{5};q^{6})_{\infty}}.$

Thus,

$\displaystyle\sum_{n=0}^{\infty}\mathrm{cp}_{5,1,6}(n)q^{n}$
$\displaystyle=\frac{(q^{6};q^{6})_{\infty}}{(q^{5};q^{6})_{\infty}(q;q^{6})_{\infty}}=\frac{(q^{2};q^{2})_{\infty}(q^{3};q^{3})_{\infty}}{(q;q)_{\infty}}$
$\displaystyle=(-q;q)_{\infty}(q^{3};q^{3})_{\infty}\equiv(q;q)_{\infty}(q^{3};q^{3})_{\infty}\pmod{2}.$
(18)

Applying Euler’s Pentagonal Number Theorem, we then have that

$\sum_{n=0}^{\infty}\mathrm{cp}_{5,1,6}(n)q^{n}\equiv\left(\sum_{j=-\infty}^{\infty}q^{j(3j+1)/2}\right)\left(\sum_{k=-\infty}^{\infty}q^{3k(3k+1)/2}\right)\pmod{2}.$

Thus, $\mathrm{cp}_{5,1,6}(n)$ must be even unless $n=j(3j+1)/2+3k(3k+1)/2$
for some integers $j$ and $k$, or equivalently, unless
$(6j+1)^{2}+3(6k+1)^{2}=24n+4$. By Lemma 4.8, this occurs if and only if
$6n+1$ is represented by the form $A^{2}+3B^{2}$. From the classical theory of
binary quadratic forms [5, Chapter 1], we know that the integers $6n+1$
representable by the form $A^{2}+3B^{2}$ are precisely those with prime
factorizations having all powers of primes $\equiv 2\pmod{3}$ occurring with
an even exponent. Since that set of representable $(6n+1)$s has density zero,
we have

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{5,1,6}(k)\text{ is
even}\\}}{n}=1$

as desired. ∎

Considering the set of even values of $\mathrm{cp}_{5,1,6}$ guaranteed by
Theorem 4.9, it is straightforward to write down arithmetic progressions on
which $\mathrm{cp}_{5,1,6}$ is always even.

######  Corollary 4.10.

For any prime $p>2,p\equiv 2\pmod{3}$, let $6\delta\equiv 1\pmod{p^{2}}$. Then

$\mathrm{cp}_{5,1,6}(p^{2}k+pt-\delta)\equiv 0\pmod{2}$

for $t=1,2,\dots,p-1$ and every nonnegative integer $k$.

Below we give two specific examples.

######  Corollary 4.11.

For $r=9,14,19,24$, we have

$\mathrm{cp}_{5,1,6}(25k+r)\equiv 0\pmod{2}$

for every nonnegative integer $k$.

######  Corollary 4.12.

For $r=9,31,42,53,64,75,86,97,108,119$, we have

$\mathrm{cp}_{5,1,6}(121k+r)\equiv 0\pmod{2}$

for every nonnegative integer $k$.

##  5 Conclusion

Conjectures 3.6, 3.7, and 4.2 and Question 4.3 remain open. Although we treat
the parity of $\mathrm{cp}_{a,a,m}(n)$ and $\mathrm{cp}_{a,m-a,m}(n)$ in some
detail above, the parity of $\mathrm{cp}_{a,b,m}(n)$ when $a\neq b$, and
$a+b\neq m$ is largely unexplored. With only a small amount of evidence, we
make the following somewhat bold conjecture.

######  Conjecture 5.1.

When $\gcd(a,b,m)=1$, $a\neq b$, and $a+b\neq m$, $\mathrm{cp}_{a,b,m}(n)$ is
even (odd) with density $\frac{1}{2}$. That is,

$\lim_{n\to\infty}\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{a,b,m}(k)\text{ is
even}\\}}{n}=\frac{1}{2}.$

In another direction, we give several congruences modulo 2 in Corollaries
4.5-4.7 and 4.10-4.12, In [1], Andrews gave a congruence modulo 5, namely
$\mathcal{EO}^{*}(10n+8)\equiv 0\pmod{5}$, which, when written in the language
of copartitions, becomes $\mathrm{cp}_{1,1,2}(5n+4)\equiv 0\pmod{5}$.
Additionally, he defined an even-odd partition crank that witnesses this
congruence. The equivalent copartition crank is the number of ground parts
minus the number of sky parts, and the copartition crank witnesses the
congruence $\mathrm{cp}_{1,1,2}(5n+4)\equiv 0\pmod{5}$. A combinatorial proof
of these congruences would be a welcome addition to the literature.

## References

  * [1] George E. Andrews, _Integer partitions with even parts below odd parts and the mock theta functions_ , Ann. Comb. 22 (2018), no. 3, 433–445.
  * [2] Joël Bellaïche, Ben Green, and Kannan Soundararajan, _Nonzero coefficients of half-integral weight modular forms ${\rm mod}\,\ell$_, Res. Math. Sci. 5 (2018), no. 1, Paper No. 6, 10. MR 3757164
  * [3] Bruce C. Berndt, Ae Ja Yee, and Alexandru Zaharescu, _On the parity of partition functions_ , Internat. J. Math. 14 (2003), no. 4, 437–459. MR 1984662
  * [4] Hannah E. Burson and Dennis Eichhorn, _Copartitions_ , arXiv:2111.04171 (2021).
  * [5] David A. Cox, _Primes of the form $x^{2}+ny^{2}$_, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989, Fermat, class field theory and complex multiplication.
  * [6] Frank Garvan and Michael J. Schlosser, _Combinatorial interpretations of Ramanujan’s tau function_ , Discrete Math. 341 (2018), no. 10, 2831 – 2840.
  * [7] O. Kolberg, _Note on the parity of the partition function_ , Math. Scand. 7 (1959), 377–378. MR 117213
  * [8] Thomas R. Parkin and Daniel Shanks, _On the distribution of parity in the partition function_ , Math. Comp. 21 (1967), 466–480. MR 227126
  * [9] J. J. Sylvester and F. Franklin, _A constructive theory of partitions, arranged in three acts, an interact and an exodion_ , American Journal of Mathematics 5, 251.

##  Appendix A Data tables

For readers interested in thinking about Conjecture 4.2 and Question 4.3, we
tabulate the proportion of even values of $\mathrm{cp}_{a,m-a,m}(k)$ for $k$
up to $32000$ for several values of $a$ and $m$. In general, we find that the
growth/behavior of the proportion seems roughly consistent when fixing $m$ and
varying $a$. For example, in Table A.1, we see that the behavior in each
column is roughly the same.

$\quad n$ | $\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{1,11,14}(k)\text{ is even}\\}}{n}$ | $\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{3,11,14}(k)\text{ is even}\\}}{n}$ | $\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{5,9,14}(k)\text{ is even}\\}}{n}$
---|---|---|---
$\quad 1000$ | $0.535$ | $0.543$ | $0.530$
$\quad 2000$ | $0.536$ | $0.545$ | $0.536$
$\quad 4000$ | $0.549$ | $0.552$ | $0.543$
$\quad 8000$ | $0.557$ | $0.553$ | $0.554$
$\quad 16000$ | $0.568$ | $0.564$ | $0.565$
$\quad 32000$ | $0.576$ | $0.572$ | $0.573$
Table A.1: The proportion of even values of $\mathrm{cp}_{a,14-a,14}(k)$ for
$1\leq k\leq n$ when $a$ is $1$, $3$, and $5$.

Since this is the case, in Table A.2 below, we just give
$\displaystyle\frac{\\#\\{1\leq k\leq n\mid\mathrm{cp}_{1,m-1,m}(k)\text{ is
even}\\}}{n}$ for several $m$ between $3$ and $32$.

$n\backslash m$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $12$
---|---|---|---|---|---|---|---|---|---
$1000$ | $0.504$ | $0.602$ | $0.503$ | $0.581$ | $0.500$ | $0.632$ | $0.519$ | $0.553$ | $0.498$
$2000$ | $0.495$ | $0.630$ | $0.511$ | $0.599$ | $0.505$ | $0.657$ | $0.500$ | $0.577$ | $0.495$
$4000$ | $0.495$ | $0.656$ | $0.509$ | $0.623$ | $0.500$ | $0.681$ | $0.505$ | $0.593$ | $0.499$
$8000$ | $0.506$ | $0.681$ | $0.509$ | $0.641$ | $0.493$ | $0.700$ | $0.497$ | $0.608$ | $0.494$
$16000$ | $0.503$ | $0.701$ | $0.508$ | $0.653$ | $0.496$ | $0.719$ | $0.497$ | $0.625$ | $0.499$
$32000$ | $0.507$ | $0.720$ | $0.501$ | $0.671$ | $0.496$ | $0.736$ | $0.502$ | $0.638$ | $0.498$

$n\backslash m$ | $14$ | $16$ | $18$ | $20$ | $22$ | $24$ | $26$ | $28$ | $30$ | $32$
---|---|---|---|---|---|---|---|---|---|---
$1000$ | $0.535$ | $0.480$ | $0.543$ | $0.523$ | $0.540$ | $0.489$ | $0.465$ | $0.490$ | $0.484$ | $0.488$
$2000$ | $0.536$ | $0.488$ | $0.550$ | $0.502$ | $0.544$ | $0.508$ | $0.507$ | $0.498$ | $0.495$ | $0.501$
$4000$ | $0.549$ | $0.497$ | $0.547$ | $0.499$ | $0.550$ | $0.503$ | $0.517$ | $0.502$ | $0.503$ | $0.503$
$8000$ | $0.557$ | $0.501$ | $0.551$ | $0.494$ | $0.566$ | $0.496$ | $0.513$ | $0.502$ | $0.507$ | $0.504$
$16000$ | $0.568$ | $0.504$ | $0.554$ | $0.496$ | $0.566$ | $0.499$ | $0.509$ | $0.501$ | $0.509$ | $0.502$
$32000$ | $0.576$ | $0.504$ | $0.556$ | $0.498$ | $0.575$ | $0.501$ | $0.505$ | $0.500$ | $0.506$ | $0.500$

Table A.2: The proportion of even values of $\mathrm{cp}_{1,m-1,m}(k)$ for
$1\leq k\leq n$.