# Partitions with parts separated by parity: conjugation, congruences and the
mock theta functions

Shishuo Fu  College of Mathematics and Statistics, Chongqing University, Huxi
Campus, Chongqing 401331, P.R. China<EMAIL_ADDRESS>and  Dazhao Tang
School of Mathematical Sciences, Chongqing Normal University, Chongqing
401331, P.R. China<EMAIL_ADDRESS>
###### Abstract.

Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we
adopt a combinatorial approach building on the map of conjugation and continue
the study of integer partitions with parts separated by parity. Our motivation
is twofold. First off, we derive results for certain restricted partitions
with even parts below odd parts. These include a Franklin-type involution
proving a parametrized identity that generalizes Andrews’ bivariate generating
function, and two families of Andrews–Beck type congruences. Secondly, we
introduce several new subsets of partitions that are stable (i.e., invariant
under conjugation) and explore their connections with three third order mock
theta functions $\omega(q)$, $\nu(q)$, and $\psi^{(3)}(q)$, introduced by
Ramanujan and Watson.

###### Key words and phrases:

Partitions, conjugation, congruences, mock theta functions, parity

###### 2010 Mathematics Subject Classification:

11P81, 11P83, 05A17

##  1\. Introduction

A _partition_ $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ of a positive integer
$n$ is a finite weakly decreasing sequence of positive integers
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r}\geq 1$ such that
$\sum_{i=1}^{r}\lambda_{i}=n$, denoted as $\lambda\vdash n$. The $\lambda_{i}$
are called the _parts_ of the partition $\lambda$. As usual, we denote the
number being partitioned $n$, and the number of parts $r$, as $|\lambda|$ and
$\\#(\lambda)$, respectively. The partition function $p(n)$ is the number of
partitions of $n$. For the sake of convenience, we denote the empty partition
of $0$ as $\epsilon$ and agree that it is contained in the set of ordinary
partitions and various subclasses of restricted partitions. Hence for example,
$p(0)=1$. Unless otherwise noted, we will follow the notations used in [1].

In 1944, Dyson [20] defined the _rank_ of a partition as the largest part
minus the number of parts, and then observed that the rank appears to give
combinatorial interpretations for the first two (i.e., (1.1) and (1.2)) of
Ramanujan’s celebrated partition congruences, namely,

$\displaystyle p(5n+4)$ $\displaystyle\equiv 0\pmod{5},$ (1.1) $\displaystyle
p(7n+5)$ $\displaystyle\equiv 0\pmod{7},$ (1.2) $\displaystyle p(11n+6)$
$\displaystyle\equiv 0\pmod{11}.$ (1.3)

Since the map of conjugation (see Sect. 2 for definition) swaps the largest
part and the number of parts between a partition $\lambda$ and its conjugate
$\lambda^{\prime}$, Dyson’s rank can be rephrased as

$\displaystyle\textrm{rank}(\lambda)=\\#(\lambda^{\prime})-\\#(\lambda).$

A similar looking partition statistic, the so-called Stanley rank, was
introduced by Stanley [39] in his study of sign-balanced, labelled posets (see
also [2, 9, 10]), and can be defined as

$\displaystyle\textrm{srank}(\lambda)=\mathcal{O}(\lambda)-\mathcal{O}(\lambda^{\prime}),$
(1.4)

where $\mathcal{O}(\lambda)$ is the number of odd parts in $\lambda$. This
statistic also refines Ramanujan’s first congruence (1.1); see [2, Corollary
1.2].

Our starting point is an observation that connects the Stanley rank with
Andrews’ recent work [4] on partitions with even parts below odd parts. We
need some further definitions before we can state this observation.

Let $\mathfrak{EO}_{n}$ denote the set of partitions of $n$ in which every
even part is less than each odd part, and let
$\mathcal{EO}(n):=|\mathfrak{EO}_{n}|$, $\mathfrak{EO}:=\bigcup_{n\geq
0}\mathfrak{EO}_{n}$. Let $\overline{\mathfrak{EO}}_{n}$ denote the set of
partitions in $\mathfrak{EO}_{n}$ in which ONLY the largest even part, if any,
appears an odd number of times, then $\overline{\mathcal{EO}}(n)$ and
$\overline{\mathfrak{EO}}$ can be defined analogously. More generally, if
$\mathfrak{P}$ denote a set of partitions with certain restrictions, then
$\mathfrak{P}_{n}:=\\{\lambda\in\mathfrak{P}:|\lambda|=n\\}$, and
$\mathcal{P}(n):=|\mathfrak{P}_{n}|$.

Recall the third order mock theta function due to Ramanujan and Watson (see,
for example, [7]), defined by

$\displaystyle\nu(q)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}}{(-q;q^{2})_{n+1}},$ (1.5)
$\displaystyle\omega(q)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{2n^{2}+2n}}{(q;q^{2})_{n+1}^{2}},$
(1.6)

where, here and throughout the rest of this paper, we always assume that $q$
is a complex number such that $|q|<1$ and adopt the following customary
abbreviations in partitions and $q$-series:

$\displaystyle(a;q)_{n}:=\prod_{j=0}^{n-1}(1-aq^{j})\qquad\textrm{and}\qquad(a;q)_{\infty}:=\prod_{j=0}^{\infty}(1-aq^{j}).$

Noting the connection with the third order mock theta function $\nu(q)$,
Andrews deduced in [4] that

$\displaystyle\overline{\mathcal{EO}}(10n+8)\equiv 0\pmod{5}.$ (1.7)

In order to explain (1.7) combinatorially as the rank does for (1.1) and
(1.2), he went on to introduce the even-odd crank for any partition
$\lambda\in\mathfrak{EO}$, to be

$\displaystyle\textrm{eoc}(\lambda)=\textrm{the~{}largest~{}even~{}part~{}of}~{}\lambda-\mathcal{O}(\lambda).$
(1.8)

Of course, $\textrm{eoc}(\lambda)$ is always an even number when
$\lambda\in\overline{\mathfrak{EO}}$. Let $N_{\textrm{eo}}(k,m,n)$ denote the
number of partitions in $\overline{\mathfrak{EO}}_{n}$ whose even-odd cranks
are congruent to $k$ modulo $m$. Andrews [4, Theorem 3.3] proved that for
$0\leq i\leq 4$,

$\displaystyle
N_{\textrm{eo}}(i,5,10n+8)=\frac{1}{5}\overline{\mathcal{EO}}(10n+8).$ (1.9)

The key observation mentioned earlier is as follows.

######  Observation 1.1.

For any partition $\lambda$, $\lambda\in\overline{\mathfrak{EO}}$ if and only
if $\lambda^{\prime}\in\overline{\mathfrak{EO}}$. In this case, we have

$\displaystyle\emph{eoc}(\lambda)=\mathcal{O}(\lambda^{\prime})-\mathcal{O}(\lambda)=-\emph{srank}(\lambda).$
(1.10)

This link sheds new light on the study of $\overline{\mathcal{EO}}(n)$, since
now we can utilize the symmetry of $\overline{\mathfrak{EO}}$, imposed by the
map of conjugation.

Motivated by (1.1), (1.2) and the statistic rank, Beck proposed some
surprisingly conjectural congruences. Let $NT(k,m,n)$ denote the total number
of parts in the partitions of $n$ with rank congruent to $k$ modulo $m$,
Andrews [6, Theorems 1 and 2] proved for any $n\geq 0$,

$\displaystyle\big{(}NT(1,5,5n+i)-NT(4,5,5n+i)\big{)}$
$\displaystyle\quad+2\big{(}NT(2,5,5n+i)-NT(3,5,5n+i)\big{)}\equiv
0\pmod{5},\quad\textrm{if}\;\;i=1\;\textrm{or}\;4,$ (1.11)
$\displaystyle\big{(}NT(1,7,7n+j)-NT(6,7,7n+j)\big{)}$
$\displaystyle\quad+\big{(}NT(2,7,7n+j)-NT(5,7,7n+j)\big{)}$
$\displaystyle\quad-\big{(}NT(3,7,7n+j)-NT(4,7,7n+j)\big{)}\equiv
0\pmod{7},\quad\textrm{if}\;\;j=1\;\textrm{or}\;5.$ (1.12)

The congruences (1.11) and (1.12) are in general called Andrews–Beck type
congruences. Motivated by these two congruences, many authors have recently
discovered Andrews–Beck type congruences for various types of partitions with
their associated statistics; see, for example, [14, 15, 16, 12, 18, 19, 27,
30, 31, 33, 34, 35, 42]. Let $NT_{\textrm{eo}}(k,m,n)$ denote the total number
of odd parts among partitions in $\overline{\mathfrak{EO}}_{n}$, whose even-
odd cranks are congruent to $k$ modulo $m$. Our first result is the following
Andrews–Beck type congruences. Unlike the methods used in the aforementioned
literature, the proofs of (1.13) and (1.14) need to take advantage of the map
of conjugation and (1.10).

######  Theorem 1.2.

For any $n\geq 0$,

$\displaystyle\big{(}NT_{\emph{eo}}(1,5,10n)-NT_{\emph{eo}}(4,5,10n)\big{)}$
$\displaystyle\quad+2\big{(}NT_{\emph{eo}}(2,5,10n)-NT_{\emph{eo}}(3,5,10n)\big{)}\equiv
0\pmod{10},$ (1.13)
$\displaystyle\big{(}NT_{\emph{eo}}(1,5,10n+8)-NT_{\emph{eo}}(4,5,10n+8)\big{)}$
$\displaystyle\quad+2\big{(}NT_{\emph{eo}}(2,5,10n+8)-NT_{\emph{eo}}(3,5,10n+8)\big{)}\equiv
0\pmod{20}.$ (1.14)

In a follow-up paper [5], Andrews further studied various types of partitions
with parts separated by parity. Motivated by this work, we also consider some
subclasses of partitions with parts separated by parity. Let $\mathcal{OE}(n)$
denote the number of partitions of $n$ in which each odd part is less than
each even part. Denote by $\overline{\mathcal{OE}}(n)$ the number of
partitions counted by $\mathcal{OE}(n)$ in which both even and odd parts
appear, and ONLY the largest odd part and the largest even part appear an odd
number of times, so in particular, $\overline{\mathcal{OE}}(n)>0$ only for odd
$n\geq 3$. For example $\overline{\mathcal{OE}}(9)=8$, with the eligible
partitions being $(8,1)$, $(6,3)$, $(6,1,1,1)$, $(4,3,1,1)$, $(4,2,2,1)$,
$(4,1,1,1,1,1)$, $(2,2,2,1,1,1)$, and $(2,1,1,1,1,1,1,1)$.

Relating to the Stanley rank, we define $\overline{\mathcal{OE}}(m,n)$ to be
the number of partitions enumerated by $\overline{\mathcal{OE}}(n)$ whose
Stanley rank equals $m$ and let

$\displaystyle\overline{\textrm{OE}}(z,q):=\sum_{\pi\in\overline{\mathfrak{OE}}}z^{\textrm{srank}(\pi)}q^{|\pi|}=\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}\overline{\mathcal{OE}}(m,n)z^{m}q^{n}$

be the bivariate generating function for $\overline{\mathcal{OE}}(m,n)$.

######  Theorem 1.3.

We have

$\displaystyle\overline{\emph{OE}}(z,q)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\dfrac{q^{4mn-1}}{(z^{2}q^{2};q^{4})_{m}(q^{2}/z^{2};q^{4})_{n}}.$
(1.15)

The rest of this paper is organized as follows. We introduce the pivotal
concept of “stable” sets of partitions in Section 2 (see Definition 2.1),
after which Observation 1.1 is explained, and a complete characterization of
the parity of $\overline{\mathcal{EO}}(n)$ is rederived with ease. Theorem 1.2
is proved in Section 3, where three further stable subsets contained in
$\mathfrak{EO}$ and their connections with mock theta functions are considered
as well. In Section 4, we prove Theorem 1.3 and explore further subsets of
$\overline{\mathfrak{OE}}$. We raise three conjectures to conclude the paper.

##  2\. Preliminaries

We set further notations and lay the groundwork in this section.

To each partition $\lambda$, we associate a graphical representation called
Ferrers graph, which is a left-justified array of unit squares such that the
$i$-th row contains $\lambda_{i}$ squares. Ferrers graph facilitates the
illustration of the following three notions that will appear frequently in
this paper (see Fig. 1). Suppose we are given a partition
$\lambda=(\lambda_{1},\ldots,\lambda_{r})$.

  * •

Conjugation: The conjugate of $\lambda$ is obtained by transposing its Ferrers
graph. Equivalently, the conjugate partition of $\lambda$ is
$\lambda^{\prime}=(\lambda_{1}^{\prime},\ldots,\lambda_{\lambda_{1}}^{\prime})$,
where $\lambda_{i}^{\prime}=|\\{1\leq j\leq r:\lambda_{j}\geq i\\}|$.

  * •

Durfee square: The Durfee square of $\lambda$ is the largest square that can
fit into the top-left corner of its Ferrers graph. If we denote the side of
the Durfee square of $\lambda$ as $d(\lambda)$, then $d(\lambda)=|\\{1\leq
j\leq r:\lambda_{j}\geq j\\}|$. Note that $\lambda$ has a unique decomposition
$\lambda=(d(\lambda);\mu,\nu)$, with $\mu$ and $\nu$ being the subpartition
below the Durfee square of $\lambda$ and the conjugate of the subpartition to
the right of the Durfee square, respectively. Clearly
$|\lambda|=d^{2}+|\mu|+|\nu|$.

  * •

Profile: The profile of a nonempty partition $\lambda$ is the sequence of
southmost and eastmost border edges in its Ferrers graph, that starts with an
east edge and ends with a north edge. We use the profile word to record the
profile of $\lambda$: $w_{\lambda}=(e_{1},n_{1},\ldots,e_{k},n_{k})$, where
for each $1\leq i\leq k\leq\min\\{r,\lambda_{1}\\}$, $e_{i}$ and $n_{i}$ are
the numbers of consecutive east and north edges appearing alternately in the
profile, respectively. It is worth noting that Dyson’s rank can be rephrased
as $\textrm{rank}(\lambda)=\sum_{i}e_{i}-\sum_{i}n_{i}$.

$\lambda=4~{}4~{}2~{}1~{}1$$\lambda^{\prime}=5~{}3~{}2~{}2$$\lambda=(2;(2,1,1),(2,2))$,$w_{\lambda}=(1,2,1,1,2,2)$
Figure 1. Ferrers graph, conjugation, Durfee square and profile

Note that both the conjugation and the Durfee square are standard notions that
have been widely applied in the study of partition theory [1]. While the
partition profile, whose definition follows [28], was involved in our previous
work [22]; see also [21, 32] for recent works demonstrating the usefulness of
this alternative way of recording a partition. A closely related notion, the
rim hook, makes an appearance in the famed Murnaghan–Nakayama rule (see, for
example, [38, Theorem 4.10.2]) and is crucial in the study of symmetric
functions in general.

As a standard tool in the realm of enumerative combinatorics, the theory of
enumeration under group action, or Pólya theory, seems to be lack of
application in the study of integer partitions. Nonetheless, one notable and
enlightening exception was the paper by Garvan, Kim and Stanton [23], in which
they utilized three dihedral groups’ actions on the partitions of $5n+4,7n+5$,
and $11n+6$, to give a uniform proof of all three Ramanujan congruences (1.1),
(1.2) and (1.3); see also Hirschhorn’s paper [25] and book [26, Chap. 4] for a
more direct version of their proof. We would also like to mention the recent
work of Kim [29], in which a $\mathbb{Z}_{2}$-action similar to conjugation
was discussed and interesting congruences were derived.

Motivated by these previous works and our Observation 1.1, we view the map of
conjugation as a $\mathbb{Z}_{2}$-action on the set of partitions. From this
perspective, it is natural to consider those subsets that are invariant under
this group action. We give a name to such subsets.

######  Definition 2.1.

A collection of partitions, say $\mathfrak{C}$, is said to be (conjugational)
stable, if the following holds true:

$\displaystyle\lambda\in\mathfrak{C}\quad\textrm{if~{}and~{}only~{}if}\quad\lambda^{\prime}\in\mathfrak{C}.$

For instance, we state without proof in Observation 1.1 that
$\overline{\mathfrak{EO}}$ is stable. To see this, simply note that for any
$\lambda\in\overline{\mathfrak{EO}}$, its profile word
$w_{\lambda}=(e_{1},n_{1},\ldots,e_{k},n_{k})$ must contain a consecutive pair
of odd numbers $(n_{i},e_{i+1})$ for some $0\leq i\leq k$ (set
$n_{0}=e_{k+1}=1$ by convention), and all remaining letters in $w_{\lambda}$
are even. This property is clearly seen to be preserved by the reversal, thus
$\lambda^{\prime}\in\overline{\mathfrak{EO}}$ as well (see Proposition 2.2
below). Furthermore, we see that the largest even part of $\lambda$ is given
by $e_{1}+e_{2}+\cdots+e_{i}$, which, upon reversal, outputs exactly
$\mathcal{O}(\lambda^{\prime})$, the number of odd parts in
$\lambda^{\prime}$, hence (1.10) follows and Observation 1.1 holds true.

From now on, we will use stability as a guideline for discovering new
subclasses of partitions with parts separated by parity. For any class of
partitions $\mathfrak{C}$, we use $\mathfrak{C}_{c}$ to denote the subset of
partitions $\lambda\in\mathfrak{C}$ that are self-conjugate, i.e.,
$\lambda=\lambda^{\prime}$. The following two facts are simple but useful.

######  Proposition 2.2.

For a partition $\lambda$ with its profile word
$w_{\lambda}=(e_{1},n_{1},\ldots,e_{k},n_{k})$, the profile word for its
conjugate is obtained from reversing $w_{\lambda}$, i.e.,

$\displaystyle w_{\lambda^{\prime}}=(n_{k},e_{k},\ldots,n_{1},e_{1}).$

In particular, $\lambda$ is self-conjugate if and only if $w_{\lambda}$ is
palindromic, i.e., $e_{1}=n_{k}$, $n_{1}=e_{k},\ldots$.

######  Proposition 2.3.

Let $\mathcal{C}(n):=|\\{\lambda\in\mathfrak{C}:|\lambda|=n\\}|$ and
$\mathcal{C}_{c}(n):=|\\{\lambda\in\mathfrak{C}_{c}:|\lambda|=n\\}|$. Then

$\displaystyle\mathcal{C}(n)\equiv\mathcal{C}_{c}(n)\pmod{2}.$ (2.1)

Based on Proposition 2.3, we give another proof of the following complete
characterization for the parity of $\overline{\mathcal{EO}}(n)$, first derived
by Passary [36, Eq. (2.1.24)].

######  Corollary 2.4.

For any $n\geq 0$,

$\displaystyle\overline{\mathcal{EO}}(n)\equiv\begin{cases}1\pmod{2},&\text{if
$n=4k(3k-1)$}~{}\textrm{for~{}some~{}}$k$,\\\
0\pmod{2},&\text{otherwise.}\end{cases}$ (2.2)

###### Proof.

Let $\overline{\mathcal{EO}}_{c}(n)$ denote the number of self-conjugate
partitions in $\overline{\mathfrak{EO}}_{n}$, then by (2.1) we have
$\overline{\mathcal{EO}}(n)\equiv\overline{\mathcal{EO}}_{c}(n)\pmod{2}$.

Now each self-conjugate partition $\lambda\in\overline{\mathfrak{EO}}_{n}$ can
be decomposed as $\lambda=(2m;\mu,\mu)$, where $2m$ is the side of $\lambda$’s
Durfee square, and $\mu$ is the subpartition below the Durfee square, which
has an odd number of largest part $2m$, with the remaining parts all being
even and all occurring an even number of times. Together with another copy of
$\mu$ to the right of the Durfee square, they are generated by $(q^{2m}\cdot
q^{2m})/(q^{8};q^{8})_{m}$. This amounts to give the generating function

$\displaystyle\sum_{n=0}^{\infty}\overline{\mathcal{EO}}_{c}(n)q^{n}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{q^{(2m)^{2}+4m}}{(q^{8};q^{8})_{m}}$
$\displaystyle=(-q^{8};q^{8})_{\infty}$ (2.3)
$\displaystyle\equiv(q^{8};q^{8})_{\infty}\pmod{2}$
$\displaystyle=\sum_{m=-\infty}^{\infty}(-1)^{m}q^{4m(3m-1)},$ (2.4)

where (2.3) and (2.4) follow from Euler’s identity [1, p. 19, Eq. (2.2.6)] and
Euler’s pentagonal number theorem [1, p. 11, Corollary 1.7], respectively. Now
(2.2) follows by comparing the coefficients. ∎

######  Remark 2.5.

Two remarks on Corollary 2.4 are in order. For one thing, in a recent paper of
Ray and Barman [37], using the theory of modular forms, they derived, among
some other things, an infinite family of congruences modulo $2$ for
$\overline{\mathcal{EO}}(n)$ (see [37, Theorem 1.1]), and a result concerning
the parity of $\overline{\mathcal{EO}}(n)$ in any arithmetic progression [37,
Theorem 1.4]. For another, the identities (2.3) and (2.4) both possess
classical combinatorial proofs. From this perspective, our proof is a
combinatorial proof.

##  3\. Stable sets in $\mathfrak{EO}$

Equipped with the combinatorial insights from section 2, we investigate in
this section various stable subsets contained in $\mathfrak{EO}$.

###  3.1. Further results for $\overline{\mathfrak{EO}}$

We begin by answering a problem raised by Andrews in [4]:

“Problem 2. Prove Proposition 3.1 combinatorially. (Hopefully more directly
than invoking [1].)”

The Proposition 3.1 referred to above is the following generating function of
$\overline{\mathcal{EO}}(m,n)$, the number of partitions in
$\overline{\mathfrak{EO}}_{n}$ whose even-odd crank equals $m$.

$\displaystyle\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}\overline{\mathcal{EO}}(m,n)z^{m}q^{n}=\dfrac{(q^{4};q^{4})_{\infty}}{(z^{2}q^{2};q^{4})_{\infty}(q^{2}/z^{2};q^{4})_{\infty}}.$
(3.1)

We construct an involution on a certain set of $2$-colored partition pairs to
establish Theorem 3.3, which is a bivariate generalization of (3.1). This
approach is reminiscent of Franklin’s proof of Euler’s pentagonal number
theorem. We need to make some further definitions.

We consider $2$-colored partitions, wherein each part of size $k$ can be
colored as either $b$ or $ab$ (when $k\geq 2$), such that $ab$-colored parts
are all distinct. We remark that this notion of $2$-colored partition is
essentially Corteel and Lovejoy’s overpartition [17], probably appended with
parts of zero (nonoverlined), but we introduce it this way for our
convenience.

Using its Ferrers graph, we realize every $2$-colored partition by assigning
the neutral weight $q$, and color weights $b$ or $ab$, such that the leftmost
cell of each part is always filled with $b$, while the rightmost cell of each
$ab$-colored part is filled with $a$, with the remaining cells all filled with
$q$. We call such cell-labelled Ferrers graph a Ferrers diagram. The weight
for each Ferrers diagram $\mu$ is the product over all the weights of its
cells, and is denoted as $w(\mu)$; see Fig. 2 for an example. Note that when
there are two parts with the same size but different color, we always put the
$ab$-colored one below the $b$-colored one. Consequently, each cell filled
with $a$ must be an outer corner cell. Denote the set of all Ferrers diagrams
as $\mathfrak{F}(a,b;q)$. From now on, we will speak of Ferrers diagram and
$2$-colored partition interchangeably. It is now routine to find its
generating function.

######  Proposition 3.1.

The weight generating function of Ferrers diagrams is given by

$\displaystyle\sum_{\mu\in\mathfrak{F}(a,b;q)}w(\mu)=\frac{(-ab;q)_{\infty}}{(b;q)_{\infty}}.$

$b$$b$$b$$b$$b$$b$$b$$a$$a$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$$q$
Figure 2. Ferrers diagram for $\mu$ with $w(\mu)=a^{2}b^{7}q^{20}$

Clearly $\mathfrak{F}(0,q;q)$ is in bijection with the set of ordinary
partitions. Now define a peculiar set of $2$-colored partition pairs

$\displaystyle\mathfrak{F}(a,b,c,d;q):=\\{(\lambda,\mu)\in\mathfrak{F}(a,b;q)\times\mathfrak{F}(c,d;q):\textrm{$\lambda=\epsilon$~{}or~{}the~{}smallest~{}part~{}of~{}$\lambda$}>\\#(\mu)\\}.$

######  Lemma 3.2.

There exists a bijection

$\displaystyle\varphi:\mathfrak{F}(0,q^{2}/z^{2},0,z^{2}q^{2};q^{4})\to\overline{\mathfrak{EO}},$

such that a pair of Ferrers diagrams
$(\lambda,\mu)\in\mathfrak{F}(0,q^{2}/z^{2},0,z^{2}q^{2};q^{4})$ weighted by
$w(\lambda)w(\mu)=z^{m}q^{n}$ corresponds to a partition counted by
$\overline{\mathcal{EO}}(m,n)$.

###### Proof.

Let $(\lambda,\mu)\in\mathfrak{F}(0,q^{2}/z^{2},0,z^{2}q^{2};q^{4})$ with
$\\#(\lambda)=s$ and $\\#(\mu)=t$, then $\lambda_{s}>t$ from the definition.
Moreover, $\lambda\in\mathfrak{F}(0,q^{2}/z^{2};q^{4})$ implies that $\lambda$
corresponds to a partition $\hat{\lambda}\vdash(4|\lambda|-2s)$ into $2s$ odd
parts, wherein each part appears an even number of times and each occurrence
is weighted by $z^{-1}$. Thus $\hat{\lambda}$ is weighted by $z^{-2s}$. On the
other hand, the conjugate partition of
$\mu\in\mathfrak{F}(0,z^{2}q^{2};q^{4})$ is seen to be corresponding to a
partition $\hat{\mu}\vdash(4|\mu|-2t)$ into even parts, with the largest part
$2t$ occurring an odd number of times while all remaining parts occurring an
even number of times. $\hat{\mu}$ is weighted by $z^{2t}$.

Now, let $\pi$ be the unique partition obtained from appending $\hat{\mu}$ to
$\hat{\lambda}$. We see that the condition $\lambda_{s}>t$ ensures that
$\pi\in\overline{\mathfrak{EO}}$. In addition, we have indeed
$\mathrm{eoc}(\pi)=2t-2s$. Conversely, it should be clear how to start with a
partition in $\overline{\mathfrak{EO}}$ and recover a unique pair in
$\mathfrak{F}(0,q^{2}/z^{2},0,z^{2}q^{2};q^{4})$. See Fig. 3 below for a
concrete example. ∎

$\frac{q^{2}}{z^{2}}$$\frac{q^{2}}{z^{2}}$$\frac{q^{2}}{z^{2}}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$z^{2}q^{2}$$z^{2}q^{2}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$q^{4}$$\lambda$$\downarrow$$\mu$$\swarrow$

$\hat{\lambda}$$+$$\hat{\mu}$$\mapsto$$\pi$

Figure 3. The correspondence $\varphi:(\lambda,\mu)\to\pi$ with
$\mathrm{eoc}(\pi)=4-6=-2$

In view of Lemma 3.2, the following theorem indeed covers Andrews’s formula
(3.1) as the special case of setting $a\to q^{2}/z^{2}$, $b\to z^{2}q^{2}$,
$q\to q^{4}$.

######  Theorem 3.3.

We have

$\displaystyle\sum_{(\lambda,\mu)\in\mathfrak{F}(0,a;q)\times\mathfrak{F}(-a,b;q)}w(\lambda)w(\mu)=\dfrac{(ab;q)_{\infty}}{(a;q)_{\infty}(b;q)_{\infty}}=\sum_{(\lambda,\mu)\in\mathfrak{F}(0,a,0,b;q)}w(\lambda)w(\mu).$
(3.2)

###### Proof.

The first equality follows directly from applying Proposition 3.1 for the pair
$(\lambda,\mu)$. To show the weight equivalence between
$\mathfrak{F}(0,a;q)\times\mathfrak{F}(-a,b;q)$ and $\mathfrak{F}(0,a,0,b;q)$,
it suffices to construct a weight-preserving, sign-reversing involution111That
is, we have $w(\lambda^{*})w(\mu^{*})=-w(\lambda)w(\mu)$ whenever
$(\lambda^{*},\mu^{*})\neq(\lambda,\mu)$., say
$*:(\lambda,\mu)\mapsto(\lambda^{*},\mu^{*})$, over
$\mathfrak{F}(0,a;q)\times\mathfrak{F}(-a,b;q)$, with
$\mathfrak{F}(0,a,0,b;q)$ being precisely the set of fixed pairs; see Fig. 4
for an example of the map $*$, where we use $\bar{a}$ in place of the weight
$-a$ to save space.

Given a pair $(\lambda,\mu)$ with $\\#(\mu)=t$, let $\lambda_{i}$ be the
largest part of $\lambda$ that is no greater than $t$, note that the leftmost
cell of $\lambda_{i}$ is filled with $a$ since
$\lambda\in\mathfrak{F}(0,a;q)$. Also let $j$ be the largest index of an
$ab$-colored part of $\mu$, note that the rightmost cell of $\mu_{j}$ is
filled with $-a$ since $\mu\in\mathfrak{F}(-a,b;q)$. If such a $\lambda_{i}$
(resp. $j$) does not exist, we simply set $\lambda_{i}=0$ (resp. $j=0$). Now
we compare $\lambda_{i}$ with $j$, and consider the following three cases.

  1. Case I

$\lambda_{i}\leq j\neq 0$. Insert $j$ as a new part into $\lambda$ to get
$\lambda^{*}$, and convert $\mu$ to $\mu^{*}$ whose parts are given by

$\displaystyle\mu^{*}_{k}=\begin{cases}\mu_{k}-1,&\textrm{if}~{}1\leq k\leq
j,\\\ \mu_{k},&\textrm{otherwise.}\\\ \end{cases}$

Note that $\mu_{j}$ is $ab$-colored hence $\mu_{j}>\mu_{j+1}$, which ensures
$\mu^{*}_{j}\geq\mu^{*}_{j+1}$, and thus $\mu^{*}$ is well-defined.

  2. Case II

$\lambda_{i}>j$. Remove part $\lambda_{i}$ from $\lambda$ to get
$\lambda^{*}$. Convert $\mu$ to $\mu^{*}$ whose parts are given by

$\displaystyle\mu^{*}_{k}=\begin{cases}\mu_{k}+1,&\textrm{if}~{}1\leq
k\leq\lambda_{i},\\\ \mu_{k},&\textrm{otherwise.}\\\ \end{cases}$

We fill the rightmost cell of $\mu^{*}_{\lambda_{i}}$ with $-a$, while other
inserted cells are filled with $q$. Note that originally $\mu_{\lambda_{i}}$
cannot be $ab$-colored, or we are not in Case II. Therefore color part
$\mu^{*}_{\lambda_{i}}$ as $ab$ is legitimate.

  3. Case III

$\lambda_{i}=j=0$. We stay put and take $(\lambda^{*},\mu^{*})=(\lambda,\mu)$.
Note that we are in this case if and only if all parts of $\lambda$ are
greater than $t$ (or $\lambda=\epsilon$), and none part of $\mu$ is
$ab$-colored. In other words, $(\lambda,\mu)\in\mathfrak{F}(0,a,0,b;q)$, as
desired.

It should be clear that the operations conducted in Cases I and II are
inverses of each other, and $*:(\lambda,\mu)\mapsto(\lambda^{*},\mu^{*})$ as
described above is an involution over
$\mathfrak{F}(0,a;q)\times\mathfrak{F}(-a,b;q)$, such that
$w(\lambda^{*})w(\mu^{*})=-w(\lambda)w(\mu)$ whenever
$(\lambda^{*},\mu^{*})\neq(\lambda,\mu)$. And the set of pairs fixed by $*$ is
exactly $\mathfrak{F}(0,a,0,b;q)$. We have now proven (3.2). ∎

$a$$a$$a$$a$$q$$q$$q$$q$$q$$q$$q$$q$$\lambda$$b$$b$$b$$b$$\bar{a}$$\bar{a}$$q$$q$$q$$q$$q$$\mu$$\stackrel{{\scriptstyle*}}{{\mapsto}}$$a$$a$$a$$q$$q$$q$$q$$q$$q$$\lambda^{*}$$b$$b$$b$$b$$\bar{a}$$\bar{a}$$\bar{a}$$q$$q$$q$$q$$q$$q$$q$$\mu^{*}$
Figure 4. An example of the involution $*$ (Case II with $\lambda_{i}=3>j=2$)

Next, we proceed to establish Theorem 1.2. The first step is to note the
vanishing of certain $N_{\textrm{eo}}(k,m,n)$, which can be readily deduced
from Lemma 3.2.

######  Corollary 3.4.

For any $n\geq 0$,

$\displaystyle N_{\emph{eo}}(0,4,4n+2)=N_{\emph{eo}}(2,4,4n+4)=0.$ (3.3)

###### Proof.

As explained in the proof of Lemma 3.2, for each $\pi=\varphi(\lambda,\mu)$
with $\\#(\lambda)=s$ and $\\#(\mu)=t$, we have

$\displaystyle\textrm{eoc}(\pi)=2t-2s\equiv 2t+2s\equiv|\pi|\pmod{4},$

which is clearly equivalent to (3.3). ∎

###### Proof of Theorem 1.2.

We first show the divisibility by $5$. Note that when $r\not\equiv 0\pmod{k}$,
$N_{\textrm{eo}}(r,k,n)=N_{\textrm{eo}}(k-r,k,n)$ via the map of conjugation,
hence for $0<r<k/2$ we have

$\displaystyle r\big{(}N_{\textrm{eo}}(r,k,n)-N_{\textrm{eo}}(k-r,k,n)\big{)}$
$\displaystyle=r\sum_{\begin{subarray}{c}\lambda\in\overline{\mathfrak{EO}}_{n}\\\
\mathrm{eoc}(\lambda)\equiv
r\pmod{k}\end{subarray}}\big{(}\mathcal{O}(\lambda)-\mathcal{O}(\lambda^{\prime})\big{)}$
$\displaystyle\equiv-r^{2}N_{\textrm{eo}}(r,k,n)\pmod{k},\qquad(\text{by~{}\eqref{eq:Andrews=Stanley}})$

In particular, if $\gcd(r,k)=1$,

$\displaystyle N_{\textrm{eo}}(r,k,n)-N_{\textrm{eo}}(k-r,k,n)\equiv-
rN_{\textrm{eo}}(r,k,n)\pmod{k}.$

Therefore,

$\displaystyle\big{(}N_{\textrm{eo}}(1,5,n)-N_{\textrm{eo}}(4,5,n)\big{)}$
$\displaystyle+2\big{(}N_{\textrm{eo}}(2,5,n)-N_{\textrm{eo}}(3,5,n)\big{)}$
$\displaystyle\quad\equiv-
N_{\textrm{eo}}(1,5,n)-4N_{\textrm{eo}}(2,5,n)\pmod{5}.$

Finally, according to [36, Theorem 2.1.5],

$\displaystyle N_{\textrm{eo}}(1,5,10n)=N_{\textrm{eo}}(2,5,10n),$

and (1.9) already gives us

$\displaystyle N_{\textrm{eo}}(1,5,10n+8)=N_{\textrm{eo}}(2,5,10n+8).$

So both congruences (1.13) and (1.14) hold modulo $5$. Since each partition
$\lambda\in\overline{\mathfrak{EO}}$ has an even number of odd parts, we get
the modulo $2$ results immediately. All it remains, is to show that

$\displaystyle NT_{\textrm{eo}}(1,5,10n+8)-NT_{\textrm{eo}}(4,5,10n+8)$
$\displaystyle\quad=\sum_{\begin{subarray}{c}\lambda\in\overline{\mathfrak{EO}}_{10n+8}\\\
\mathrm{eoc}(\lambda)\equiv
1\pmod{5}\end{subarray}}\big{(}\mathcal{O}(\lambda)-\mathcal{O}(\lambda^{\prime})\big{)}=-\sum_{\begin{subarray}{c}\lambda\in\overline{\mathfrak{EO}}_{10n+8}\\\
\mathrm{eoc}(\lambda)\equiv
1\pmod{5}\end{subarray}}\mathrm{eoc}(\lambda)\equiv 0\pmod{4}.$ (3.4)

To this end, we write $n=4m+i$ with $i=0,1,2,3$, and discuss the following two
cases.

  1. (1)

The case $i\equiv 0\pmod{2}$. In this case $10n+8\equiv 0\pmod{4}$ so each
$\lambda\in\overline{\mathfrak{EO}}_{10n+8}$ satisfies
$\textrm{eoc}(\lambda)\equiv 0\pmod{4}$ by (3.3), thus (3.4) holds.

  2. (2)

The case $i\equiv 1\pmod{2}$. In this case $10n+8\equiv 2\pmod{4}$ so each
$\lambda\in\overline{\mathfrak{EO}}_{10n+8}$ satisfies
$\textrm{eoc}(\lambda)\equiv 2\pmod{4}$ by (3.3), and the total number of
summands is

$\displaystyle
N_{\textrm{eo}}(1,5,40m+10i+8)=\frac{1}{5}\overline{\mathcal{EO}}(40m+10i+8)\equiv
0\pmod{2},\quad\text{(by~{}\eqref{cong:bEOparity})}$

i.e., it is an even number. So (3.4) still holds.

This completes the proof. ∎

###  3.2. Three more stable subsets of $\mathfrak{EO}$

With the discussion of $\overline{\mathfrak{EO}}$ after Definition 2.1 in
mind, other stable subsets of $\mathfrak{EO}$ besides
$\overline{\mathfrak{EO}}$ naturally present themselves. We study three of
them in this subsection. First note that $\mathfrak{EO}$ itself is not stable,
as can be seen from Fig. 1, wherein $\lambda^{\prime}\in\mathfrak{EO}$ but
$\lambda\not\in\mathfrak{EO}$. The three stable subsets of $\mathfrak{EO}$ are
defined as follows, the first of which is the largest one, in the sense that
there exists no stable subset $\mathfrak{C}$, such that
$\mathfrak{EO}^{(1)}\subsetneq\mathfrak{C}\subsetneq\mathfrak{EO}$.

$\displaystyle\mathfrak{EO}^{(1)}$
$\displaystyle:=\\{\lambda\in\mathfrak{EO}\colon\textrm{at~{}most~{}one~{}part~{}of~{}$\lambda$~{}aont}\\},$
$\displaystyle\mathfrak{EO}^{(2)}$
$\displaystyle:=\\{\lambda\in\mathfrak{EO}\colon\textrm{only~{}the~{}smallest~{}odd~{}part~{}of~{}$\lambda$~{}aont}\\},$
$\displaystyle\mathfrak{EO}^{(3)}$
$\displaystyle:=\\{\lambda\in\mathfrak{EO}\colon\textrm{all~{}parts~{}of~{}$\lambda$~{}odd~{}and~{}only~{}the~{}largest~{}odd~{}part~{}of~{}$\lambda$
aont}\\}.$

Here “aont” stands for “appears an odd number of times”. Note that
$\epsilon\not\in\mathfrak{EO}^{(2)}\cup\mathfrak{EO}^{(3)}$. Interpreting the
defining conditions for $\mathfrak{EO}^{(1)}$, $\mathfrak{EO}^{(2)}$ and
$\mathfrak{EO}^{(3)}$ in terms of the profile words for the respective
partitions, we obtain the following connection with mock theta functions
$\nu(q)$ and $\omega(q)$.

######  Theorem 3.5.

The three subsets $\mathfrak{EO}^{(1)}$, $\mathfrak{EO}^{(2)}$ and
$\mathfrak{EO}^{(3)}$ are all stable. Furthermore, we have

$\displaystyle\sum_{n=1}^{\infty}\mathcal{EO}^{(2)}(n)q^{n}$
$\displaystyle=\dfrac{1}{2}q\big{(}\nu(q)+\nu(-q)\big{)},$ (3.5)
$\displaystyle\sum_{n=1}^{\infty}\mathcal{EO}^{(3)}(n)q^{n}$
$\displaystyle=q\\!\>\omega(q^{2}),$ (3.6)

where $\nu(q)$ and $\omega(q)$ are defined as in (1.5) and (1.6).

###### Proof.

Given a profile word $w_{\lambda}=(e_{1},n_{1},\ldots,e_{k},n_{k})$ associated
with partition $\lambda$. It can be characterized respectively as having at
most one odd number among all $n_{i}$’s when $\lambda\in\mathfrak{EO}^{(1)}$,
having a consecutive odd pair $(e_{i},n_{i})$ when
$\lambda\in\mathfrak{EO}^{(2)}$, and having only $e_{1}$ and $n_{k}$ as odd
components when $\lambda\in\mathfrak{EO}^{(3)}$. Each of the above
characterizations is clearly invariant under the reversal $w_{\lambda}\to
w_{\lambda^{\prime}}$, hence all three subsets $\mathfrak{EO}^{(1)}$,
$\mathfrak{EO}^{(2)}$ and $\mathfrak{EO}^{(3)}$ are stable.

To obtain (3.5), we observe that there is a natural bijection between the set
$\mathfrak{EO}^{(2)}_{n}$ and $\overline{\mathfrak{EO}}_{n-1}$. For a given
partition $\lambda\in\mathfrak{EO}^{(2)}_{n}$ with $n\geq 1$, subtract one
from the last smallest odd part of $\lambda$ and denote this new partition by
$\mu$. Obviously, $\mu\in\overline{\mathfrak{EO}}_{n-1}$. Conversely, for a
given partition $\mu\in\overline{\mathfrak{EO}}_{n-1}$, adding one to its
first largest even part222If $\mu$ has none even part, simply append $1$ as a
new part. recovers for us a partition $\lambda\in\mathfrak{EO}^{(2)}_{n}$.
Therefore,

$\displaystyle\sum_{n=1}^{\infty}\mathcal{EO}^{(2)}(n)q^{n}=\sum_{n=1}^{\infty}\overline{\mathcal{EO}}(n-1)q^{n}=q\dfrac{(q^{4};q^{4})_{\infty}^{3}}{(q^{2};q^{2})_{\infty}^{2}}=\dfrac{1}{2}q\big{(}\nu(q)+\nu(-q)\big{)},$

where the second and last equalities follow from Corollary 3.2 and Theorem 1.1
in [4].

To handle the connection with $\omega(q)$ and prove (3.6), we recall a
variation of the standard Ferrers graph introduced by Andrews in [3], called
the “odd Ferrers graph”. It consists of a Ferrers graph using $2$’s with a
surrounding border of $1$’s; see Fig. 5 for an illustration along with its
“$2$-dilation”. As noted in [3, p. 60], the generating function for odd
Ferrers graphs is $q\\!\>\omega(q)$. Now we immediately have (3.6) by noticing
that each partition in $\mathfrak{EO}^{(3)}$ can be uniquely represented by a
$2$-dilated odd Ferrers graph with its top-left corner cell subtracted by $1$.
∎

$1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$4$$4$$4$$4$$4$$4$$4$$4$$4$$4$$2$-dilation$\longrightarrow$
Figure 5. An odd Ferrers graph and its $2$-dilation

######  Remark 3.6.

Several remarks on Theorem 3.5 are necessary. Firstly, the coefficients of
$\nu(-q)$ and $\omega(q)$ also possess their own partition-theoretic
interpretations, so it is interesting to give a bijective proof of (3.5) and
(3.6). Secondly, there are various investigations on arithmetic properties of
coefficients of $\nu(q)$ and $\omega(q)$ (see, for example, [24, 11, 8]). This
suggests that there should be various congruences for functions
$\mathcal{EO}^{(2)}(n)$ and $\mathcal{EO}^{(3)}(n)$, inherited from those of
$\nu(q)$ and $\omega(q)$. Finally, it is not easy to find the generating
function for $\mathcal{EO}^{(1)}(n)$ following a similar line of proving (3.5)
and (3.6).

Let $\nu(-q):=\sum_{n\geq 0}p_{\nu}(n)q^{n}$, then it was shown by Andrews,
Dixit, and Yee [7, Theorem. 4.1] that $p_{\nu}(n)$ counts the number of
distinct partitions of $n$ in which all odd parts are less than twice the
smallest part.333Actually, when such a nonempty partition contains no odd
parts, it will be counted twice in $p_{\nu}(n)$. For instance, $p_{\nu}(6)=4$,
counting both partitions $(6)$ and $(4,2)$ twice. This correction has been
confirmed with Yee [43]. Thanks to the new connections made in Theorem 3.5, we
are rewarded with an alternative interpretation of $p_{\nu}(n)$. Note that the
even $n$ case of the following result is essentially rephrasing Theorem 1.1 of
[4]. From this perspective, the next corollary is a companion to Theorem 1.1
of [4].

######  Corollary 3.7.

For any $n\geq 0$, $p_{\nu}(n)$ also counts the number of partitions of $n$ in
$\mathfrak{EO}$, such that when $n$ is even, then only the largest even part
appears an odd number of times; when $n$ is odd, then all parts are odd and
only the largest odd part appears an odd number of times.

###### Proof.

According to [41, p. 72], we find that

$\displaystyle\nu(q)+q\\!\>\omega(q^{2})=\dfrac{(q^{4};q^{4})_{\infty}^{3}}{(q^{2};q^{2})_{\infty}^{2}}.$
(3.7)

Replacing $q$ by $-q$ yields

$\displaystyle\nu(-q)-q\\!\>\omega(q^{2})=\dfrac{(q^{4};q^{4})_{\infty}^{3}}{(q^{2};q^{2})_{\infty}^{2}}.$
(3.8)

In view of (3.5)–(3.8),

$\displaystyle\sum_{n=1}^{\infty}\mathcal{EO}^{(2)}(n)q^{n}+\sum_{n=1}^{\infty}\mathcal{EO}^{(3)}(n)q^{n+1}=q\nu(-q)=\sum_{n=0}^{\infty}p_{\nu}(n)q^{n+1},$

from which we obtain

$\displaystyle\mathcal{EO}^{(2)}(n+1)+\mathcal{EO}^{(3)}(n)=p_{\nu}(n),\quad\text{for
}n\geq 0.$

Finally, we observe that the powers of $q$ in both
$q\big{(}\nu(q)+\nu(-q)\big{)}/2$ and $q\\!\>\omega(q^{2})$ are odd, which
results in $\mathcal{EO}^{(2)}(2m)=\mathcal{EO}^{(3)}(2m)=0$ for every integer
$m\geq 0$. Now we apply the identity
$\mathcal{EO}^{(2)}(n+1)=\overline{\mathcal{EO}}(n)$ to finish the proof. ∎

##  4\. Stable sets in $\mathfrak{OE}$

In the first subsection, we present a proof of Theorem 1.3 and characterize
completely the parity of $\overline{\mathcal{OE}}(n)$. We consider further
subsets of $\overline{\mathfrak{OE}}$ in subsection 4.2.

###  4.1. Results for $\overline{\mathfrak{OE}}$

For a partition $\lambda\in\overline{\mathfrak{OE}}$, its profile word
$w_{\lambda}$ begins and ends with an odd number $e_{1}$ and $n_{k}$
respectively, and contains precisely one consecutive odd pair
$(n_{i},e_{i+1})$, with remaining letters all being even. This
characterization is seen to be invariant under conjugation thus
$\overline{\mathfrak{OE}}$ is stable. Then, with Observation 1.1 in mind, we
see (1.15), which we prove next, parallels Andrews’ generating function (3.1)
of $\overline{\mathcal{EO}}(m,n)$ nicely.

###### Proof of Theorem 1.3.

Recall that for each $\lambda\in\overline{\mathfrak{OE}}$, it contains both
even and odd parts, each odd part is smaller than each even part, and ONLY the
largest odd part and the largest even part appears an odd number of times.
Suppose $\lambda$ (resp. $\lambda^{\prime}$) has $2n-1$ (resp. $2m-1$) even
parts, for certain $n,m\geq 1$. From these constraints we can uniquely dissect
$\lambda$ into five pieces; see Fig. 6 for an illustration. The top-left
rectangle contains $(2n-1)\times(2m-1)$ cells which contribute
$q^{(2m-1)(2n-1)}$ to the generating function; a horizontal strip representing
the largest odd part of $\lambda$ with contribution $zq^{2m-1}$; a vertical
strip representing the largest odd part of $\lambda^{\prime}$ with
contribution $z^{-1}q^{2n-1}$; a subpartition $\alpha$ below the horizontal
strip; and a subpartition $\beta$ to the right of the vertical strip. Note
that $\alpha$ is a partition into odd parts each occurring an even number of
times, with the largest part no greater than $2m-1$, and its contribution is
$z^{\\#(\alpha)}q^{|\alpha|}$. Similarly, $\beta$’s conjugation
$\beta^{\prime}$ is also a partition into odd parts each occurring an even
number of times, with the largest part no greater than $2n-1$, and its
contribution is $z^{-\\#(\beta^{\prime})}q^{|\beta^{\prime}|}$. Finally, it
suffices to note that all such $\alpha$’s (resp. $\beta$’s) are generated by
$1/(z^{2}q^{2};q^{4})_{m}$ (resp. $1/(q^{2}/z^{2};q^{4})_{n}$). Putting
together all five pieces we arrive at (1.15). ∎

$\beta$$\alpha$$2n-1$$2m-1$ Figure 6. Dissection of a partition
$\lambda\in\overline{\mathfrak{OE}}$

Next, we study the parity of $\overline{\mathcal{OE}}(n)$, in the hope of
getting a complete characterization analogous to Corollary 2.4.

Another third order mock theta function due to Watson [41, p. 62] is given by

$\displaystyle\psi^{(3)}(q)=\sum_{n=1}^{\infty}\dfrac{q^{n^{2}}}{(q;q^{2})_{n}}:=\sum_{n=1}^{\infty}p_{\psi}(n)q^{n}.$
(4.1)

Utilizing some techniques from analytic and algebraic number theory, Wang [40,
Theorem 4.2] established the following complete characterization modulo 2 for
$p_{\psi}(n)$:

$\displaystyle p_{\psi}(m)\equiv
1\pmod{2}~{}\Longleftrightarrow~{}24m-1=p^{4\alpha+1}k^{2}\;\;\textrm{for~{}some~{}prime~{}$p$~{}coprime~{}to~{}$k$}.$
(4.2)

Relying on Wang’s result, we are able to fully characterize
$\overline{\mathcal{OE}}(n)$ modulo $2$.

######  Theorem 4.1.

For any $n\geq 1$,

$\displaystyle\overline{\mathcal{OE}}(n)\equiv\begin{cases}1\pmod{2},&\textrm{if~{}$n\equiv
3\pmod{4}$}\;\;\textrm{and}\;\;6n+5=p^{4\alpha+1}k^{2}\\\
&\textrm{for~{}some~{}prime~{}$p$~{}coprime~{}to~{}$k$},\\\
0\pmod{2},&\textrm{otherwise}.\end{cases}$ (4.3)

###### Proof.

By Proposition 2.3, it suffices to consider only self-conjugate partitions in
$\overline{\mathfrak{OE}}$. Their generating function is given by

$\displaystyle\sum_{n=0}^{\infty}\overline{\mathcal{OE}}_{c}(n)q^{n}$
$\displaystyle=q^{-1}\sum_{n=1}^{\infty}\frac{q^{4n^{2}}}{(q^{4};q^{8})_{n}}=q^{-1}\sum_{m=1}^{\infty}p_{\psi}(m)q^{4m}\qquad\textrm{(by~{}\eqref{def:psi})}.$

It follows that

$\displaystyle\overline{\mathcal{OE}}(n)\equiv\begin{cases}p_{\psi}(m)\pmod{2},&\textrm{if}\;\;n=4m-1,\\\
\quad 0\quad\;\pmod{2},&\textrm{otherwise.}\end{cases}$ (4.4)

Now (4.3) follows from (4.4) and (4.2) immediately. ∎

With the aid of Theorem 4.1, it is possible to deduce Ramanujan-type
congruence for a certain arithmetic progression. We record one example below.

######  Corollary 4.2.

For any $n>0$ and $n\not\equiv 0\pmod{5}$,

$\displaystyle\overline{\mathcal{OE}}(20n-5)$ $\displaystyle\equiv 0\pmod{2}.$
(4.5)

###### Proof.

Writing $n=5m+i$ for $i=1,2,3,4$, we see that (4.5) is equivalent to

$\displaystyle\overline{\mathcal{OE}}(100m+15)\equiv\overline{\mathcal{OE}}(100m+35)$
$\displaystyle\equiv 0\pmod{2},$
$\displaystyle\overline{\mathcal{OE}}(100m+55)\equiv\overline{\mathcal{OE}}(100m+75)$
$\displaystyle\equiv 0\pmod{2}.$

In view of the relation (4.4), we only need to show that

$\displaystyle p_{\psi}(25m+4)\equiv p_{\psi}(25m+9)\equiv
p_{\psi}(25m+14)\equiv p_{\psi}(25m+19)\equiv 0\pmod{2}.$ (4.6)

Here we only prove the congruence $p_{\psi}(25m+4)\equiv 0\pmod{2}$, because
the proofs of the remaining cases are quite similar.

For any $m\geq 0$, note that $24(25m+4)-1=5(120m+19)$, $5\nmid(120m+19)$ and
$120m+19$ can not be a square (a square is congruent to $0$ or $1$ modulo
$4$). According to (4.2), we obtain the desired congruence. ∎

######  Remark 4.3.

Quite recently, Chen and Garvan [13, Eq. (4.31)] derived an infinite family of
congruences modulo $4$ satisfied by $p_{\psi}(n)$. One easily derives from
their results the following strengthening of (4.6). For any $n\geq 0$,

$\displaystyle p_{\psi}(25n+9)\equiv p_{\psi}(25n+14)\equiv 0\pmod{4}.$

From this perspective, it might be interesting to pursue mod $4$ results for
$\overline{\mathcal{OE}}(n)$.

###  4.2. Subsets of $\overline{\mathfrak{OE}}$

Given a partition $\lambda$, let $\mathcal{E}(\lambda)$ be the number of even
parts in $\lambda$. We further refine $\overline{\mathfrak{OE}}$ as follows.
Let $k\geq 0$ and

$\displaystyle\overline{\mathfrak{OE}}_{k}:=\\{\lambda\in\overline{\mathfrak{OE}}:\mathcal{E}(\lambda)-\mathcal{E}(\lambda^{\prime})=2k\\}.$

Noting that each partition $\lambda\in\overline{\mathfrak{OE}}$ must be a
partition of an odd number, we define $p_{k}(0)=p_{k}(1)=0$, and for all
$n\geq 2$ let

$\displaystyle p_{k}(n)$
$\displaystyle:=|\\{\lambda\in\overline{\mathfrak{OE}}_{k}:|\lambda|=2n-1\\}|.$

The following explicit generating functions for $p_{0}(n)$ and $p_{1}(n)$
presage their connection with the mock theta function $\omega(q)$ (see
Corollary 4.6).

######  Corollary 4.4.

There holds

$\displaystyle\sum_{n=0}^{\infty}p_{0}(n)q^{n}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{q^{2n^{2}}}{(q;q^{2})_{n}^{2}},$ (4.7)
$\displaystyle\sum_{n=0}^{\infty}p_{1}(n)q^{n}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{q^{2n^{2}+2n}}{(q;q^{2})_{n}(q;q^{2})_{n+1}}.$
(4.8)

###### Proof.

The length of the vertical (resp. horizontal) strip that arises in the proof
of Theorem 1.3 is precisely $\mathcal{E}(\lambda)$ (resp.
$\mathcal{E}(\lambda^{\prime})$). Therefore, by setting $n=m$ and putting
$z\to 1$, $q\to q^{1/2}$ in (1.15), we immediately arrive at (4.7). Similarly,
with $n=m+1$ we can deduce (4.8). ∎

######  Remark 4.5.

Combining (4.1) and (4.7), we obtain $p_{0}(2n)\equiv p_{\psi}(n)\pmod{2}$
immediately. Alternatively, this congruence could be combinatorially justified
as we derive (4.4).

Let $p_{\omega}(n)$ denote the number of partitions of $n$ in which each odd
part is less than twice the smallest part. Just like the partition-theoretical
interpretation for $p_{\nu}(n)$, it was first noticed in the same paper by
Andrews, Dixit, and Yee [7, Theorem 3.1] that

$\displaystyle q\\!\>\omega(q)=\sum_{n=1}^{\infty}p_{\omega}(n)q^{n}.$

######  Corollary 4.6.

For any $n\geq 1$,

$\displaystyle p_{0}(n)+p_{1}(n-1)+1=p_{\omega}(n).$ (4.9)

###### Proof.

It follows form (4.7) and (4.8) that

$\displaystyle\sum_{n=0}^{\infty}p_{0}(n)q^{n}+\sum_{n=0}^{\infty}p_{1}(n)q^{n+1}+\sum_{n=1}^{\infty}q^{n}$
$\displaystyle\quad=\sum_{n=0}^{\infty}\frac{q^{2n^{2}+4n+2}}{(q;q^{2})_{n+1}^{2}}+\sum_{n=1}^{\infty}\frac{q^{2n^{2}+2n+1}}{(q;q^{2})_{n}(q;q^{2})_{n+1}}+\dfrac{q}{1-q}$
$\displaystyle\quad=\sum_{n=0}^{\infty}\frac{q^{2n^{2}+4n+2}}{(q;q^{2})_{n+1}^{2}}+\sum_{n=1}^{\infty}\frac{q^{2n^{2}+2n+1}(1-q^{2n+1})}{(q;q^{2})_{n+1}^{2}}+\dfrac{q}{1-q}$
$\displaystyle\quad=\sum_{n=1}^{\infty}\dfrac{q^{2n^{2}+2n+1}}{(q;q^{2})_{n+1}^{2}}+\dfrac{q^{2}}{(1-q)^{2}}+\dfrac{q}{1-q}$
$\displaystyle\quad=\sum_{n=0}^{\infty}\dfrac{q^{2n^{2}+2n+1}}{(q;q^{2})_{n+1}^{2}}=q\\!\>\omega(q)=\sum_{n=1}^{\infty}p_{\omega}(n)q^{n},$

Then (4.9) follows by comparing the coefficients of $q^{n}$ on both sides of
the identity above. ∎

######  Problem 4.7.

Prove (4.9) combinatorially via their partition-theoretical interpretations.

##  5\. Final remarks

We conclude this paper with several conjectures to motivate further
investigation.

######  Conjecture 5.1.

Let $\ell>3$ be a prime number and $\ell\not\equiv 23\pmod{24}$. Then for any
$n\geq 0$,

$\displaystyle p_{0}\big{(}2\ell^{2}n+2\ell j+2\delta_{\ell}\big{)}\equiv
0\pmod{4},$

where $0\leq j\leq\ell-2$ and $\delta_{\ell}$ is the least positive integer
such that $24\delta_{\ell}\equiv 1\pmod{\ell}$.

######  Conjecture 5.2.

We have

$\displaystyle\lim_{n\rightarrow\infty}\dfrac{\\#\\{m|\>p_{0}(m)\equiv
0\pmod{2},\;\;0\leq m<n\\}}{n}=\dfrac{1}{5}.$

######  Conjecture 5.3.

  1. (i)

For any $n\geq 3$,

$\displaystyle\overline{\mathcal{OE}}(2n+1)>\overline{\mathcal{EO}}(2n).$

  2. (ii)

For any $n\geq 1$,

$\displaystyle p_{0}(n)\geq p_{1}(n),$

where the strict inequality holds if $n\neq 1$, $10$ or $13$.

## Acknowledgements

Shishuo Fu was partially supported by the National Natural Science Foundation
of China (No. 12171059), the Natural Science Foundation Project of Chongqing
(No. cstc2021jcyj-msxmX0693), and the Mathematical Research Center of
Chongqing University. Dazhao Tang was partially supported by the National
Natural Science Foundation of China (No. 12201093), the Natural Science
Foundation Project of Chongqing CSTB (No. CSTB2022NS
CQ–MSX0387), the Science and Technology Research Program of Chongqing
Municipal Education Commission (No. KJQN202200509), and the Doctoral start-up
research grant (No. 21XLB038) of Chongqing Normal University.

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