# New Infinite Families of Congruences Modulo Powers of 2 for 2–Regular
Partitions with Designated Summands

James A. Sellers  Department of Mathematics and Statistics, University of
Minnesota Duluth, Duluth, MN 55812, USA<EMAIL_ADDRESS>
###### Abstract.

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects
which they called partitions with designated summands. These are built by
taking unrestricted integer partitions and designating exactly one of each
occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also
studied such partitions wherein all parts must be odd. Recently, Herden,
Sepanski, Stanfill, Hammon, Henningsen, Ickes, and Ruiz proved a number of
Ramanujan–like congruences for the function $PD_{2}(n)$ which counts the
number of partitions of weight $n$ with designated summands wherein all parts
must be odd. In this work, we prove some of the results conjectured by Herden,
et. al. by proving the following two infinite families of congruences
satisfied by $PD_{2}(n)$: For all $\alpha\geq 0$ and $n\geq 0,$

$\displaystyle PD_{2}(2^{\alpha}(4n+3))$ $\displaystyle\equiv$ $\displaystyle
0\pmod{4}\ \ \ \ \ {\textrm{a}nd}$ $\displaystyle PD_{2}(2^{\alpha}(8n+7))$
$\displaystyle\equiv$ $\displaystyle 0\pmod{8}.$

All of the proof techniques used herein are elementary, relying on classical
$q$–series identities and generating function manipulations.

###### Key words and phrases:

partitions, congruences, designated summands, generating functions,
dissections

###### 2010 Mathematics Subject Classification:

11P83, 05A17

##  1\. Introduction

In 2002, Andrews, Lewis, and Lovejoy [1] introduced the combinatorial objects
which they called partitions with designated summands. These are built by
taking unrestricted integer partitions and designating exactly one of each
occurrence of a part. For example, there are 10 partitions with designated
summands of weight 4:

$4^{\prime},\ \ \ 3^{\prime}+1^{\prime},\ \ \ 2^{\prime}+2,\ \ \
2+2^{\prime},\ \ \ 2^{\prime}+1^{\prime}+1,\ \ \ 2^{\prime}+1+1^{\prime}$
$1^{\prime}+1+1+1,\ \ \ 1+1^{\prime}+1+1,\ \ \ 1+1+1^{\prime}+1,\ \ \
1+1+1+1^{\prime}$

Andrews, Lewis, and Lovejoy denoted the number of partitions with designated
summands of weight $n$ by the function $PD(n).$ Hence, using this notation and
the example above, we know $PD(4)=10.$

In the same paper, Andrews, Lewis, and Lovejoy [1] also considered the
restricted partitions with designated summands wherein all parts must be odd,
and they denoted the corresponding enumeration function by $PDO(n).$ Thus,
from the example above, we see that $PDO(4)=5,$ where we have counted the
following five objects:

$3^{\prime}+1^{\prime},\ \ \ 1^{\prime}+1+1+1,\ \ \ 1+1^{\prime}+1+1,\ \ \
1+1+1^{\prime}+1,\ \ \ 1+1+1+1^{\prime}$

Since then, numerous authors have studied arithmetic properties of $PD(n)$ as
well as the family of functions $PD_{k}(n)$ which denotes the number of
$k$–regular partitions of $n$ with designated summands. (Recall that a
partition is called $k$–regular if none of the parts of the partition are
divisible by $k.$) Note that $PDO(n)=PD_{2}(n)$ in this notation.

Beginning with [1], a wide variety of Ramanujan–like congruences have been
proven for $PD(n)$ and $PD_{k}(n)$ for infinitely many values of $k$. See [2,
3, 6, 7, 8, 10, 13, 15] for such work.

Recently, Herden, et. al. [11] proved a number of arithmetic properties
satisfied by the functions $PD_{2^{\ell}}(n)$ and $PD_{3^{\ell}}(n)$ for a
variety of values of $\ell$. At the end of their paper, they shared the
following conjecture which will serve as the starting point for the work in
this paper:

######  Conjecture 1.1.

For $n\geq 0,$ we have

$\displaystyle PD_{2}(16n+12)$ $\displaystyle\equiv$ $\displaystyle
0\pmod{4},$ $\displaystyle PD_{2}(24n+20)$ $\displaystyle\equiv$
$\displaystyle 0\pmod{4},$ $\displaystyle PD_{2}(25n+5)$ $\displaystyle\equiv$
$\displaystyle 0\pmod{4},$ $\displaystyle PD_{2}(32n+24)$
$\displaystyle\equiv$ $\displaystyle 0\pmod{4},$ $\displaystyle
PD_{2}(48n+26)$ $\displaystyle\equiv$ $\displaystyle 0\pmod{4}$

It is intriguing to note that the fourth arithmetic progression which appears
above, $32n+24$, equals $2(16n+12)$, i.e., $32n+24$ is twice the first
arithmetic progression above. This is a hint of something much larger; indeed,
these two congruences are true, and they belong to an easily–described
infinite family of congruences modulo 4:

######  Theorem 1.2.

For all $\alpha\geq 0$ and all $n\geq 0,$

$PD_{2}(2^{\alpha}(4n+3))\equiv 0\pmod{4}.$

In fact, there is an additional family of congruences of a similar form modulo
8 which we will also prove below.

######  Theorem 1.3.

For all $\alpha\geq 0$ and all $n\geq 0,$

$PD_{2}(2^{\alpha}(8n+7))\equiv 0\pmod{8}.$

Our main goal in this paper is to prove Theorems 1.2 and 1.3. All of the proof
techniques used herein are elementary, relying on classical $q$–series
identites and generating function manipulations.

##  2\. Preliminaries

As noted in [1], the generating function for $PD_{2}(n)$ is given by

(1)
$\sum_{n=0}^{\infty}PD_{2}(n)q^{n}=\frac{f_{4}f_{6}^{2}}{f_{1}f_{3}f_{12}}$

where $f_{r}=(1-q^{r})(1-q^{2r})(1-q^{3r})(1-q^{4r})\dots$ is the usual
$q$–Pochhammer symbol.

In order to prove our results, we will require several elementary generating
function dissection tools. For the most part, these will involve well–known
2–dissection results that will allow us to manipulate (1) to our advantage. We
share these results here.

######  Lemma 2.1.

We have

$\displaystyle\frac{1}{f_{1}^{4}}$
$\displaystyle=\frac{f_{4}^{14}}{f_{2}^{14}f_{8}^{4}}+4q\frac{f_{4}^{2}f_{8}^{4}}{f_{2}^{10}}.$

###### Proof.

This identity can be found in the work of Brietzke, da Silva, and Sellers [4,
Equation (18)]. ∎

######  Lemma 2.2.

We have

$f_{1}^{2}=\frac{f_{2}f_{8}^{5}}{f_{4}^{2}f_{16}^{2}}-2q\frac{f_{2}f_{16}^{2}}{f_{8}}.$

###### Proof.

See the work of da Silva and Sellers [9, Equation (12), Lemma 1]. ∎

######  Lemma 2.3.

We have

$\displaystyle\frac{1}{f_{1}f_{3}}$
$\displaystyle=\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}+q\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}},$
$\displaystyle f_{1}f_{3}$
$\displaystyle=\frac{f_{2}f_{8}^{2}f_{12}^{4}}{f_{4}^{2}f_{6}f_{24}^{2}}-q\frac{f_{4}^{4}f_{6}f_{24}^{2}}{f_{2}f_{8}^{2}f_{12}^{2}}.$

###### Proof.

Both of these identities can be found in Hirschhorn [12]; the first is
Equation (30.12.3) while the second is Equation (30.12.1). ∎

Finally, we state without proof the well–known result regarding congruences of
the functions $f_{k}$ modulo powers of a prime (which, in essence, relies on
the divisibility properties of binomial coefficients).

######  Lemma 2.4.

For any prime $p,$ any $j\geq 1,$ and any $m\geq 1,$ $f_{m}^{p^{j}k}\equiv
f_{pm}^{p^{j-1}k}\pmod{p^{j}}$.

We close this section by listing a number of dissection results for the
generating function for $PD_{2}$ which are easily found via elementary
generating function dissections.

######  Theorem 2.5.

We have

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{4}^{2}f_{6}^{4}}{f_{1}^{2}f_{3}^{2}f_{12}^{2}},\textrm{\
\ \ and}$ $\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n+1)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{2}^{6}f_{12}^{2}}{f_{1}^{4}f_{4}^{2}f_{6}^{2}}.$

###### Proof.

See [1, Theorem 21]. ∎

######  Theorem 2.6.

We have

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{4}^{4}f_{6}^{8}}{f_{1}^{4}f_{3}^{4}f_{12}^{4}}+q\frac{f_{2}^{12}f_{12}^{4}}{f_{1}^{8}f_{4}^{4}f_{6}^{4}},$
$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+1)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{2}^{12}f_{6}^{2}}{f_{1}^{8}f_{3}^{2}f_{4}^{4}},$
$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{n}$ $\displaystyle=$
$\displaystyle 2\frac{f_{2}^{6}f_{6}^{2}}{f_{1}^{6}f_{3}^{2}},\textrm{\ \ \
and}$ $\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+3)q^{n}$ $\displaystyle=$
$\displaystyle 4\frac{f_{4}^{4}f_{6}^{2}}{f_{1}^{4}f_{3}^{2}}.$

###### Proof.

Using Lemma 2.3 and Theorem 2.5, we have

(2) $\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{4}^{2}f_{6}^{4}}{f_{1}^{2}f_{3}^{2}f_{12}^{2}}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{f_{1}f_{3}}\right)^{2}\frac{f_{4}^{2}f_{6}^{4}}{f_{12}^{2}}$
$\displaystyle=$
$\displaystyle\left(\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}+q\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)^{2}\frac{f_{4}^{2}f_{6}^{4}}{f_{12}^{2}}$
$\displaystyle=$
$\displaystyle\left(\frac{f_{8}^{4}f_{12}^{10}}{f_{2}^{4}f_{4}^{2}f_{6}^{8}f_{24}^{4}}+2q\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}\cdot\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}+q^{2}\frac{f_{4}^{10}f_{24}^{4}}{f_{2}^{8}f_{6}^{4}f_{8}^{4}f_{12}^{2}}\right)\frac{f_{4}^{2}f_{6}^{4}}{f_{12}^{2}}.$

From (2), we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n)q^{2n}$ $\displaystyle=$
$\displaystyle\frac{f_{8}^{4}f_{12}^{8}}{f_{2}^{4}f_{6}^{4}f_{24}^{4}}+q^{2}\frac{f_{4}^{12}f_{24}^{4}}{f_{2}^{8}f_{8}^{4}f_{12}^{4}}$

or

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{4}^{4}f_{6}^{8}}{f_{1}^{4}f_{3}^{4}f_{12}^{4}}+q\frac{f_{2}^{12}f_{12}^{4}}{f_{1}^{8}f_{4}^{4}f_{6}^{4}}.$

Also from (2), we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{2n+1}$ $\displaystyle=$
$\displaystyle
2q\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}\cdot\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\cdot\frac{f_{4}^{2}f_{6}^{4}}{f_{12}^{2}}$
$\displaystyle=$ $\displaystyle
2q\frac{f_{4}^{6}f_{12}^{2}}{f_{2}^{6}f_{6}^{2}}$

after simplification. And this implies

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{n}$ $\displaystyle=$
$\displaystyle 2\frac{f_{2}^{6}f_{6}^{2}}{f_{1}^{6}f_{3}^{2}}$

which also appears in [3, Theorem 1.3]. Indeed, the generating function
results for $PD_{2}(4n+1)$ and $PD_{2}(4n+3)$ follow in similar fashion and
can also be found in [3, Theorem 1.3]. ∎

With these tools in hand, we proceed to our elementary proofs of Theorems 1.2
and 1.3.

##  3\. An Infinite Family of Congruences Modulo 4

Our focus in this section is on Theorem 1.2. In order to prove this result, we
begin with several generating function dissections for $PD_{2}(n)$ modulo 4.
We begin by proving the 2-dissection for $PD_{2}(n)$ modulo 4.

######  Theorem 3.1.

We have

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n)q^{n}$ $\displaystyle\equiv$
$\displaystyle\frac{f_{4}^{2}}{f_{1}^{2}f_{3}^{2}}\pmod{4},\textrm{\ \ \ and}$
$\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n+1)q^{n}$ $\displaystyle\equiv$
$\displaystyle f_{6}^{2}\pmod{4}.$

###### Proof.

Thanks to Lemma 2.4 and Theorem 2.5, we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{4}^{2}f_{6}^{4}}{f_{1}^{2}f_{3}^{2}f_{12}^{2}}$
$\displaystyle\equiv$
$\displaystyle\frac{f_{2}^{4}f_{12}^{2}}{f_{1}^{2}f_{3}^{2}f_{12}^{2}}\pmod{4}$
$\displaystyle\equiv$
$\displaystyle\frac{f_{4}^{2}}{f_{1}^{2}f_{3}^{2}}\pmod{4}.$

Again using Lemma 2.4, Theorem 2.5, and elementary simplifications, we have

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n+1)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{2}^{6}f_{12}^{2}}{f_{1}^{4}f_{4}^{2}f_{6}^{2}}$
$\displaystyle\equiv$ $\displaystyle\frac{f_{6}^{4}}{f_{6}^{2}}\pmod{4}$
$\displaystyle=$ $\displaystyle f_{6}^{2}.$

∎

Thanks to the fact that $\sum_{n=0}^{\infty}PD_{2}(2n+1)q^{n}\equiv
f_{6}^{2}\pmod{4}$, which is a function of $q^{6}$, we immediately have the
following corollary:

######  Corollary 3.2.

For all $n\geq 0,$

$\displaystyle PD_{2}(4n+3)$ $\displaystyle\equiv 0$ $\displaystyle\pmod{4},$
$\displaystyle PD_{2}(6n+3)$ $\displaystyle\equiv 0$
$\displaystyle\pmod{4},\textrm{\ \ \ and}$ $\displaystyle PD_{2}(6n+5)$
$\displaystyle\equiv 0$ $\displaystyle\pmod{4}.$

###### Proof.

We note that, once expanded, the power series representation of $f_{6}^{2}$
must only contain powers of $q$ whose exponents are multiples of 6. Therefore,
we immediately know that $PD_{2}(2(2n+1)+1),$ $PD_{2}(2(3n+1)+1),$ and
$PD_{2}(2(3n+2)+1)$ must all be congruent to 0 $\pmod{4}.$ ∎

We note that the above congruences, along with several others, appear in [3,
Theorem 1.4].

We next transition to the necessary 4–dissections of the generating function
for $PD_{2}(n)$ modulo 4.

######  Theorem 3.3.

We have

(3) $\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n)q^{n}$
$\displaystyle\equiv\left(\frac{f_{2}^{3}}{f_{6}}\right)^{2}+qf_{12}^{2}\pmod{4}\textrm{\
\ \ and}$ $\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{n}$
$\displaystyle\equiv 2f_{2}^{3}f_{6}\pmod{4}.$

###### Proof.

From Theorem 3.1, we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n)q^{n}$ $\displaystyle\equiv$
$\displaystyle\frac{f_{4}^{2}}{f_{1}^{2}f_{3}^{2}}\pmod{4}$ $\displaystyle=$
$\displaystyle f_{4}^{2}\left(\frac{1}{f_{1}f_{3}}\right)^{2}.$

Thus, thanks to Lemma 2.3, we know

(4) $\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n)q^{n}$ $\displaystyle\equiv$
$\displaystyle f_{4}^{2}\left(\frac{1}{f_{1}f_{3}}\right)^{2}\pmod{4}$
$\displaystyle=$ $\displaystyle
f_{4}^{2}\left(\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}+q\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)^{2}$
$\displaystyle=$ $\displaystyle
f_{4}^{2}\left(\frac{f_{8}^{4}f_{12}^{10}}{f_{2}^{4}f_{4}^{2}f_{6}^{8}f_{24}^{4}}+q^{2}\frac{f_{4}^{10}f_{24}^{4}}{f_{2}^{8}f_{6}^{4}f_{8}^{4}f_{12}^{2}}+2q\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}\cdot\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)$
$\displaystyle=$
$\displaystyle\frac{f_{8}^{4}f_{12}^{10}}{f_{2}^{4}f_{6}^{8}f_{24}^{4}}+q^{2}\frac{f_{4}^{12}f_{24}^{4}}{f_{2}^{8}f_{6}^{4}f_{8}^{4}f_{12}^{2}}+2q\frac{f_{4}^{6}f_{12}^{4}}{f_{2}^{6}f_{6}^{6}}.$

Collecting those terms above which are even functions of $q$ and replacing
$q^{2}$ by $q$, we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n)q^{n}$ $\displaystyle\equiv$
$\displaystyle\frac{f_{4}^{4}f_{6}^{10}}{f_{1}^{4}f_{3}^{8}f_{12}^{4}}+q\frac{f_{2}^{12}f_{12}^{4}}{f_{1}^{8}f_{3}^{4}f_{4}^{4}f_{6}^{2}}\pmod{4}$
$\displaystyle\equiv$
$\displaystyle\frac{f_{2}^{8}f_{6}^{10}}{f_{2}^{2}f_{6}^{4}f_{6}^{8}}+q\frac{f_{4}^{6}f_{6}^{8}}{f_{4}^{2}f_{6}^{2}f_{4}^{4}f_{6}^{2}}\pmod{4}$
$\displaystyle\equiv$
$\displaystyle\frac{f_{2}^{6}}{f_{6}^{2}}+qf_{6}^{4}\pmod{4}$
$\displaystyle\equiv$
$\displaystyle\left(\frac{f_{2}^{3}}{f_{6}}\right)^{2}+qf_{12}^{2}\pmod{4}$

after simplification. Returning to (4) and focusing on the odd powers of $q$,
we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{2n+1}$ $\displaystyle\equiv$
$\displaystyle 2q\frac{f_{4}^{6}f_{12}^{4}}{f_{2}^{6}f_{6}^{6}}\pmod{4}$

which implies

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{n}$ $\displaystyle\equiv$
$\displaystyle 2\frac{f_{2}^{6}f_{6}^{4}}{f_{1}^{6}f_{3}^{6}}\pmod{4}$
$\displaystyle\equiv$ $\displaystyle
2\frac{f_{2}^{6}f_{6}^{4}}{f_{2}^{3}f_{6}^{3}}\pmod{4}$ $\displaystyle\equiv$
$\displaystyle 2f_{2}^{3}f_{6}\pmod{4}$

using Lemma 2.4. ∎

Thanks to Theorem 3.3, we have the following corollaries:

######  Corollary 3.4.

For all $n\geq 0,$ $PD_{2}(8n+6)\equiv 0\pmod{4}.$

###### Proof.

From Theorem 3.3 we know

$\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{n}\equiv 2f_{2}^{3}f_{6}\pmod{4}$

where $f_{2}^{3}f_{6}$ is an even function of $q.$ Therefore, for all $n\geq
0,$

$PD_{2}(4(2n+1)+2)=PD_{2}(8n+6)\equiv 0\pmod{4}.$

∎

Note that the result in Corollary 3.4 also appears in [3, Theorem 1.5].

######  Corollary 3.5.

We have

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n)q^{n}$ $\displaystyle\equiv$
$\displaystyle\left(\frac{f_{1}^{3}}{f_{3}}\right)^{2}\pmod{4}\textrm{\ \ \
and}$ $\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n+4)q^{n}$
$\displaystyle\equiv$ $\displaystyle f_{6}^{2}\pmod{4}.$

###### Proof.

Thanks to Theorem 3.3 we know

$\sum_{n=0}^{\infty}PD_{2}(4n)q^{n}\equiv\left(\frac{f_{2}^{3}}{f_{6}}\right)^{2}+qf_{12}^{2}\pmod{4}$

and the right–hand side above is already 2–dissected. Therefore, the corollary
follows by simply splitting the result for the generating function congruence
for $PD_{2}(4n)$ into its even and odd parts and replacing $q^{2}$ by $q$. ∎

Corollary 3.5 also leads us to our next Ramanujan–like divisibility property.

######  Corollary 3.6.

For all $n\geq 0,$ $PD_{2}(16n+12)\equiv 0\pmod{4}.$

###### Proof.

From Corollary 3.5, we see that

$\sum_{n=0}^{\infty}PD_{2}(8n+4)q^{n}\equiv f_{6}^{2}\pmod{4}.$

Because the right–hand side of this congruence is an even function of $q$, we
know that, for all $n\geq 0$,

$PD_{2}(8(2n+1)+4)=PD_{2}(16n+12)\equiv 0\pmod{4}.$

∎

Note that the congruence in Corollary 3.6 was one of the congruences
conjectured in [11].

We now need only one additional tool in order to complete our proof of Theorem
1.2. The following theorem provides an “internal congruence” modulo 4 which is
satisfied by $PD_{2}(n)$.

######  Theorem 3.7.

For all $n\geq 0,$ $PD_{2}(4n)\equiv PD_{2}(n)\pmod{4}.$

###### Proof.

From (3) above, we know that

$\sum_{n=0}^{\infty}PD_{2}(4n)q^{n}\equiv\left(\frac{f_{2}^{3}}{f_{6}}\right)^{2}+qf_{12}^{2}\pmod{4}.$

From [11, Lemma 4], we also know

$\sum_{n=0}^{\infty}PD_{2}(n)q^{n}\equiv\left(\frac{f_{2}^{3}}{f_{6}}\right)^{2}+qf_{12}^{2}\pmod{4}.$

The result follows. ∎

We now have all of the tools needed to provide a proof of Theorem 1.2.

###### Proof.

(of Theorem 1.2) Note that the $\alpha=0$ case is proven in Corollary 3.2,
while the $\alpha=1$ case appears in Corollary 3.4. We then see that, thanks
to Theorem 3.7, we have (mod 4)

$0\equiv PD_{2}(4n+3)\equiv PD_{2}(4(4n+3))\equiv PD_{2}(4^{2}(4n+3))\equiv
PD_{2}(4^{3}(4n+3))\dots$

which takes care of all of the cases with an even power of $\alpha.$ For the
cases where $\alpha$ is odd, we note that $8n+6=2(4n+3),$ and this then gives
us (mod 4)

$0\equiv PD_{2}(2(4n+3))\equiv PD_{2}(4\cdot 2(4n+3))\equiv PD_{2}(4^{2}\cdot
2(4n+3))\equiv PD_{2}(4^{3}\cdot 2(4n+3))\dots$

∎

We now have a proof of an infinite family of congruences modulo 4 satisfied by
$PD_{2}$ which encompasses two of the conjectured congruences in [11].

We close this section by noting that, thanks to Corollary 3.2 and Theorem 3.7,
we now know the following:

######  Theorem 3.8.

For all $\alpha\geq 0$ and all $n\geq 0,$

$PD_{2}(4^{\alpha}(6n+5))\equiv 0\pmod{4}.$

This is another satisfying infinite family of congruences modulo 4, and we
note that the $\alpha=1$ case of this family is the second conjecture that
appeared in the list at the end of [11].

##  4\. An Infinite Family of Congruences Modulo 8

We now transition to the proof of a related family of Ramanujan–like
congruences satisfied by $PD_{2},$ this time modulo 8, which was stated in
Theorem 1.3. As a reminder, that theorem states that, for all $\alpha\geq 0$
and all $n\geq 0,$

$PD_{2}(2^{\alpha}(8n+7))\equiv 0\pmod{8}.$

We will prove Theorem 1.3 in an analogous manner to that used above; namely,
we will directly prove a few “base” cases of the congruence, and then we will
prove an internal congruence modulo 8 which will serve as the mechanism by
which we can allow $\alpha$ to grow arbitrarily large.

To begin this process, we prove the $\alpha=0$ case of the above theorem.

######  Theorem 4.1.

For all $n\geq 0,$ $PD_{2}(8n+7)\equiv 0\pmod{8}.$

###### Proof.

Thanks to Theorem 2.5 as well as Lemma 2.1, we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(2n+1)q^{n}$ $\displaystyle=$
$\displaystyle\frac{1}{f_{1}^{4}}\left(\frac{f_{2}^{6}f_{12}^{2}}{f_{4}^{2}f_{6}^{2}}\right)$
$\displaystyle=$
$\displaystyle\left(\frac{f_{4}^{14}}{f_{2}^{14}f_{8}^{4}}+4q\frac{f_{4}^{2}f_{8}^{4}}{f_{2}^{10}}\right)\left(\frac{f_{2}^{6}f_{12}^{2}}{f_{4}^{2}f_{6}^{2}}\right).$

Thus, we see that

$\sum_{n=0}^{\infty}PD_{2}(4n+3)q^{2n+1}=4q\frac{f_{8}^{4}f_{12}^{2}}{f_{2}^{4}f_{6}^{2}}$

or

$\sum_{n=0}^{\infty}PD_{2}(4n+3)q^{n}=4\frac{f_{4}^{4}f_{6}^{2}}{f_{1}^{4}f_{3}^{2}}\equiv
4\frac{f_{4}^{4}f_{6}^{2}}{f_{2}^{2}f_{6}}\pmod{8}.$

Because the function
$\frac{f_{4}^{4}f_{6}^{2}}{f_{2}^{2}f_{6}}=\frac{f_{4}^{4}f_{6}}{f_{2}^{2}}$
is an even function of $q$, we immediately conclude that, for all $n\geq 0,$
$PD_{2}(4(2n+1)+3)=PD_{2}(8n+7)\equiv 0\pmod{8}.$ ∎

Next, we prove the $\alpha=1$ case of Theorem 1.3.

######  Theorem 4.2.

For all $n\geq 0,$ $PD_{2}(16n+14)\equiv 0\pmod{8}.$

###### Proof.

Using Theorem 2.6 and Lemmas 2.3 and 2.1, we have

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{n}$ $\displaystyle=$
$\displaystyle 2\frac{f_{2}^{6}f_{6}^{2}}{f_{1}^{6}f_{3}^{2}}$
$\displaystyle=$ $\displaystyle
2f_{2}^{6}f_{6}^{2}\left(\frac{1}{f_{1}^{4}}\right)\left(\frac{1}{f_{1}f_{3}}\right)^{2}$
$\displaystyle=$ $\displaystyle
2f_{2}^{6}f_{6}^{2}\left(\frac{f_{4}^{14}}{f_{2}^{14}f_{8}^{4}}+4q\frac{f_{4}^{2}f_{8}^{4}}{f_{2}^{10}}\right)\left(\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}+q\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)^{2}$
$\displaystyle\equiv$ $\displaystyle
2\left(\frac{f_{4}^{14}f_{6}^{2}}{f_{2}^{8}f_{8}^{4}}\right)\left(\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}+q\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)^{2}\pmod{8}.$

Expanding the above and keeping only those terms where the powers of $q$ are
odd yields

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n+6)q^{2n+1}$ $\displaystyle\equiv$
$\displaystyle
2\left(\frac{f_{4}^{14}f_{6}^{2}}{f_{2}^{8}f_{8}^{4}}\right)\left(2q\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}\cdot\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4q\frac{f_{4}^{18}f_{12}^{4}}{f_{2}^{14}f_{6}^{4}f_{8}^{4}}\pmod{8}$

which implies that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n+6)q^{n}$ $\displaystyle\equiv$
$\displaystyle
4\frac{f_{2}^{18}f_{6}^{4}}{f_{1}^{14}f_{3}^{4}f_{4}^{4}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4\frac{f_{2}^{11}f_{6}^{2}}{f_{4}^{4}}\pmod{8}.$

Since the last expression above is an even function of $q$, we immediately
know that, for all $n\geq 0$,

$PD_{2}(8(2n+1)+6)=PD_{2}(16n+14)\equiv 0\pmod{8}.$

∎

We next prove the $\alpha=2$ case of Theorem 1.3.

######  Theorem 4.3.

For all $n\geq 0,$ $PD_{2}(32n+28)\equiv 0\pmod{8}.$

###### Proof.

Thanks to Theorem 2.6, we know

(5) $\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n)q^{n}$ $\displaystyle=$
$\displaystyle\frac{f_{4}^{4}f_{6}^{8}}{f_{1}^{4}f_{3}^{4}f_{12}^{4}}+q\frac{f_{2}^{12}f_{12}^{4}}{f_{1}^{8}f_{4}^{4}f_{6}^{4}}$
$\displaystyle\equiv$
$\displaystyle\left(\frac{f_{4}}{f_{1}f_{3}}\right)^{4}+q\left(\frac{f_{12}}{f_{6}}\right)^{4}\pmod{8}\
\ \textrm{using Lemma \ref{general_congs_result}}$ (6) $\displaystyle=$
$\displaystyle
f_{4}^{4}\left(\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}+q\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)^{4}+q\frac{f_{12}^{4}}{f_{6}^{4}}.$

Thus,

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n+4)q^{2n+1}$ $\displaystyle\equiv$
$\displaystyle
f_{4}^{4}\left(4q\left(\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}\right)^{3}\left(\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)\right)$
$\displaystyle+f_{4}^{4}\left(4q^{3}\left(\frac{f_{8}^{2}f_{12}^{5}}{f_{2}^{2}f_{4}f_{6}^{4}f_{24}^{2}}\right)\left(\frac{f_{4}^{5}f_{24}^{2}}{f_{2}^{4}f_{6}^{2}f_{8}^{2}f_{12}}\right)^{3}\right)+q\frac{f_{12}^{4}}{f_{6}^{4}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4q\frac{f_{4}^{9}f_{8}^{6}f_{12}^{15}f_{24}^{2}}{f_{2}^{10}f_{4}^{3}f_{6}^{14}f_{8}^{2}f_{12}f_{24}^{6}}+4q^{3}\frac{f_{4}^{19}f_{8}^{2}f_{12}^{5}f_{24}^{6}}{f_{2}^{14}f_{4}f_{6}^{10}f_{8}^{6}f_{12}^{3}f_{24}^{2}}+q\frac{f_{12}^{4}}{f_{6}^{4}}\pmod{8}.$

This means

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n+4)q^{n}$ $\displaystyle\equiv$
$\displaystyle
4\frac{f_{2}^{9}f_{4}^{6}f_{6}^{15}f_{12}^{2}}{f_{1}^{10}f_{2}^{3}f_{3}^{14}f_{4}^{2}f_{6}f_{12}^{6}}+4q\frac{f_{2}^{19}f_{4}^{2}f_{6}^{5}f_{12}^{6}}{f_{1}^{14}f_{2}f_{3}^{10}f_{4}^{6}f_{6}^{3}f_{12}^{2}}+\frac{f_{6}^{4}}{f_{3}^{4}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4\frac{f_{2}^{9}f_{2}^{12}f_{6}^{15}f_{6}^{4}}{f_{2}^{5}f_{2}^{3}f_{6}^{7}f_{2}^{4}f_{6}f_{6}^{12}}+4q\frac{f_{2}^{19}f_{2}^{4}f_{6}^{5}f_{6}^{12}}{f_{2}^{7}f_{2}f_{6}^{5}f_{2}^{12}f_{6}^{3}f_{6}^{4}}+\frac{f_{6}^{4}}{f_{3}^{4}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4\frac{f_{2}^{9}}{f_{6}}+4qf_{2}^{3}f_{6}^{5}+f_{6}^{4}\left(\frac{f_{12}^{14}}{f_{6}^{14}f_{24}^{4}}+4q^{3}\frac{f_{12}^{2}f_{24}^{4}}{f_{6}^{10}}\right)\pmod{8}$

using Lemma 2.1 with $q$ replaced by $q^{3}.$ Retaining only those terms where
the powers of $q$ are odd yields

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n+12)q^{2n+1}$ $\displaystyle\equiv$
$\displaystyle
4qf_{2}^{3}f_{6}^{5}+4q^{3}\frac{f_{6}^{4}f_{12}^{2}f_{24}^{4}}{f_{6}^{10}}\pmod{8}$

which means

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n+12)q^{n}$ $\displaystyle\equiv$
$\displaystyle
4f_{1}^{3}f_{3}^{5}+4q\frac{f_{6}^{2}f_{12}^{4}}{f_{3}^{6}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4f_{1}^{2}f_{1}f_{3}^{4}f_{3}+4q\frac{f_{6}^{2}f_{6}^{8}}{f_{6}^{3}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4f_{1}f_{2}f_{3}f_{6}^{2}+4qf_{6}^{7}\pmod{8}$ $\displaystyle\equiv$
$\displaystyle
4f_{2}f_{6}^{2}\left(\frac{f_{2}f_{8}^{2}f_{12}^{4}}{f_{4}^{2}f_{6}f_{24}^{2}}-q\frac{f_{4}^{4}f_{6}f_{24}^{2}}{f_{2}f_{8}^{2}f_{12}^{2}}\right)+4qf_{6}^{7}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4\frac{f_{2}^{2}f_{6}f_{8}^{2}f_{12}^{4}}{f_{4}^{2}f_{24}^{2}}+4q\frac{f_{4}^{4}f_{6}^{3}f_{24}^{2}}{f_{8}^{2}f_{12}^{2}}+4qf_{6}^{7}\pmod{8}$

where in the penultimate line above we have used Lemma 2.3. This implies that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(32n+28)q^{2n+1}$ $\displaystyle\equiv$
$\displaystyle
4q\frac{f_{4}^{4}f_{6}^{3}f_{24}^{2}}{f_{8}^{2}f_{12}^{2}}+4qf_{6}^{7}\pmod{8}$

or

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(32n+28)q^{n}$ $\displaystyle\equiv$
$\displaystyle
4\frac{f_{2}^{4}f_{3}^{3}f_{12}^{2}}{f_{4}^{2}f_{6}^{2}}+4f_{3}^{7}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4\frac{f_{4}^{2}f_{3}^{3}f_{3}^{8}}{f_{4}^{2}f_{3}^{4}}+4f_{3}^{7}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle 4f_{3}^{7}+4f_{3}^{7}\pmod{8}$
$\displaystyle=$ $\displaystyle 0.$

The result follows. ∎

We now prove one more Ramanujan–like congruence modulo 8 which is the
$\alpha=3$ case of Theorem 1.3.

######  Theorem 4.4.

For all $n\geq 0,$ $PD_{2}(64n+56)\equiv 0\pmod{8}.$

###### Proof.

Thanks to (6), we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n)q^{2n}$ $\displaystyle\equiv$
$\displaystyle\frac{f_{4}^{4}f_{8}^{8}f_{12}^{20}}{f_{2}^{8}f_{4}^{4}f_{6}^{16}f_{24}^{8}}+6q^{2}\frac{f_{4}^{4}f_{8}^{4}f_{12}^{10}}{f_{2}^{4}f_{4}^{2}f_{6}^{8}f_{24}^{4}}\cdot\frac{f_{4}^{10}f_{24}^{4}}{f_{2}^{8}f_{6}^{4}f_{8}^{4}f_{12}^{2}}$
$\displaystyle+q^{4}\frac{f_{4}^{4}f_{4}^{20}f_{24}^{8}}{f_{2}^{16}f_{6}^{8}f_{8}^{8}f_{12}^{4}}\pmod{8}.$

Thus,

(7) $\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n)q^{n}$ $\displaystyle\equiv$
$\displaystyle\frac{f_{4}^{8}f_{6}^{20}}{f_{1}^{8}f_{3}^{16}f_{12}^{8}}+6q\frac{f_{2}^{14}f_{6}^{10}}{f_{1}^{12}f_{2}^{2}f_{3}^{12}f_{6}^{2}}+q^{2}\frac{f_{2}^{24}f_{12}^{8}}{f_{1}^{16}f_{3}^{8}f_{4}^{8}f_{6}^{4}}\pmod{8}$
$\displaystyle\equiv$
$\displaystyle\frac{f_{8}^{4}f_{6}^{20}}{f_{2}^{4}f_{6}^{8}f_{6}^{16}}+6q\frac{f_{2}^{14}f_{6}^{10}}{f_{2}^{6}f_{2}^{2}f_{6}^{6}f_{6}^{2}}+q^{2}\frac{f_{2}^{24}f_{12}^{8}}{f_{2}^{8}f_{6}^{4}f_{4}^{8}f_{6}^{4}}\pmod{8}$
$\displaystyle\equiv$
$\displaystyle\frac{f_{8}^{4}}{f_{2}^{4}f_{6}^{4}}+6q\frac{f_{2}^{14}f_{6}^{10}}{f_{2}^{8}f_{6}^{8}}+q^{2}\frac{f_{2}^{24}f_{24}^{4}}{f_{2}^{8}f_{12}^{4}f_{4}^{8}}\pmod{8}.$

This means that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n+8)q^{2n+1}$ $\displaystyle\equiv$
$\displaystyle 6q\frac{f_{2}^{14}f_{6}^{10}}{f_{2}^{8}f_{6}^{8}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle 6qf_{2}^{6}f_{6}^{2}\pmod{8}.$

Dividing both sides of this congruence by $q$ and replacing $q^{2}$ by $q$
yields

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n+8)q^{n}$ $\displaystyle\equiv$
$\displaystyle 6f_{1}^{6}f_{3}^{2}\pmod{8}$ $\displaystyle=$ $\displaystyle
6(f_{1}^{2})^{2}(f_{1}f_{3})^{2}.$

We can then apply Lemma 2.2 and Lemma 2.3 to get

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n+8)q^{n}$ $\displaystyle\equiv$
$\displaystyle 6(f_{1}^{2})^{2}(f_{1}f_{3})^{2}\pmod{8}$ $\displaystyle\equiv$
$\displaystyle
6\left(\frac{f_{2}f_{8}^{5}}{f_{4}^{2}f_{16}^{2}}-2q\frac{f_{2}f_{16}^{2}}{f_{8}}\right)^{2}\left(\frac{f_{2}f_{8}^{2}f_{12}^{4}}{f_{4}^{2}f_{6}f_{24}^{2}}-q\frac{f_{4}^{4}f_{6}f_{24}^{2}}{f_{2}f_{8}^{2}f_{12}^{2}}\right)^{2}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
6\left(\frac{f_{2}f_{8}^{5}}{f_{4}^{2}f_{16}^{2}}\right)^{2}\left(\frac{f_{2}^{2}f_{8}^{4}f_{12}^{8}}{f_{4}^{4}f_{6}^{2}f_{24}^{4}}-2q\frac{f_{2}f_{8}^{2}f_{12}^{4}}{f_{4}^{2}f_{6}f_{24}^{2}}\cdot\frac{f_{4}^{4}f_{6}f_{24}^{2}}{f_{2}f_{8}^{2}f_{12}^{2}}+q^{2}\frac{f_{4}^{8}f_{6}^{2}f_{24}^{4}}{f_{2}^{2}f_{8}^{4}f_{12}^{4}}\right)\pmod{8}.$

Therefore,

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(32n+24)q^{2n+1}$ $\displaystyle\equiv$
$\displaystyle-12q\left(\frac{f_{2}f_{8}^{5}}{f_{4}^{2}f_{16}^{2}}\right)^{2}\frac{f_{2}f_{8}^{2}f_{12}^{4}}{f_{4}^{2}f_{6}f_{24}^{2}}\cdot\frac{f_{4}^{4}f_{6}f_{24}^{2}}{f_{2}f_{8}^{2}f_{12}^{2}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle
4q\frac{f_{2}^{2}f_{8}^{10}f_{12}^{4}}{f_{4}^{2}f_{12}^{2}f_{16}^{4}}\pmod{8}$
$\displaystyle\equiv$ $\displaystyle 4q\frac{f_{12}^{2}f_{16}}{f_{4}}\pmod{8}$

after simplification. Therefore,

$\sum_{n=0}^{\infty}PD_{2}(32n+24)q^{2n+1}\equiv
4\frac{f_{6}^{2}f_{8}}{f_{2}}\pmod{8}.$

Because the above is an even function of $q,$ we know that, for all $n\geq 0,$

$PD_{2}(32(2n+1)+24)=PD_{2}(64n+56)\equiv 0\pmod{8}.$

∎

The last item that we need in order to complete the proof of Theorem 1.3 is
the following internal congruence.

######  Theorem 4.5.

For all $n\geq 0,$ $PD_{2}(16n)\equiv PD_{2}(4n)\pmod{8}.$

###### Proof.

Thanks to (7), we see that

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n)q^{2n}$ $\displaystyle\equiv$
$\displaystyle\frac{f_{8}^{4}}{f_{2}^{4}f_{6}^{4}}+q^{2}\frac{f_{2}^{24}f_{24}^{4}}{f_{2}^{8}f_{12}^{4}f_{4}^{8}}\pmod{8}$
$\displaystyle\equiv$
$\displaystyle\frac{f_{8}^{4}}{f_{2}^{4}f_{6}^{4}}+q^{2}\frac{f_{2}^{24}f_{24}^{4}}{f_{2}^{8}f_{12}^{4}f_{2}^{16}}\pmod{8}$
$\displaystyle\equiv$
$\displaystyle\frac{f_{8}^{4}}{f_{2}^{4}f_{6}^{4}}+q^{2}\frac{f_{24}^{4}}{f_{12}^{4}}\pmod{8}$

which means

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n)q^{n}$ $\displaystyle\equiv$
$\displaystyle\frac{f_{4}^{4}}{f_{1}^{4}f_{3}^{4}}+q\frac{f_{12}^{4}}{f_{6}^{4}}\pmod{8}$
$\displaystyle\equiv$
$\displaystyle\sum_{n=0}^{\infty}PD_{2}(4n)q^{n}\textrm{\ \ thanks to
(\ref{4n_mod8_overall})}.$

∎

We now have all of the tools needed to provide a proof of Theorem 1.3.

###### Proof.

(of Theorem 1.3) Note that the $\alpha=0,1,2,3$ cases are proven in Theorems
4.1, 4.2, 4.3, and 4.4 respectively. We then see that, thanks to Theorems 4.3
and 4.5, and the fact that $32n+28=4(8n+7)$, we have (mod 8)

$0\equiv PD_{2}(4(8n+7))\equiv PD_{2}(4^{2}(8n+7))\equiv
PD_{2}(4^{3}(8n+7))\dots$

which takes care of all of the cases with an even power of $\alpha.$ For the
cases where $\alpha$ is odd, we use Theorems 4.4 and 4.5 and the fact that

$64n+56=4(2(8n+7))$

to yield (mod 8)

$0\equiv PD_{2}(4(2(8n+7)))\equiv PD_{2}(4^{2}(2(8n+7)))\equiv
PD_{2}(4^{3}(2(8n+7)))\dots$

∎

##  5\. Closing Thoughts

We close by providing an elementary proof of the last congruence in Conjecture
1.1:

######  Theorem 5.1.

For all $n\geq 0,$ $PD_{2}(48n+26)\equiv 0\pmod{4}.$

###### Proof.

From Theorem 3.3, we know

$\sum_{n=0}^{\infty}PD_{2}(4n+2)q^{n}\equiv 2f_{2}^{3}f_{6}\pmod{4}.$

Thus, we see that

$\sum_{n=0}^{\infty}PD_{2}(8n+2)q^{n}\equiv 2f_{1}^{3}f_{3}\pmod{4}.$

Using Lemmas 2.2 and 2.3, we then have

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(8n+2)q^{n}$ $\displaystyle\equiv$
$\displaystyle 2f_{1}^{3}f_{3}\pmod{4}$ $\displaystyle=$ $\displaystyle
2(f_{1}f_{3})f_{1}^{2}$ $\displaystyle=$ $\displaystyle
2\left(\frac{f_{2}f_{8}^{2}f_{12}^{4}}{f_{4}^{2}f_{6}f_{24}^{2}}-q\frac{f_{4}^{4}f_{6}f_{24}^{2}}{f_{2}f_{8}^{2}f_{12}^{2}}\right)\left(\frac{f_{2}f_{8}^{5}}{f_{4}^{2}f_{16}^{2}}-2q\frac{f_{2}f_{16}^{2}}{f_{8}}\right)^{2}$
$\displaystyle\equiv$ $\displaystyle
2\left(\frac{f_{2}f_{8}^{2}f_{12}^{4}}{f_{4}^{2}f_{6}f_{24}^{2}}-q\frac{f_{4}^{4}f_{6}f_{24}^{2}}{f_{2}f_{8}^{2}f_{12}^{2}}\right)\left(\frac{f_{2}f_{8}^{5}}{f_{4}^{2}f_{16}^{2}}\right)^{2}\pmod{4}$
$\displaystyle\equiv$ $\displaystyle
2\frac{f_{2}^{2}f_{8}^{7}f_{12}^{4}}{f_{4}^{4}f_{6}f_{24}^{2}}-2q\frac{f_{2}f_{4}^{4}f_{6}f_{8}^{5}f_{24}^{2}}{f_{2}f_{4}^{2}f_{8}^{2}f_{12}^{2}f_{16}^{2}}\pmod{4}.$

This implies

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n+10)q^{2n+1}$ $\displaystyle\equiv$
$\displaystyle
2q\frac{f_{2}f_{4}^{4}f_{6}f_{8}^{5}f_{24}^{2}}{f_{2}f_{4}^{2}f_{8}^{2}f_{12}^{2}f_{16}^{2}}\pmod{4}$

or

$\displaystyle\sum_{n=0}^{\infty}PD_{2}(16n+10)q^{n}$ $\displaystyle\equiv$
$\displaystyle
2\frac{f_{2}^{4}f_{3}f_{4}^{5}f_{12}^{2}}{f_{2}^{2}f_{4}^{2}f_{6}^{2}f_{8}^{2}}\pmod{4}$
$\displaystyle\equiv$ $\displaystyle
2\frac{f_{2}^{4}f_{3}f_{2}^{10}f_{6}^{4}}{f_{2}^{2}f_{2}^{4}f_{6}^{2}f_{2}^{8}}\pmod{4}$
$\displaystyle\equiv$ $\displaystyle
2\frac{f_{3}f_{6}^{4}}{f_{6}^{2}}\pmod{4}$ $\displaystyle=$ $\displaystyle
2f_{3}f_{6}^{2}.$

Since the last function above is a function of $q^{3},$ we know that, for all
$n\geq 0,$

$PD_{2}(16(3n+1)+10)=PD_{2}(48n+26)\equiv 0\pmod{4}.$

∎

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Research Isaiah Lyons-Galante and Morteza Karimzadeh are with the University of Colorado Boulder. E-mails: isaiah.lyons-galante |<EMAIL_ADDRESS>Samantha Molnar, Graham Johnson, and Kenny Gruchalla are with the National Renewable Energy Lab. E-mails: sam.molnar | graham.johnson |<EMAIL_ADDRESS>
# Alternatives to Contour Visualizations for Power Systems Data

Isaiah Lyons-Galante0009-0000-4615-3721  Morteza Karimzadeh0000-0002-6498-1763
Samantha Molnar0000-0002-9589-4670  Graham Johnson0000-0002-4456-6909  Kenny
Gruchalla0000-0002-8155-1175

###### Abstract

Electrical grids are geographical and topological structures whose voltage
states are challenging to represent accurately and efficiently for visual
analysis. The current common practice is to use colored contour maps, yet
these can misrepresent the data. We examine the suitability of four
alternative visualization methods for depicting voltage data in a
geographically dense distribution system—Voronoi polygons, H3 tessellations,
S2 tessellations, and a network-weighted contour map. We find that Voronoi
tessellations and network-weighted contour maps more accurately represent the
statistical distribution of the data than regular contour maps.

###### keywords:

Visualization, Power systems, Tessellation

Voltage on 24,000+ buses represented with Voronoi polygons that accurately
depict the system voltage.

Introduction

Ensuring a reliable power supply and maintaining voltage within acceptable
limits is a regulatory obligation for electricity utility companies [2]. The
voltage on each electrical junction (bus) in the grid should be within 5% of
the expected value (0.95-1.05 per unit, or p.u.) to ensure hardware functions
properly. A single utility can have millions of buses to manage, necessitating
effective visualization. In recent years, increased renewable energy
penetration has driven power systems models to grow larger and more complex,
which further drives the need to accurately visualize these systems.
Integration of renewable sources and storage makes the spatial and topological
relationships between network nodes are especially important because the grid
is affected by both geographical and electrical phenomena such as changing
weather or electric vehicle charging. Electrical grid bus voltages present a
unique geovisualization problem because they are both spatial and topological,
and because they have a highly variable spatial density. This makes it
difficult to aggregate and visualize accurately.

Colored contour maps have become a standard practice for visualizations among
the power systems community [8]. They assign color to each pixel based on a
weighted-average of the voltage values of the $n$ closest buses. Each
neighboring bus voltage value is weighted proportionally to the inverse
distance to the pixel [18]. Since every pixel is colored, this method produces
a continuous distribution of bus values across a geographical area (see Figure
1). They are common in research [9], operations [4], planning [10, 13], and
commercial power system tools [16]. However, colored contour maps have
limitations in accurately representing power systems data [8]: the algorithm
confounds the geographical and topological proximity of bus values,
potentially leading to misinterpretation. Furthermore, aggregation in these
maps can result in artifacts and the loss of extreme values that may be
crucial for identifying critical areas. The contour map smoothing algorithm
can cover up spatial discontinuities present in the data. The violin plot at
the bottom of Figure 1 shows how the voltage data’s statistical distribution
differs from that depicted on the contour map. Finally, building the contour
maps can be computationally expensive for large data. This can limit the
visualization systems in scale and in scope of applications.

This paper asks, are there alternative visualization methods to contour maps
that can more accurately and efficiently show the state of a grid? Here, we
propose alternatives to contour maps and evaluate their effectiveness in
representing bus voltage in a dense distribution system. We create and compare
four alternative strategies to contouring for visualizing distribution grid
voltages: (a) Voronoi tessellation, (b) hexagonal tessellations, (c) four-
sided tessellations with multiple resolutions, and (d) networked contours. We
evaluate all methods on their effectiveness at accurately depicting the
statistical distribution of voltage data, preserving anomalous voltage values,
and preserving areas of high variability. We also compare the computational
efficiency of each method by using the theoretical algorithm complexity and
actual compute times. Finally, we discuss challenges and opportunities with
each method, and opportunities for further exploration.

##  1 Related Work

While colored contour maps have widespread use for power system data, further
research has explored both contour variations and some alternatives. This
includes using force-directed network layouts [19], using marker clusters in
visual analytics [11] using pseudo-geographical mosaics [14], and varying the
kernel used for contouring [18]. However, each of these methods either do not
represent the geography of the grid or are prone to the limitations of
contouring discussed above.

Perhaps the simplest method is directly representing the bus values with a
geometric primitive (a mark or a glyph) that encodes the value by size and
color [3] (see Figure 2). In an empirical research study, power system
engineers were significantly better at identifying voltage anomalies using
glyph-based representations than colored contour maps of bus values [8].
However, glyphs are prone to limitations on large networks. In low-density
regions, the white space between glyphs can be undesirable and has the
potential to be misinterpreted. In high-density regions, where the separation
between buses approaches the size of a pixel, the glyphs become over-plotted
and obscure some of the variation in the grid.

Figure 1: Voltage represented in a contour map. Values at each cell are
computed with an inverse distance weighting function considering the 100
nearest neighbors. Below is a violin plot that compares the distribution of
voltage values represented in the visualization (top) versus the distribution
of voltage values in the actual data (bottom). The range of the distribution
is reduced as values are smoothed in the contour map.

Figure 2: Voltage represented in circular glyphs. Color represents the nominal
voltage value, and glyph size represents the deviation from nominal. The
visualized voltage distribution is shifted away from 1.0 p.u.

Tessellating is an alternative method for aggregating spatial data. A
tessellation is the covering of a surface with one or more geometric shapes
without gaps or overlaps. One common technique is with Voronoi polygons, where
each polygon contains a single point [6], in our context, a bus.
Alternatively, a surface can be tessellated with a regular polygon, typically
triangles, quadrilaterals, or hexagons. For geospatial tessellation, the most
popular GitHub library for hexagons is H3 [17] and for quadrilaterals is S2
[5]. However, there is minimal research on the suitability of these geographic
tessellations to visualize topological power systems data.

Unlike previous work building on contour maps, this paper explores alternative
geographical visualizations that allow for spatial discontinuities or that
consider topological relationships between nodes, while still creating a
continuous covering. We also present the use statistical measures to evaluate
the effectiveness of each visualization.

##  2 Methods

Overview: To generate our alternative visualizations, we vary both the (a)
geographic tessellation shape, (b) the number of layers, and (c) the smoothing
algorithm. Within (a) we try three tessellation shapes: Voronoi polygons,
hexagons, and quadrilaterals. For (b), we use between 1-3 layers of tiles to
show local variability while preserving a full-area covering. Within (c), we
try a smoothing algorithm that considers both spatial and network distance.
Finally, we evaluate the effectiveness of each with statistical measures like
median, kurtosis, range, and standard deviation.

Dataset: The data used for the visualizations are from the SFO SmartDS
synthetic model [15]. This synthetic data was created for high-fidelity power
distribution system simulations. It constitutes over 24,000 buses in a few
square miles in East Oakland. The voltage data are the output of an actual
model simulation. We consider the voltage on every bus at a single snapshot in
time. Voltage is measured as the actual voltage divided by the expected
voltage, or per unit (p.u.), with 1.0 being the expected value. We filter out
outlier voltage values of 0 p.u. (about 30 buses), and we average the voltage
values on buses with multiple phases (about 3000 buses). We used the same data
as Gruchalla et al. [7] to allow for comparisons across studies. The code used
for analysis is available on GitHub [12]. Below is a description of how each
alternative visualization is created.

Voronoi Tessellation: A polygon is created around each bus such that all
points within the polygon share that point as their nearest bus. The color of
each polygon is then selected depending on the voltage value of the bus within
the polygon. The vector nature of the Voronoi polygons means that the
resolution of the visualization is limited only by number of pixels in the
final print.

H3 Tessellations: We extract the H3 tile index of each bus at H3 resolutions
10-12, ranging from about $15,000m^{2}$ down to $300m^{2}$ per cell. The
lowest resolution (10) has tiles large enough to ensure that the tessellated
surface has no gaps while still having at least one bus in every tile.
Meanwhile, the highest resolution (12) has tiles about 50X smaller, close in
size to the spacing of buses in dense areas. This means that most of these
tiles contain just 1 or 2 buses, which helps preserve anomalous values. The
three resolutions of hexes are then colored by their mean voltage and plotted
on top of one another, with the highest resolution on top.

S2 Tessellations: We extract the S2 index of each bus from resolutions 16-18,
ranging from about $25,000m^{2}$ down to about $1500m^{2}$. Like with H3, the
lowest resolution is selected to ensure the entire space is covered with no
gaps with at least one bus per tile, along with the next two higher
resolutions. The highest resolution is only 16X smaller, meaning most of these
tiles contain several buses. Note that S2 resolutions do not correspond 1:1
with H3 resolutions. Next, we compute the summary voltage statistics of mean
and standard deviation (SD) for each S2 tile. The tiles are colored by their
mean voltage, and plotted in layers, but this time with the lowest resolution
on top. Then, any tile from the lower resolutions with a SD > 0.003 is
filtered out. This exposes the higher resolution tiles beneath, providing more
detail in the areas of high variation. Unlike with H3 tiles, the S2 tiles of
higher resolutions can nest neatly inside lower resolutions. This allows for a
single layer with multiple resolutions to cover the entire study area
seamlessly.

Networked Contour Map: First, we create a network distance matrix of all nodes
in the grid using Dijkstra’s algorithm. We then recalculate the voltage at
each bus $i$ as a weighted average of itself and its $n$ network-nearest
neighbors, including itself, that are $h$ network hops away. This is
expression in equation (1) below:

$V_{i}^{{}^{\prime}}=\frac{\sum_{j=1}^{n}V_{j}w_{ij}}{\sum_{j=1}^{n}w_{ij}}$
(1)

where $w_{ij}=\frac{1}{h_{ij}+1}$. Next, we overlay a regular grid of cells on
the study area. For this dataset, the grid has about 100,000 cells, making
each cell have an area of roughly $100m^{2}$, chosen to approximate the size
of a display pixel in the final print. For each cell $x$ in the grid, the
nearest $k$ bus values at euclidean distance from bus $i$ at $d_{xi}$, are
averaged and mapped to a color. This is expressed in equation (2) below:

$V_{x}=\frac{\sum_{i=1}^{k}V_{i}^{{}^{\prime}}w_{xi}}{\sum_{i=1}^{k}w_{xi}}$
(2)

where $w_{xi}=\frac{1}{d_{xi}}$. The two key parameters here are $n$ and $k$.
For $n=1$ and $k=1$, we have essentially a pixelated Voronoi tessellation. For
$n=1$ and $k=100$, we have a regular colored contour map. Here, we want to
emphasize network distance over spatial distance, and so we chose $n=10$ and
$k=1$ for the result in the next section.

Figures: The buses are represented geographically as small points to
illustrate their spatial relationships. Thin gray lines represent the lines
that connect them to illustrate their topological relationships, all overlaid
on the colored tiles. The coloring is bounded to an area within 100 meters of
the grid lines to avoid showing color where there are no buses. The color
scale used is blue-red diverging with white in the middle for nominal values
of 1.0 per unit [p.u.]. The colors correspond to absolute voltage values to
allow for identification of anomalous values and for comparability with prior
research. A background street map is added for visual scale.

Evaluation: Visualizations methods are then compared with key statistics such
as the range, median, and kurtosis. The distribution of the actual data is
compared with the distribution of the visualized data using a kernel density
estimate [1] for a visual comparison of the fidelity of each visualization.
Computational complexity is determined both theoretically and experimentally
by recording the duration in seconds that the visualization took to create,
both before and after considering the voltage data. Durations are based on a
3.1 GHz Quad-Core Intel Core i7 processor. The list of evaluation criteria
are:

  1. 1.

Accurately depicts the statistical distribution of the real data in color
space (median and kurtosis)

  2. 2.

Preserves areas of high variability (standard deviation)

  3. 3.

Highlights anomalous values (range)

  4. 4.

Is computationally efficient to create (complexity and runtime)

##  3 Results

Figure 3: Voronoi tessellation showing the bus voltage value within each
polygon. Each bus voltage is represented in its own polygon.

Figure 4: Hexagonal multi-layered tessellation created with H3 with each
hexagon showing the mean bus voltage value of its area. High resolutions are
plotted on top of low resolutions.

The resulting figure from each method is included above (see Figures 3-6). For
each figure, the summary statistics are compiled (see Table 1 for a
quantitative comparison of each. Finally, the evaluation criteria are applied
to each of the visualizations with a pass or fail (see Table 2). Below, we
discuss each of the resulting visualizations one at a time. We acknowledge
that though we have tried to use as realistic of a dataset as possible, these
results are specific to our dataset and would likely vary for a dataset with a
different number of buses, different topology or geography, different voltage
distributions, and for visualizations with different resolutions.

In the Voronoi tessellation (see Figure 3), we see much more of the data
variability preserved than in the contour map. Network-adjacent polygons blend
together to form linear segments following the network branches, while sharp
boundaries form between spatially-close but topologically-distant branches.
These combine to help illustrate the topology of the grid. Outliers are not
smoothed but are limited to the number of pixels in their respective polygons.
The distribution mean shifts slightly left because some of the lower voltage
buses are in spatially-sparse regions, leading to larger Voronoi polygons and,
therefore, visual over representation.

In the H3 tessellation (see Figure 4), we again see more of the data
variability than in the contour map. However, some of the data’s variability
is lost, as outlying buses are aggregated. The maximum actual bus value of
1.073 is averaged with an adjacent bus of 1.020, bringing the maximum hex
voltage down to 1.047. Without the grid lines overlaid, the network structure
is not obvious. However, the regular hexagonal tiles do create a homogeneous
spatial unit for understanding the variation throughout the grid. Though the
algorithmic complexity is the same as the regular contour, the computation
time is significantly less because the resolution is low. A comparably sized
resolution would likely lead to the same runtime.

In the S2 tessellation (see Figure 5), we see subareas of high variation
represented with more cells. However, there is influential aggregation
happening, and outlier voltages are aggregated with other buses, as with H3.
The four-sided cells also do not communicate the topology of the grid. The
distribution tails are slightly reduced. The computation time is essentially
the same as that for the H3 tessellation.

In the networked contour map (see Figure 6), we can see the branches of the
grid emerge in a way similar to the Voronoi contour map, and we have smoother
gradients along the network lines. However, this has come at the cost of
aggregating some of the outliers (while preserving others). While the
statistical distribution of data is more faithful than the original contour
map, the distribution tails are still reduced and anomalous buses all but
eliminated. The computation time is larger because of the network distance
calculations that need to be done before computing the contour map.

Figure 5: Quadrilateral multi-resolution tessellation created with S2 with
each four-sided tile showing the mean bus voltage value of its area. Areas
with high voltage variance are broken down into higher resolutions.

Figure 6: Networked contour map created by first taking a weighted average of the voltage value at each bus with its 10 closest topological neighbors. After voltages are recalculated, the study area with gridded with cells. Each cell is colored with the voltage value from its closest spatial neighbor. Table 1: Summary statistics and complexity for each visualization. The "Data" column gives us the true statistics of the voltage data against which each visualization is compared. Each summary statistic corresponds to an evaluation criteria outlined in the Methods section. Metric | Data |  Contours |  Glyphs |  Voronoi | H3 | S2 |  Networked Contours
---|---|---|---|---|---|---|---
Median (p.u.) | 1.0188 | 1.0183 | 1.0213 | 1.0173 | 1.0183 | 1.0189 | 1.0179
Kurtosis | 16.632 | 4.296 | 53.810 | 13.520 | 12.655 | 10.585 | 9.203
Standard Deviation (p.u.) | 0.0057 | 0.0048 | 0.0062 | 0.0059 | 0.0049 | 0.0049 | 0.0054
Minimum (p.u.) | 0.9459 | 0.9564 | 0.9459 | 0.9459 | 0.9460 | 0.9689 | 0.9561
Maximum (p.u.) | 1.0734 | 1.0486 | 1.0734 | 1.0734 | 1.0468 | 1.0645 | 1.0502
Pre-processing time (s) | NA | 350 | 0 | 150 | <10 | <10 | 430
Rendering time (s) | NA | <10 | <10 | <10 | 40 | 30 | 30
Pre-processing complexity | NA | $O(n\,m)$ | $O(1)$ | $O(n\log{}n)$ | $O(nm)$ | $O(nm)$ | $O(n\,m)+O(n\log{}n)$
Rendering complexity | NA | $O(m)$ | $O(n)$ | $O(n)$ | $O(m)$ | $O(m)$ | $O(m)$
Table 2: Evaluation criteria results for each visualization. A visualization receives a green "✓" if the corresponding statistics are close in value to the true data, or if the rendering is relatively efficient. Otherwise, it is given a red "$x$". Metric | C | G | V | H3 | S2 | N
---|---|---|---|---|---|---
Accurate distribution | ✓ | $x$ | ✓ | ✓ | ✓ | ✓
Preserves areas of high variance | $x$ | ✓ | ✓ | $x$ | $x$ | ✓
Preserves tails / outliers | $x$ | ✓ | ✓ | $x$ | $x$ | $x$
Computationally efficient | $x$ | ✓ | ✓ | $x$ | $x$ | $x$

##  4 Discussion

Given the four criteria for evaluating the visualizations, the top performing
visualization is the Voronoi tessellation for this dataset. Anomalies are
depicted because each bus has its own polygon. The statistical distribution of
the visualized versus actual data does shift the mean slightly but does not
bias against the tails, which allows for the identification of areas with high
variability. Finally, the Voronoi structure is static; once calculated we can
efficiently render new bus values onto the polygons. The potential limitation,
however, is for small-scale maps, where multiple polygons may form a single
pixel or small pixels that are not visually discernible. We would then be
prone to the same over-plotting limitations that apply to glyphs, and a
certain degree of smoothing would become necessary. Nonetheless, the Voronoi
tessellation is able to both faithfully represent bus values while providing a
full contour-like covering for quick assessment of network trends, making it a
strong alternative to contour maps and glyph maps.

The H3 and S2 tessellations offer promising alternatives to contour maps for
their improved representation of the statistical data distribution and regular
spatial units, but they are prone to the same issues of aggregating outlying
values and failing to relay the structure of the network. And in our judgment,
the S2 mapping is aesthetically noisy. Further research could try selecting
the highest deviation value for each cell in order to preserve outliers.

Finally, the network-weighted contour map builds on the regular contour map by
allowing for discontinuities that convey the network structure, however is
prone to smoothing some of the larger outliers based on the choice of n and k,
which might be inevitable in small-scale maps. Further research should compare
the networked contour map side by side with the traditional contour map for
given values of $k$ and $n$ to control for effective size of the kernel.

As mentioned, these results hold true for this dataset of about 24,000 buses
in an urban area with a relatively high but variable density. Lower density
regions or smaller grids, where there are many pixels per bus, can afford to
depict each bus with methods such as glyphs or Voronoi polygons, which
preserves the full variability of the original data. However, larger grids
with 100,000+ buses may have more buses than pixels, and are forced to
aggregate the data in some way. Here, we can look to tiling or to contour
maps. Using a small kernel or high resolution tiles helps preserve the
variability of the original dataset. Using tiles or networked contour maps
allows for discrete boundaries in the visualization, helping to preserving the
spatial discontinuities in the data. Future research should evaluate these
visualization methods on a large dataset that necessitates aggregation.

Additionally, future research should also explore adding in time-series data
to the network through animations or other methods. Further topological
information could be conveyed by overlaying multiple types of visualizations
or by designing linked views. Finally, follow up research should conduct a
human factors study by surveying the preferences of power systems users or by
testing human readability of each of these visualizations. We hope to inspire
further investigation into the proposed methods and new alternative methods to
better display power systems data in an intuitive, efficient, and beautiful
way.

##  5 Conclusion

In summary, we took on the challenge of visualizing the current state of an
urban electrical grid with over 24,000 buses. Instead of the standard method
of colored contour maps, we tried alternative methods that vary the tile shape
and size, add multiple layers, and consider the network topology. We evaluated
the effectiveness of each of these visualizations by their ability to
accurately depict the data with the same statistical distribution and to do so
with computational efficiency. We found that a Voronoi tessellation works well
for this dataset that has more pixels than buses, and that a networked contour
map holds promise this and larger datasets. We look forward to further
research that considers time series, linked views, and a human factors study.

###### Acknowledgements.

The authors would like to thank Kristi Potter for her contribution to this
research. This work was authored by the National Renewable Energy Laboratory,
managed and operated by Alliance for Sustainable Energy, LLC for the U.S.
Department of Energy (DOE) under Contract No. DE-AC36-08G028308. Funding
provided by the U.S. Department of Energy Office of Energy Efficiency and
Renewable Energy. The views expressed in the article do not necessarily
represent the views of the DOE or the U.S. Government. The U.S. Government
retains and the publisher, by accepting the article for publication,
acknowledges that the U.S. Government retains a nonexclusive, paid-up,
irrevocable, worldwide license to publish or reproduce the published form of
this work, or allow others to do so, for U.S. Government purposes.

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