# Weighted Cylindric Partitions

Walter Bridges  [W.B.] University of Cologne, Department of Mathematics and
Computer Science, Weyertal 86-90, 50931 Cologne, Germany  wbridges@uni-
koeln.de and  Ali K. Uncu  [A.K.U.] Johann Radon Institute for Computational
and Applied Mathematics, Austrian Academy of Science, Altenbergerstraße 69,
A-4040 Linz, Austria<EMAIL_ADDRESS>[A.K.U.] University of Bath,
Faculty of Science, Department of Computer Science, Bath, BA2 7AY, UK
<EMAIL_ADDRESS>
###### Abstract.

Recently Corteel and Welsh outlined a technique for finding new sum-product
identities by using functional relations between generating functions for
cylindric partitions and a theorem of Borodin. Here, we extend this framework
to include very general product-sides coming from work of Han and Xiong. In
doing so, we are led to consider structures such as weighted cylindric
partitions, symmetric cylindric partitions and weighted skew double shifted
plane partitions. We prove some new identities and obtain new proofs of known
identities, including the Göllnitz–Gordon and Little Göllnitz identities as
well as some beautiful Schmidt-type identities of Andrews and Paule.

###### Key words and phrases:

Cylindric partitions, skew double shifted plane partitions, partition
diamonds, partition identities, weighted partition identities

###### 2010 Mathematics Subject Classification:

Primary 05A15; Secondary 05A17, 11B65, 11P81, 11P84

Research of the first author is partially supported by the DFG (Projektnummer
281071066 TRR 191). Research of the second author is partly supported by EPSRC
grant number EP/T015713/1 and partly by FWF grant P-34501N

##  1\. Introduction

Sum-product identities lie at the heart of the theory of integer partitions
and $q$-series. For example, the two Rogers–Ramanujan identities state that
for $\varepsilon\in\\{0,1\\}$, we have

(1.1) $\sum_{n\geq 0}\frac{q^{n^{2}+\varepsilon
n}}{(q;q)_{n}}=\frac{1}{(q^{1+\varepsilon},q^{4-\varepsilon};q^{5})_{\infty}},$

where throughout the article we use the $q$-Pochhammer notation,
$(a;q)_{n}:=\prod_{j=1}^{n}(1-aq^{j})$ for $n\geq 1$ with $(a;q)_{0}:=1$ and
$(q;q)_{n}^{-1}=0$ for $n\leq-1$. The Rogers–Ramanujan identities lie deep
enough within mathematics to have quite a variety of proofs—combinatorial
[10], so-called “motivated proofs” (via recurrences) [3], and via affine Lie
algebras [15], to name just a few. We refer the reader to Sills’ recent book
[18] for a thorough history. The Rogers–Ramanujan identities can be
interpreted in terms of integer partitions, and such identities in which a
$q$-hypergeometric series equals a product as simple as the right-hand side of
(1.1) are rare. Thus, in addition to discovering new identities, it is
worthwhile to provide new connections to combinatorial structures beyond
integer partitions.

Corteel and Welsh [8] recently showed how cylindric partitions can be used in
the discovery of new sum-product identities. (We give precise definitions of
these objects and some generalizations in Section 2.) Cylindric partition
generating functions for any profile were shown by Borodin [6] to equal
infinite products, and “sum-sides” were then found by Corteel and Welsh to
satisfy $q$-difference equations, which they showed occur naturally for
cylindric partitions. This led to alternative proofs of all four of
Andrews–Schilling–Warnnar’s $A_{2}$ Rogers–Ramanujan identities [4], as well
as a proof of a new fifth identity which was conjectured in [9]. One such
identity Corteel–Welsh proved in [8, Theorem 1.1] is as follows:

(1.2) $\sum_{n_{1},n_{2}\geq
0}\frac{q^{n_{1}^{2}+n_{2}^{2}-n_{1}n_{2}+n_{1}+n_{2}}}{(q;q)_{n_{1}}}\begin{bmatrix}2n_{1}\\\
n_{2}\end{bmatrix}_{q}=\frac{1}{(q^{2},q^{3},q^{3},q^{4},q^{4},q^{5};q^{7})_{\infty}}.$

Here the Gaussian polynomials are defined as

$\begin{bmatrix}n\\\
m\end{bmatrix}_{q}=\left\\{\begin{array}[]{cl}\displaystyle\frac{(q;q)_{n}}{(q;q)_{m}(q;q)_{n-m}},&\text{if
}n\geq m\geq 0,\\\\[-6.45831pt] \\\ 0,&\text{otherwise.}\end{array}\right.$

Recent work of Corteel, Dousse and the second author [7], as well as Warnaar
[20], uses Corteel–Welsh’s machinery to prove and conjecture many other sum-
product identities.

In the present article, we extend the search for sum-product identities to
certain generalizations of cylindric partitions, recently proved to also have
infinite product generating functions by Han and Xiong [13]. Essentially, one
can write any product $(q^{b_{1}},\dots,q^{b_{r}};q^{b_{r}})^{-1}_{\infty}$ as
a generating function for some weighted cylindric partitions (see Remark
2.11). By then extending Corteel and Welsh’s machinery to this more general
setting, we were able to prove the following identities.

######  Theorem 1.1.

For $z\in\mathbb{C}$ and $|q|<1$, we have

$\sum_{n\geq
0}(-1)^{n}q^{4n^{2}}\frac{(q^{2},-q^{4};q^{4})_{n}}{(q^{4};q^{4})_{2n}}\left(1-\frac{q^{4n+1}z}{1+q^{4n+2}}\right)z^{2n}=(zq,-zq^{3};q^{4})_{\infty}.$

We offer a combinatorial proof of a specialization of Theorem 1.1 in Section
4. Next, we have the following family of four identities.

######  Theorem 1.2.

For $|q|<1$, we have

(1.3) $\displaystyle\sum_{n,m\geq
0}(-1)^{m}q^{3\binom{n+1}{2}-3m(m+1)}\frac{(-q,-q^{5};q^{6})_{m}}{(q^{6};q^{6})_{m}(q^{3};q^{3})_{n-2m}}=\frac{(q^{4},q^{8};q^{12})_{\infty}}{(q^{6};q^{12})_{\infty}},$
(1.4) $\displaystyle\sum_{n,m\geq 0}(-1)^{m+1}q^{3{n-1\choose
2}+2n-3m(m+1)-1}\frac{(-q,-q^{5};q^{6})_{m}}{(q^{6};q^{6})_{m}(q^{3};q^{3})_{n-2m}}(1-q^{3n+1}+q^{3n-6m})=\frac{(q;q^{6})_{\infty}(q^{10};q^{12})_{\infty}}{(q^{5};q^{6})_{\infty}},$
(1.5) $\displaystyle\sum_{n,m\geq 0}(-1)^{m+1}q^{3{n+1\choose
2}-3m(m+1)-1}\frac{(-q,-q^{5};q^{6})_{m}}{(q^{6};q^{6})_{m}(q^{3};q^{3})_{n-2m}}(1-q^{3n+1}+q^{3n-6m})=\frac{(q^{2},q^{10};q^{12})_{\infty}}{(q^{6};q^{12})_{\infty}},$
(1.6) $\displaystyle\sum_{n,m\geq 0}(-1)^{m+1}q^{3{n+1\choose
2}-2n-3m(m+1)-1}\frac{(-q,-q^{5};q^{6})_{m}}{(q^{6};q^{6})_{m}(q^{3};q^{3})_{n-2m}}$
$\displaystyle\hskip
113.81102pt\times(1+q^{3n-12m-3}(1+q^{1+6m})(q^{3n}-q^{6m}(1+q^{3}))=\frac{(q^{2},q^{5},q^{11};q^{12})_{\infty}}{(q;q^{6})_{\infty}}.$

We further use our machinery to offer new proofs of the Little Göllnitz and
Göllnitz–Gordon identities [2, Theorem 7.11], and [12].

######  Theorem 1.3 (Little Göllnitz and Göllnitz–Gordon Identities).

We have

(1.7) $\displaystyle\sum_{n\geq
0}\frac{\left(-q;q^{2}\right)_{n}}{\left(q^{2};q^{2}\right)_{n}}q^{n^{2}+2n}=\frac{1}{\left(q^{3},q^{4},q^{5};q^{8}\right)_{\infty}},$
(1.8) $\displaystyle\sum_{n\geq
0}\frac{\left(-q;q^{2}\right)_{n}}{\left(q^{2};q^{2}\right)_{n}}q^{n^{2}+n}=\frac{1}{\left(q^{3};q^{4}\right)_{\infty}(q^{2};q^{8})_{\infty}},$
(1.9) $\displaystyle\sum_{n\geq
0}\frac{\left(-q;q^{2}\right)_{n}}{\left(q^{2};q^{2}\right)_{n}}q^{n^{2}}=\frac{1}{\left(q,q^{4},q^{7};q^{8}\right)_{\infty}},$
(1.10) $\displaystyle\sum_{n\geq
0}\frac{\left(-q^{-1};q^{2}\right)_{n}}{\left(q^{2};q^{2}\right)_{n}}q^{n^{2}+n}=\frac{1}{\left(q;q^{4}\right)_{\infty}\left(q^{6};q^{8}\right)_{\infty}}.$

As a final application, we give refinements of the following Schmidt-type
partition identities appearing in recent work of Andrews and Paule [5].
Identity (1.11) was first conjectured in [17] by Schmidt, who himself
subsequently gave a proof. The first two identities also independently
appeared in work by the second author [19] in the context of certain weighted
partition identities. Interested readers are invited to look into these works.

A partition $\lambda$ of an integer $n$ is a sequence of positive integers
whose parts satisfy

$\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{\ell}\qquad\text{and}\qquad|\lambda|:=\sum_{j=1}^{\ell}\lambda_{j}=n.$

We call such $\lambda$ a partition of size $n$.

Let $\mathcal{P}$ (resp. $\mathcal{D}$) denote the set of partitions (resp.
partitions with distinct parts—when the inequalities between the parts are all
strict). Also let $\diamondsuit$ denote the set of partition diamonds, i.e.
finite sequences of integers $\lambda_{1},\lambda_{2},\dots,$ where

$\lambda_{1}\geq\begin{matrix}\lambda_{2}\\\\[-8.61108pt] \\\
\lambda_{3}\end{matrix}\geq\lambda_{4}\geq\begin{matrix}\lambda_{5}\\\\[-8.61108pt]
\\\
\lambda_{6}\end{matrix}\geq\lambda_{7}\geq\begin{matrix}\lambda_{8}\\\\[-8.61108pt]
\\\
\lambda_{9}\end{matrix}\geq\lambda_{10}\geq\begin{matrix}\lambda_{11}\\\\[-8.61108pt]
\\\ \lambda_{12}\end{matrix}\geq\lambda_{13}\geq\dots.$

######  Theorem 1.4 (Theorems 1, 2 and 4 of [5], respectively).

We have

(1.11)
$\displaystyle\sum_{\lambda\in\mathcal{D}}q^{\lambda_{1}+\lambda_{3}+\lambda_{5}+\dots}=\frac{1}{(q;q)_{\infty}},$
(1.12)
$\displaystyle\sum_{\lambda\in\mathcal{P}}q^{\lambda_{1}+\lambda_{3}+\lambda_{5}+\dots}=\frac{1}{(q;q)_{\infty}^{2}},$
(1.13)
$\displaystyle\sum_{\lambda\in\diamondsuit}q^{\lambda_{1}+\lambda_{4}+\lambda_{7}+\dots}=\frac{(-q;q)_{\infty}}{(q;q)^{3}_{\infty}}.$

We prove the following two-variable versions.

######  Theorem 1.5.

We have

(1.14)
$\displaystyle\sum_{\lambda\in\mathcal{D}}z^{\lambda_{1}}q^{\lambda_{1}+\lambda_{3}+\lambda_{5}+\dots}=\sum_{n\geq
0}\frac{z^{2n}q^{n(n+1)}}{(zq;q)_{n}(zq;q)_{n+1}},$ (1.15)
$\displaystyle\sum_{\lambda\in\mathcal{D}}z^{\lambda_{1}}q^{\lambda_{2}+\lambda_{4}+\lambda_{6}+\dots}=1+\sum_{n\geq
1}\frac{z^{2n-1}q^{n(n-1)}}{(z,zq;q)_{n}},$ (1.16)
$\displaystyle\sum_{\lambda\in\mathcal{P}}z^{\lambda_{1}}q^{\lambda_{1}+\lambda_{3}+\lambda_{5}+\dots}=\frac{1}{(zq;q)_{\infty}^{2}},$
(1.17)
$\displaystyle\sum_{\lambda\in\mathcal{P}}z^{\lambda_{1}}q^{\lambda_{2}+\lambda_{4}+\lambda_{6}+\dots}=\frac{1}{(1-z)(zq;q)_{\infty}^{2}},$
(1.18)
$\displaystyle\sum_{\lambda\in\diamondsuit}z^{\lambda_{1}}q^{\lambda_{1}+\lambda_{4}+\lambda_{7}+\dots}=\frac{(-zq;q)_{\infty}}{(zq;q)^{3}_{\infty}}.$

It is easy to observe that, as $z\mapsto 1$, (1.16) and (1.18) imply (1.12)
and (1.13), respectively.

Identities (1.14)–(1.18) have natural combinatorial interpretations; for
example, (1.14) implies the following.

######  Corollary 1.6.

The number of partitions into distinct parts with largest part $m$ and
$n=\lambda_{1}+\lambda_{3}+\lambda_{5}+\dots$ equals the number of partitions
of $n$ into unrestricted parts with largest hook length $m$.

Note that (1.11) follows from Corollary 1.6. Similarly, setting $z\mapsto zq$
in (1.15), one has the following combinatorial interpretation.

######  Corollary 1.7.

For $n\geq 1$, the number of partitions into distinct parts with largest part
$m$ and $n=\lambda_{1}+(\lambda_{2}+\lambda_{4}+\lambda_{6}+\dots)$ equals the
number of partitions of $n+1$ into unrestricted parts greater than 1 with
largest hook length $m+1$.

In Section 2, we define our objects of study—weighted cylindric partitions,
symmetric cylindric partitions, and skew double shifted plane partitions—and
state the product generating functions for these that follow from Han and
Xiong’s work in [13]. In Section 3, we record our Corteel–Welsh-type
recurrences for two variable generating functions. In Section 4, we use these
recurrences to prove Theorems 1.1, 1.2 and Theorem 1.3; Theorem 1.2 is proved
with the aid of Mathematica. In Section 5, we prove Theorem 1.5. We conclude
in Section 6 by mentioning a few open problems and avenues for further work.
Appendix A contains a short proof of the modularity of the products in
Proposition 2.2. In Appendix B, we briefly mention the new computer algebra
implementations developed in connection to this work.

## Acknowledgements

We would like to thank George E. Andrews, Jehanne Dousse, Guoniu Han, Ralf
Hemmecke, and Peter Paule for their interest and encouragement. We would like
to particularly thank Christian Krattenthaler and Wadim Zudilin for the
discussion and their valuable feedback.

The first author is partially supported by the SFB/TRR 191 “Symplectic
Structures in Geometry, Algebra and Dynamics”, funded by the DFG
(Projektnummer 281071066 TRR 191). Research of the second author is partly
supported by EPSRC grant number EP/T015713/1 and partly by FWF grant P-34501N.

##  2\. Definitions and product sides

Cylindric partitions were introduced by Gessel and Krattenthaler [11] as a
type of repeating grid pattern of positive integers that are weakly decreasing
along rows and columns; see Figure 1. Here, we will define them using the
diagonal integer partitions. For two partitions $\lambda$ and $\mu,$ we write
$\lambda\succeq\mu$ if

(2.1) $\lambda_{1}\geq\mu_{1}\geq\lambda_{2}\geq\mu_{2}\geq\dots.$

######  Definition 2.1.

A cylindric partition of width $h$ with profile
$\delta=(\delta_{1},\dots,\delta_{h})\in\\{\pm 1\\}^{h}$ is an $(h+1)$-tuple
of integer partitions $(\lambda^{0},\dots,\lambda^{h})$ such that
$\lambda^{0}=\lambda^{h}$ and $\lambda^{j-1}\succeq\lambda^{j}$ (resp.
$\lambda^{j-1}\preceq\lambda^{j}$) if $\delta_{j}=-1$ (resp. if
$\delta_{j}=1$). We define the size of $\lambda$ as

$|\lambda|:=\sum_{j=0}^{h-1}|\lambda^{j}|.$

We define the rank of a cylindric partition $\lambda$ to be the number of
$(-1)$’s in the profile. We write $\mathcal{CP}_{\delta}$ for the set of
cylindric partitions of profile $\delta$, including the “empty cylindric
partition” of profile $\delta.$

534421435223211 Figure 1. A cylindric partition of width $9,$ profile
$(-1,1,-1,1,1,-1,1,-1)$, rank 4 and size 42. Here $\lambda^{0}$ and
$\lambda^{h}$ are highlighted. The purple line indicates the profile, from
bottom-left to top-right.

Graphically, a profile $\delta$ starts on the top left corner of the starting
diagonal $\lambda^{0}$ and ends on the top left corner of the last diagonal
$\lambda^{h}$.

Borodin proved that the generating function for cylindric partitions of any
profile is always an infinite product. We use the formulation in [13].

######  Proposition 2.2 ([6], Proposition on p. 8).

Let $\delta=(\delta_{1},\dots,\delta_{h})$ be a profile and define the
multiset

$W_{3}(\delta):=\\{h\\}\cup\\{j-i:1\leq i<j\leq
h,\delta_{i}>\delta_{j}\\}\cup\\{h-(j-i):1\leq i<j\leq
h,\delta_{i}<\delta_{j}\\}.$

Then

(2.2)
$\mathrm{CP}_{\delta}(q):=\sum_{\lambda\in\mathcal{CP}_{\delta}}q^{|\lambda|}=\prod_{k\in
W_{3}(\delta)}\frac{1}{(q^{k};q^{h})_{\infty}}.$

######  Remark 2.3.

This product is always symmetric—that is, every $k<h$ appears as many times in
$W_{3}(\delta)$ as $h-k$. So, up to a rational power of $q$,
$\mathrm{CP}_{\delta}(q)$ is a modular form. To the best of our knowledge a
proof of this non-trivial fact was lacking in the literature, so we provide
one in Appendix A.

To look for “sum-side” companions to the above products, Corteel and Welsh
introduced the two variable generating function,
$\mathrm{CP}_{\delta}(z;q):=\sum_{\lambda\in\mathcal{CP}_{\delta}}z^{\mathrm{max}(\lambda)}q^{|\lambda|},$
where $\mathrm{max}(\lambda)$ denotes the size of the largest part among all
partitions in $\lambda$. They proved general systems of recurrences for these
functions and found sum-generating functions that solved all cases for width 7
and rank 3. The identity (1.2) in the introduction is
$(q;q)_{\infty}\mathrm{CP}_{\delta}(1;q)$ with $\delta=(-1,-1,-1,1,1,1,1)$.
Identities for all rank 3, width 8 cases were subsequently proved by Corteel,
Dousse and the second author in [7]. Warnaar [20] further conjectured
identities for all rank 3, width $h$ profiles with $3\nmid h$.

To allow for products much more general than those in (2.2), we consider one
natural restriction and one generalization of cylindric partitions defined by
Han and Xiong in [13].

######  Definition 2.4.

A symmetric cylindric partition $\lambda$ is a cylindric partition of the form
$(\lambda^{h},\dots,\lambda^{1},\lambda^{0},\lambda^{1},\dots,\lambda^{h})$
with profile
$\delta=(-\delta_{h},\dots,-\delta_{1},\delta_{1},\dots,\delta_{h}).$ We
define the size as

$|\lambda|:=|\lambda^{0}|+|\lambda^{h}|+2\sum_{j=1}^{h-1}|\lambda^{j}|.$

Let $\mathcal{SCP}_{\delta}$ be the set of symmetric cylindric partitions of
profile $\delta$, including the “empty partition”.

######  Remark 2.5.

Han and Xiong defined the size of symmetric cylindric partitions instead as
$|\lambda|:=|\lambda^{0}|+2\sum_{j=1}^{h}|\lambda^{j}|$ and stated their
product formula for this notion of size. But we will consider general weighted
sizes below that include both of these.

4553152243211 Figure 2. A symmetric cylindric partition of width $6,$ profile
$(-1,1,1,-1,-1,1)$ and size 38. The green highlighted partition indicates the
line of symmetry.

######  Remark 2.6.

When viewing a symmetric cylindric partition on the face of a cylinder, one
notices that there are two axes of symmetry, namely at $\lambda^{0}$ and at
$\lambda^{h}$. Thus, it is easy to see that
$\mathrm{SCP}_{\delta}(q)=\mathrm{SCP}_{-\mathrm{rev}(\delta)}(q),$ where
$\mathrm{rev}(\delta)$ is the reverse of $\delta.$

If the restriction $\lambda^{0}=\lambda^{h}$ which creates a repeating pattern
in a cylindric partition is dropped, then the resulting structures were called
skew double shifted plane partitions by Han and Xiong [13].

######  Definition 2.7.

A skew double shifted plane partition (DSPP) of width $h$ with profile
$\delta=(\delta_{1},\dots,\delta_{h})\in\\{\pm 1\\}^{h}$ is an $(h+1)$-tuple
of integer partitions $(\lambda^{0},\dots,\lambda^{h})$ such that
$\lambda^{j-1}\succeq\lambda^{j}$ (resp. $\lambda^{j-1}\preceq\lambda^{j}$) if
$\delta_{j}=-1$ (resp. if $\delta_{j}=1$). We define the size as

$|\lambda|:=\sum_{j=0}^{h}|\lambda^{j}|.$

Let $\mathcal{DSPP}_{\delta}$ be the set of skew double shifted plane
partitions of profile $\delta$, including the “empty partition”.

7764465532143121 Figure 3. A DSPP of width $6,$ profile $(-1,-1,1,-1,-1,1)$
and size 61.

Han and Xiong [13] proved a general lemma in the theory of symmetric functions
and used it to prove product formulas analogous to Borodin’s for both
symmetric cylindric partitions and DSPPs. It follows directly from their work
that generating functions are products even with a general weighted size. For
a DSPP $\lambda$ of width $h$ and a vector ${\bf
a}=(a_{0},\dots,a_{h})\in\mathbb{R}_{\geq 0}^{h+1}$, we define

$|\lambda|_{{\bf a}}:=\sum_{j=0}^{h}a_{j}|\lambda^{j}|.$

Similarly, for a cylindric partition $\lambda$ of width $h$ and a vector ${\bf
a}=(a_{0},\dots,a_{h-1})\in\mathbb{R}_{\geq 0}^{h}$, we define

$|\lambda|_{{\bf a}}:=\sum_{j=0}^{h-1}a_{j}|\lambda^{j}|.$

Throughout, let $A_{k}:=\sum_{j=0}^{k-1}a_{j}.$ We say that ${\bf a}$ is the
standard weight when ${\bf a}=(1,1,\dots,1)$.

######  Proposition 2.8.

Let $\delta=(\delta_{1},\dots,\delta_{h})$ be a profile of width $h$, and let
${\bf a}=(a_{0},\dots,a_{h-1})\in\mathbb{R}_{\geq 0}^{h}$. Define the set

$\displaystyle W^{{\bf a}}_{3}(\delta)$
$\displaystyle:=\\{A_{h}\\}\cup\\{A_{j}-A_{i}:1\leq i<j\leq
h,\delta_{i}<\delta_{j}\\}\cup\\{A_{h}-(A_{j}-A_{i}):1\leq i<j\leq
h,\delta_{i}>\delta_{j}\\}.$

If $0\notin W_{3}^{{\bf a}}(\delta)$, then

(2.3) $\mathrm{CP}_{\delta}^{{\bf
a}}(q):=\sum_{\lambda\in\mathcal{CP}_{\delta}}q^{|\lambda|_{{\bf
a}}}=\prod_{k\in W^{{\bf
a}}_{3}(\delta)}\frac{1}{(q^{k};q^{A_{h}})_{\infty}}.$

The proof of Proposition 2.8 directly follows from [13, Theorem 3.1] by
setting $u_{i}:=q^{a_{i}}.$ In the same spirit, we get the following.

######  Proposition 2.9.

Let $\delta=(\delta_{1},\dots,\delta_{h})$ be a profile of width $h$, and let
${\bf a}=(a_{0},\dots,a_{h})\in\mathbb{R}_{\geq 0}^{h+1}$. Define the sets

$\displaystyle W^{{\bf a}}_{1}(\delta):=$
$\displaystyle\\{A_{h+1}\\}\cup\\{A_{i}:\delta_{i}=-1\\}\cup\\{A_{h+1}-A_{i}:\delta_{i}=1\\},$
$\displaystyle W^{{\bf a}}_{2}(\delta):=$ $\displaystyle\\{A_{i}+A_{j}:1\leq
i<j\leq h,\ \delta_{i}=\delta_{j}=-1\\}$
$\displaystyle\cup\\{2A_{h+1}-A_{i}-A_{j}:1\leq i<j\leq h,\
\delta_{i}=\delta_{j}=1\\}$ $\displaystyle\cup\\{2A_{h+1}-(A_{j}-A_{i}):1\leq
i<j\leq h,\ \delta_{i}<\delta_{j}\\}$ $\displaystyle\cup\\{A_{j}-A_{i}:1\leq
i<j\leq h,\ \delta_{i}>\delta_{j}\\}.$

If $0\notin W_{1}^{{\bf a}}(\delta)\cup W_{2}^{{\bf a}}(\delta)$, then

(2.4) $\mathrm{DSPP}_{\delta}^{{\bf
a}}(q):=\sum_{\lambda\in\mathcal{DSPP}_{\delta}}q^{|\lambda|_{{\bf
a}}}=\prod_{\begin{subarray}{c}k\in W^{{\bf a}}_{1}(\delta)\\\ \ell\in W^{{\bf
a}}_{2}(\delta)\end{subarray}}\frac{1}{(q^{k};q^{A_{h+1}})_{\infty}(q^{\ell};q^{2A_{h+1}})_{\infty}}.$

Note that for the width 0 and 1 profiles these generating functions with
standard weights equals the partition function

$\mathrm{DSPP}_{\emptyset}^{(1)}(q)=\mathrm{DSPP}_{(1)}^{(1,1)}(q)=\frac{1}{(q;q)_{\infty}}.$

This corresponence can easily be seen through the diagrams of the counted
objects.

######  Remark 2.10.

In contrast to the generating functions for cylindric partitions, neither of
the products (2.3) or (2.4) are symmetric.

######  Remark 2.11.

We may now write any product of the following form as a weighted cylindric
partition generating function; precisely, for any real numbers $0<b_{1}\leq
b_{2}\leq\dots\leq b_{r+1}$, we have

$\mathrm{CP}_{(-1,-1,\dots,-1,1)}^{(b_{1},b_{2}-b_{1},\dots,b_{r+1}-b_{r})}(q)=\frac{1}{(q^{b_{1}},q^{b_{2}},\dots,q^{b_{r}},q^{b_{r+1}};q^{b_{r+1}})_{\infty}}.$

For example, this approach provides a combinatorial interpretation for a
specilization of the reciprocal of the Ramanujan theta function [18, (1.59).
p.37]

$\sum_{n=-\infty}^{\infty}a^{n+1\choose 2}b^{n\choose
2}=(-a,-b,ab;ab)_{\infty},$

as a generating function for the number of weighted cylindric partitions with
profile $(1,1,-1)$. Precisely, let $b_{2}>b_{1}>0$ then

$\mathrm{CP}_{(-1,-1,1)}^{(b_{1}b_{2}-b_{1},b_{1})}(q)=\frac{1}{(q^{b_{1}},q^{b_{2}},q^{b_{1}+b_{2}};q^{b_{1}+b_{2}})_{\infty}}=\left(\sum_{n=-\infty}^{\infty}(-1)^{n}q^{b_{1}{n+1\choose
2}+b_{2}{n\choose 2}}\right)^{-1}.$

The profile $(-1,-1,\dots,-1,-1,1)$ of width $h$ has rank $h-1$ and does not
lead to interesting recurrences and sum-sides in general; however, there are
often multiple combinations of weights and profiles with higher rank that
yield the same product. In general, by picking different profiles and weights
we can discover connections between weighted counts of different classes of
cylindric partitions. To demonstrate, from Proposition 2.8, we can easily
confirm that

(2.5) $\displaystyle\mathrm{CP}_{(-1,-1,1)}^{(1,3,1)}(q)$
$\displaystyle=\mathrm{CP}_{(-1,-1,1,1)}^{(1,3,1,5)}(q)=\mathrm{CP}_{(-1,1,-1,1,1)}^{(1,4,4,1,5)}(q)=\mathrm{CP}_{(-1,1,-1,1,-1)}^{(5,4,1,1,4)}(q)=\frac{(q^{2},q^{3};q^{5})}{(q;q)_{\infty}},$
(2.6) $\displaystyle\mathrm{CP}_{(-1,1,1)}^{(2,2,1)}(q)$
$\displaystyle=\mathrm{CP}_{(-1,1,-1,1)}^{(2,2,3,3)}(q)=\mathrm{CP}_{(-1,1,-1,1,1)}^{(2,3,3,2,5)}(q)=\mathrm{CP}_{(-1,1,-1,1,-1)}^{(5,3,2,2,3)}(q)=\frac{(q,q^{4};q^{5})}{(q;q)_{\infty}}.$

Similar to finding weighted correspondences between cylindric partitions with
different profiles, we can also find weighted correspondences between skew
double shifted plane partitions with different profiles. Furthermore, we can
find weighted relations between CPs and DSPPs.

For example, Propositions 2.8 and 2.9 is enough to see that

(2.7)
$\mathrm{CP}^{(1,1,0,0)}_{(-1,1,1,1)}(q)=\mathrm{DSPP}^{(0,1,0)}_{(1,-1)}(q)=\frac{1}{(q;q)^{3}_{\infty}(q;q^{2})_{\infty}}.$

To give an explicit example, we note that the coefficient of the $q^{3}$ term
of the $q$-series of (2.7) is 36. We explicitly present these weighted CPs and
DSPPs in Figure 4, where the yellow diagonals for the cylindric partitions
highlight the same diagonal’s repetition due to cylindricity (only one of the
diagonals contribute to the total size) and the gray diagonals are weighted
with 0 (the numbers on these diagonals do not contribute to the total size of
the object). The total value in the boxes with the white backgrounds (each
counted with weight 1 in this case) makes the size of these objects. There are
some explicitly stated 0 parts. On top of that every empty box in these
diagrams can be thought to have 0s inside and they add nothing to the total
size. Finally, we compress multiple objects with braces of a range of possible
numbers to avoid repetition the presentation.

CPs: | 33 0,…,3 0,…,3 | 22 1 or 2 100 | 22 1 or 2 1 1 or 2 0 or 1
---|---|---|---
| 16 | 2 | 8
CPs: | 2121 1 or 2 1 or 2 | 11110 0 or 1 1111 | 11111 0 or 1 0 or 1 1111
| 4 | 2 | 4
DSPPs: | 3 0,…,3 0,…,3 | 2 1 or 2 1 or 2 1 0 or 1 0 or 1 | 1111111 0 or 1 0 or 1
| 16 | 16 | 4
Figure 4. Total size 3 CPs and DSPPs counted with the weights of (2.7). The
number under each diagram represents the total number of distinct objects
these diagrams represent.

As in [13], symmetric cylindric partitions can be viewed as weighted
DSPPs—that is, for $\delta=(\delta_{1},\dots,\delta_{h})$ we have

(2.8)
$\mathrm{SCP}_{(-\mathrm{rev}(\delta),\delta)}(q)=\mathrm{DSPP}_{\delta}^{(1,2,2,\dots,2,1)}(q).$

Consider those cylindric partitions with symmetric profile
$(-\mathrm{rev}(\delta),\delta)$ which are themselves not symmetric, i.e. the
set
$\mathrm{CP}_{(-\mathrm{rev}(\delta),\delta)}\setminus\mathrm{SCP}_{(-\mathrm{rev}(\delta),\delta)}$.
One can show that

$\displaystyle
W_{3}((-\delta,\delta))=W_{1}^{(1,2,\dots,2,1)}(\delta)\cup\frac{1}{2}W_{2}^{(1,2,\dots,2,1)}(\delta)\cup\frac{1}{2}W_{2}^{(1,2,\dots,2,1)}(\delta)$

as a disjoint union of multisets, and this leads directly to a generating
function with manifestly positive coefficients.

######  Corollary 2.12.

Let $\delta=(\delta_{1},\dots,\delta_{h})$ be a profile. Then with notation as
in Proposition 2.9, we have

$\sum_{\lambda\in\mathcal{CP}_{(-\mathrm{rev}(\delta),\delta)}\setminus\mathcal{SCP}_{(-\mathrm{rev}(\delta),\delta)}}q^{|\lambda|}=\prod_{\begin{subarray}{c}k\in
W_{1}^{(1,2,\dots,2,1)}(\delta)\\\ \ell\in
W_{2}^{(1,2,\dots,2,1)}(\delta)\end{subarray}}\frac{1}{(q^{k};q^{2h})_{\infty}(q^{\ell};q^{4h})_{\infty}}\left(\prod_{\ell\in
W_{2}^{(1,2,\dots,2,1)}(\delta)}\frac{(-q^{\ell/2};q^{2h})_{\infty}}{(q^{\ell/2};q^{2h})_{\infty}}-1\right).$

In Section 3, we introduce and prove recurrences for two variable analogues of
$\mathrm{CP}_{\delta}(z;q)$ with weighted size,

(2.9) $\mathrm{CP}_{\delta}^{{\bf
a}}(z;q):=\sum_{\lambda\in\mathcal{CP}_{\delta}}z^{\mathrm{max}(\lambda)}q^{|\lambda|_{{\bf
a}}},\qquad\mathrm{DSPP}_{\delta}^{{\bf
a}}(z;q):=\sum_{\lambda\in\mathcal{DSPP}_{\delta}}z^{\mathrm{max}(\lambda)}q^{|\lambda|_{{\bf
a}}}.$

Theorem 1.1 follows from solving the symmetric cylindric cases of width 4 (or
equivalently, the DSPP cases of width 2 and weight $(1,2,1)$), Theorem 1.2
from the symmetric cylindric cases of width 6 (or equivalently, the DSPP cases
of width 3 and weight $(1,2,2,1)$), and Theorem 1.3 from the standard weight
DSPP cases of width 3.

##  3\. Systems of recurrences for cylindric partitions and DSPPs

To demonstrate Corteel–Welsh’s recurrence and our generalizations, consider
the following toy example: If $\mathrm{lg}(\lambda)$ denotes the largest part
of the integer partition $\lambda$, then one can prove the following
recurrence for
$P(z;q):=\sum_{\lambda\in\mathcal{P}}z^{\mathrm{lg}(\lambda)}q^{|\lambda|}$,

(3.1) $P(z;q)=\sum_{\begin{subarray}{c}\mu\in\mathcal{P}\\\ m\geq
0\end{subarray}}z^{\mathrm{\lg}(\mu)+m}q^{m+|\mu|}=\frac{P(zq;q)}{1-zq}.$

The proof consists taking a partition $\lambda$ on the left-hand side and
removing $\mathrm{lg}(\lambda)$ to create a new partition $\mu$; one then has
$\mathrm{lg}(\lambda)=\mathrm{lg}(\mu)+m$ for some $m\geq 0$. From (3.1), it
is easy to derive Euler’s product-sum identity ([2], Corollary 2.2)

$P(z;q)=\sum_{n\geq 0}\frac{z^{n}q^{n}}{(q;q)_{n}}=\frac{1}{(zq;q)_{\infty}}.$

If we begin instead with a cylindric partition generated by
$\mathrm{CP}_{\delta}(z;q)$ and remove some largest parts in the “corners”
where they occur, then the resulting cylindric partition has a new profile.
Since subsets of largest parts can be removed in multiple ways, an inclusion-
exclusion process leads to the following analogue of (3.1).

######  Proposition 3.1 ([8], Proposition 3.1, reformulated).

Let $\delta=(\delta_{1},\dots,\delta_{h})$ be a profile, and for convenience
define $\delta_{0}:=\delta_{h}$. Define

$I_{\delta}:=\\{0\leq j\leq h-1:(\delta_{j},\delta_{j+1})=(1,-1)\\}.$

For a subset $\emptyset\subsetneq J\subseteq I_{\delta}$, define a new profile
$\sigma_{J}(\delta)$ by swapping the signs of $(\delta_{j},\delta_{j+1})$ for
$j\in J$. Then

(3.2) ${\rm CP}_{\delta}(z;q)=\sum_{\emptyset\subsetneq J\subseteq
I_{\delta}}(-1)^{|J|-1}\frac{{\rm
CP}_{\sigma_{J}(\delta)}\left(zq^{|J|};q\right)}{1-zq^{|J|}}.$

Similarly, we have the following systems of recurrences for weighted cylindric
partitions and weighted DSPPs. The proofs are simple adjustments of the proof
of Proposition 3.1 in [8].

######  Proposition 3.2.

Let $\delta=(\delta_{1},\dots,\delta_{h})$ be a profile and let ${\bf
a}\in\mathbb{R}_{\geq 0}^{h}$. Then with the same notation as in Proposition
3.1, we have

(3.3) ${\rm CP}^{{\bf a}}_{\delta}(z;q)=\sum_{\emptyset\subsetneq J\subseteq
I_{\delta}}(-1)^{|J|-1}\frac{{\rm CP}^{{\bf
a}}_{\sigma_{J}(\delta)}\left(zq^{\sum_{j\in
J}a_{j}};q\right)}{1-zq^{\sum_{j\in J}a_{j}}}.$

Note that for DSPPs, the set $\hat{I}_{\delta}$ and maps $\hat{\sigma}_{J}$
below are subtly different.

######  Proposition 3.3.

Let $\delta=(\delta_{1},\dots,\delta_{h})$ be a profile and let ${\bf
a}\in\mathbb{R}_{\geq 0}^{h+1}$. For convenience, define
$\delta_{h+1}:=-\delta_{h}$ and $\delta_{0}:=-\delta_{1}$. Define

$\hat{I}_{\delta}:=\\{0\leq j\leq h:(\delta_{j},\delta_{j+1})=(1,-1)\\}.$

For a subset $\emptyset\subsetneq J\subseteq I_{\delta}$, define a new profile
$\hat{\sigma}_{J}(\delta)$ by swapping the signs of
$(\delta_{j},\delta_{j+1})$ for $j\in J$. Then

(3.4) ${\rm DSPP}_{\delta}^{{\bf a}}(z;q)=\sum_{\emptyset\subsetneq
J\subseteq\hat{I}_{\delta}}(-1)^{|J|-1}\frac{{\rm DSPP}^{{\bf
a}}_{\hat{\sigma}_{J}(\delta)}\left(zq^{\sum_{j\in
J}a_{j}};q\right)}{1-zq^{\sum_{j\in J}a_{j}}}.$

######  Remark 3.4.

For cylindric partitions, the maps $\sigma_{J}$ permute the entries of
profiles, so the rank—the number of $(-1)$’s—is preserved. Hence, there is a
system of $\binom{h}{k}$ recurrences (3.3) for width $h$, rank $k$ profiles
$\delta.$

By contrast, for DSPPs the maps $\sigma_{J}$ act transitively on all profiles
of width $h$. Thus, in general there is a single system of $2^{h}$ recurrences
(3.4). However, for some weights, like those corresponding to symmetric
cylindric partitions, the number of equations can be reduced through certain
symmetries.

Following [8], our analysis will simplify if the denominators in (3.3) and
(3.4) are removed by defining

$G_{\delta}(z):=(zq;q)_{\infty}\mathrm{CP}^{{\bf a}}_{\delta}(z;q).$

(We will suppress the weight ${\bf a}$ and the second variable $q$ in
$G_{\delta}$ to save space when they are clear from context.) Then (3.3)
becomes

(3.5) $G_{\delta}(z)=\sum_{\emptyset\subsetneq J\subseteq
I_{\delta}}(-1)^{|J|-1}(zq;q)_{\sum_{j\in
J}a_{j}-1}G_{\sigma_{J}(\delta)}\left(zq^{\sum_{j\in J}a_{j}}\right).$

Likewise, with $a_{j}\geq 1$ for all $j$ and

$H_{\delta}(z):=(zq;q)_{\infty}\mathrm{DSPP}^{{\bf a}}_{\delta}(z;q),$

(3.4) becomes

(3.6) $H_{\delta}(z)=\sum_{\emptyset\subsetneq J\subseteq
I_{\delta}}(-1)^{|J|-1}(zq;q)_{\sum_{j\in
J}a_{j}-1}H_{\sigma_{J}(\delta)}\left(zq^{\sum_{j\in J}a_{j}}\right).$

##  4\. Proofs of Theorems 1.1, 1.2, and 1.3

Although the presented results can be performed by hand, to avoid error-prone
and tedious calculations, we mainly use two symbolic computation
implementations in Mathematica to carry out our calculations: the package
qFunctions by Ablinger and the second author [1], and the package
HolonomicFunctions by Koutchan [14]. These implementations are distributed
through the RISC (Research Institute for Symbolic Computation, Johannes Kepler
University, Linz) openly for researchers to benefit from. These
implementations and more can be downloaded through
https://risc.jku.at/software/.

Koutchan’s HolonomicFunctions [14] implementation is well established and
offers state of the art technology in holonomic functions research (in our
context, functions that satisfy linear $q$-recurrence relations). In
particular, this package includes functionality that automatically uncouples a
coupled system of recurrences and it also has the creative telescoping
algorithm to automatically produce, proof, and certify recurrence relations
for given closed hypergeometric formulas.

The qFunctions package [1] by Ablinger and the second author offers many tools
to symbolically manipulate (such as by doing substitutions) linear functional
relations of $q$-holonomic functions and to symbolically create the
$q$-difference relations that Corteel–Welsh [8] originally presented. We
include a short Appendix to list the new functionality (that is soon to be
added to the main qFunctions release) which implements new functions related
to this paper.

It might be possible to prove some of these identities using hypergeometric
means directly. However, we would like to present proofs solely through
coupled system of $q$-difference equations, which were introduced in Section
3. This is to emphasize the technique of how one can go about reducing the
coupled system and under fortunate circumstances guess a sum representation of
the generating functions for the cylindric partitions and other objects. These
proofs are in the spirit of [8] and [7].

###  4.1. Proof of Theorem 1.1 and related results:

Earlier we defined the symmetric cylindric partitions with regards to DSPPs
(2.8). We extract Theorems 1.1 and 1.2 through the study of the generating
functions of the number of symmetric cylindric partitions width 4 and 6,
respectively.

######  Lemma 4.1.

For the profiles with width 4, we have

$\displaystyle\mathrm{SCP}_{(-1,-1,1,1)}(z;q)$
$\displaystyle=\frac{(-zq^{4},zq^{2};q^{4})_{\infty}}{(zq;q)_{\infty}},$
$\displaystyle\mathrm{SCP}_{(1,-1,1,-1)}(z;q)$
$\displaystyle=\frac{(-zq^{3},zq;q^{4})_{\infty}}{(zq;q)_{\infty}},$
$\displaystyle\mathrm{SCP}_{(-1,1,-1,1)}(z;q)$
$\displaystyle=\frac{(-zq,zq^{3};q^{4})_{\infty}}{(zq;q)_{\infty}}.$

###### Proof.

We get the following coupled system of $q$-difference equations when we apply
(3.6) to the width 2 profiles with $a=(1,2,1)$:

(4.1) $\displaystyle H_{(1,1)}(z)$ $\displaystyle=H_{(1,-1)}(zq),$ (4.2)
$\displaystyle H_{(1,-1)}(z)$ $\displaystyle=(1-zq)H_{(-1,1)}(zq^{2}),$ (4.3)
$\displaystyle H_{(-1,1)}(z)$
$\displaystyle=2H_{(1,1)}(zq)-(1-zq)H_{(1,-1)}(zq^{2}),$

where
$H_{\delta}(z):=(zq;q)_{\infty}\mathrm{SCP}_{(-\mathrm{rev}(\delta),\delta)}(z)$.

Recall that $\mathrm{SCP}_{(1,1,-1,-1)}(z;q)=\mathrm{SCP}_{(-1,-1,1,1)}(z;q)$,
hence, we do not need to consider $H_{(-1,-1)}(z)$ as a separate entity.

We can uncouple these $q$-difference equations for the $H_{(-1,1)}(z)$
function by substituting (4.1) followed by substituting (4.2) in 4.3. This
yields

$\displaystyle H_{(-1,1)}(z)$
$\displaystyle=2(1-zq^{3})H_{(-1,1)}(zq^{4})-(1+zq)(1-zq^{3})H_{(-1,1)}(zq^{4})$
(4.4) $\displaystyle=(1+zq)(1-zq^{3})H_{(-1,1)}(zq^{4}).$

Iterating (4.4) shows that

(4.5) $H_{(-1,1)}(z)=(-zq,zq^{3};q^{4})_{\infty}.$

Substituting (4.5) in (4.2) shows that
$H_{(1,-1)}(z)=(-zq^{3},zq;q^{4})_{\infty}$ and finally substituting this
product in (4.1) proves that $H_{(1,1)}(z)=(-zq^{4},zq^{2};q^{4})_{\infty}.$
Writing the definitions of $H_{\delta}(z)$s in their respective product
representations proves Lemma 4.1. ∎

An alternate proof of Lemma 4.1, which will also lead to the proof of Theorem
1.1, can be given through uncoupling the recurrence system. Let

$H_{\delta}(z;q):=\sum_{n\geq 0}h_{\delta}(n)z^{n}.$

Then the $q$-difference equation system that the $H_{\delta}(z)$ functions
satisfy is equivalent to the following coupled system of recurrences.

$\displaystyle h_{(1,1)}(n)$ $\displaystyle=q^{n}h_{(1,-1)}(n),$
$\displaystyle h_{(1,-1)}(n)$
$\displaystyle=q^{2n}h_{(-1,1)}(n)-q^{2n-1}h_{(-1,1)}(n-1),$ $\displaystyle
h_{(-1,1)}(n)$
$\displaystyle=2q^{n}h_{(1,1)}(n)-q^{2n}h_{(1,-1)}(n)+q^{2n-1}(n-1).$

The initial conditions $h_{\delta}(0)=1$ and $h_{\delta}(m)=1$ for all
negative integer $m$, defines these sequences uniquely. This system of
recurrences can be uncoupled and we see that these coefficients satisfy the
second order recurrences

(4.6) $\displaystyle(1-q^{4n+8})h_{(1,1)}(n+2)$
$\displaystyle=-q^{4n+6}(1-q^{2})h_{(1,1)}(n+1)-q^{4n+6}h_{(1,1)}(n),$ (4.7)
$\displaystyle(1-q^{4n+8})h_{(1,-1)}(n+2)$
$\displaystyle=-q^{4n+5}(1-q^{2})h_{(1,-1)}(n+1)-q^{4n+4}h_{(1,-1)}(n),$ (4.8)
$\displaystyle(1-q^{4n+8})h_{(-1,1)}(n+2)$
$\displaystyle=-q^{4n+5}(1-q^{2})h_{(-1,1)}(n+1)+q^{4n+4}h_{(-1,1)}(n).$

We can switch from these uncoupled $q$-recurrence relations to the equivalent
$q$-difference equations. This yields another proof of Lemma 4.1 after
confirming that the $q$-difference equations one get from (4.6)-(4.8) are
homogeneous two term relations (one of which is (4.4)).

However, one can also look at the recurrences (4.6)-(4.8) and some initial
conditions to guess (and later prove) a closed formula for these sequences. In
our cases we have

(4.9) $\displaystyle h_{(1,1)}(n)=(-1)^{\lceil
n/2\rceil}q^{n(n+1)}\frac{(q^{2};q^{4})_{\lceil
n/2\rceil}(-q^{4};q^{4})_{\lfloor n/2\rfloor}}{(q^{4};q^{4})_{n}},$ (4.10)
$\displaystyle h_{(1,-1)}(n)=(-1)^{\lceil
n/2\rceil}q^{n^{2}}\frac{(q^{2};q^{4})_{\lceil
n/2\rceil}(-q^{4};q^{4})_{\lfloor n/2\rfloor}}{(q^{4};q^{4})_{n}},$ and (4.11)
$\displaystyle h_{(-1,1)}(n)=(-1)^{\lfloor
n/2\rfloor}q^{n^{2}}\frac{(q^{2};q^{4})_{\lceil
n/2\rceil}(-q^{4};q^{4})_{\lfloor n/2\rfloor}}{(q^{4};q^{4})_{n}},$

where $\lceil\cdot\rceil$ and $\lfloor\cdot\rfloor$ are the classical ceil and
floor functions. The right-hand side expressions satisfy the same initial
conditions as the expressions on the left; they vanish for every negative
integer $n$, and when $n=0$ they become 1. We can easily show that the
expressions (4.9)-(4.11) satisfy (4.6)-(4.8), respectively. For example,
letting $n\mapsto 2n$, in (4.11), looking at the difference of the two sides
of (4.6) we see that everything vanishes:

$\displaystyle(1-$ $\displaystyle
q^{8n+8})(-1)^{n+1}q^{(2n+2)(2n+3)}\frac{(q^{2};q^{4})_{n+1}(-q^{4};q^{4})_{n+1}}{(q^{4};q^{4})_{2n+2}}$
$\displaystyle+q^{8n+6}(1-q^{2})(-1)^{n+1}q^{(2n+1)(2n+2)}\frac{(q^{2};q^{4})_{n+1}(-q^{4};q^{4})_{n}}{(q^{4};q^{4})_{2n+1}}+q^{8n+6}(-1)^{n}q^{2n(2n+1)}\frac{(q^{2},-q^{4};q^{4})_{n}}{(q^{4};q^{4})_{2n}}$
$\displaystyle=$
$\displaystyle(-1)^{n+1}q^{(2n+1)(2n+2)}\frac{(q^{2};q^{4})_{n+1}(-q^{4};q^{4})_{n}}{(q^{4};q^{4})_{2n+1}}(q^{4n+4}(1+q^{4n+4})+q^{8n+6}(1-q^{2}))$
$\displaystyle+q^{8n+6}(-1)^{n}q^{2n(2n+1)}\frac{(q^{2},-q^{4};q^{4})_{n}}{(q^{4};q^{4})_{2n}}$
$\displaystyle=$ $\displaystyle
q^{8n+6}(1+q^{4n+2})(-1)^{n+1}q^{2n(2n+1)}\frac{(q^{2};q^{4})_{n+1}(-q^{4};q^{4})_{n}}{(q^{4};q^{4})_{2n+1}}+q^{8n+6}(-1)^{n}q^{2n(2n+1)}\frac{(q^{2},-q^{4};q^{4})_{n}}{(q^{4};q^{4})_{2n}}=0.$

This proves that the right-side expression in (4.9) satisfies (4.6). By
carrying the same calculations for $h_{(1,1)}(2n+1)$ and for the other two
sequences we prove that (4.9)-(4.11) satisfy (4.6)-(4.8), respectively.
Lastly, we check the initial conditions, and that finishes the proof the
equalities in (4.9)-(4.11).

Rewriting $H_{(-1,1)}(z)$ using the formula for $h_{(-1,1)}(n)$ in (4.5)
yields

(4.12) $H_{(-1,1)}(z)=\sum_{n\geq 0}h_{(-1,1)}(n)z^{n}=\sum_{n\geq
0}(-1)^{\lfloor n/2\rfloor}q^{n^{2}}\frac{(q^{2};q^{4})_{\lceil
n/2\rceil}(-q^{4};q^{4})_{\lfloor
n/2\rfloor}}{(q^{4};q^{4})_{n}}z^{n}=(-zq,zq^{3};q^{4})_{\infty}.$

Splitting the even and odd cases of the summand in (4.12) and regrouping the
terms proves Theorem 1.1. Moreover, the analogous calculations for
$H_{(1,1)}(z)$ and $H_{(1,-1)}(z)$ also lead to Theorem 1.1.

It should be noted that one can give a shorter proof to Theorem 1.1 without
resorting to the coupled system of symmetric cylindric partitions of width 4.
For example, one can use the $q$-binomial theorem to expand
$(zq;q^{4})_{\infty}$ and $(-zq^{3};q^{4})_{\infty}$ separately, multiply
these series expansions, collect and compare the terms of $z^{N}$ of this
expansion with the $z^{N}$ term of the left-side sum in Theorem 1.1. Then one
can find the recurrences for these compared terms with $q$-Zeilberger
algorithm and finish the proof. Another alternative to recurrences after the
comparison is to observe that these the convolution sums one gets from the
$q$-binomial theorems are equivalent to formulas (21) and (22) in [16]
(corresponding to even and odd exponents of $z$ in the expansions).

Setting $z=q$ followed by $q^{2}\mapsto q$ in Theorem 1.1 gives the following
corollary.

######  Corollary 4.2.

(4.13) $\sum_{n\geq
0}(-1)^{n}q^{2n^{2}+n}\frac{(q;q^{2})_{n+1}(-q^{2};q^{2})_{n}}{(q^{2};q^{2})_{2n+1}}=(q,-q^{2};q^{2})_{\infty}.$

The identity (4.13) can also be proven through combinatorial arguments. The
right-side product is the generating function for the number of partitions
into distinct parts with the extra weight $-1$ to the number of odd parts in
the partition.

One can interpret the left-hand side summand as the generating function of
partitions into $2n$ or $2n+1$ distinct parts, where the partitions are
counted with the extra weight $-1$ to the number of odd parts. This
interpretation can be seen starting from the front factor
$(-1)^{n}q^{2n^{2}+n}$. The term

$2n^{2}+n=1+2+3+\dots+(2n-1)+2n$

is the generating function for partitions into exactly $2n$ consecutive parts.
Furthermore, half of these $2n$ parts are odd and the $(-1)^{n}$ reflects the
weight on the number of odd parts. We interpret the $(-q^{2};q^{2})_{n}$ as
the generating function of Young diagrams with columns even length $\leq 2n$
where each column length only appears once. Similarly $(q;q^{2})_{n+1}$ is the
generating function of Young diagrams with columns odd length $\leq 2n+1$
where each column length only appears once, where in this case we also have
the extra weight $-1$ to the number of columns. Putting together these three
terms, we can see that

$(-1)^{n}q^{2n^{2}+n}(q;q^{2})_{n+1}(-q^{2};q^{2})_{n}$

is the generating function of partitions into $2n$ or $2n+1$ parts, where the
smallest part is at most 2 if there are $2n$ parts, it is exactly 1 if there
are $2n+1$ parts, and the difference between consecutive parts is at most 2.
Moreover, each insertion of an odd length column in the original Young diagram
of the partition with size $2n^{2}+n$ changes the number of odd parts by one,
and each of these column also comes with a $-1$ weight itself that accounts
for this change. Hence, combining the $-1$ weights we see that the exponent of
$-1$ the number of odd parts in the outcome partition. Finally, the term
$1/(q^{2};q^{2})_{2n+1}$ is the generating function of Young diagrams with
even length $\leq 2n+1$ columns where each column appears even number of
times. With this final observation, we can now interpret the summand

$(-1)^{n}q^{2n^{2}+n}\frac{(q;q^{2})_{n+1}(-q^{2};q^{2})_{n}}{(q^{2};q^{2})_{2n+1}}$

as the generating function for partitions into $2n$ or $2n+1$ distinct parts
with an extra alternating weight on the number of odd parts in the partitions.
Summing over all possible $n$ we see that the left-hand side sum counts the
same partitions as the right-hand side product.

###  4.2. Proof of Theorem 1.2:

Similar to the previous Subsection 4.1, we study the generating functions
width 6 profiled symmetric cylindric partitions. Applying the normalized
weighted DSPP recurrence (3.6) to the width 3 profiles with $a=(1,2,2,1)$
gives the coupled $q$-difference equation system

(4.14) $\displaystyle H_{\alpha}(z)$ $\displaystyle=H_{\beta}(zq),$ (4.15)
$\displaystyle H_{\beta}(z)$ $\displaystyle=(1-zq)H_{\gamma}(zq^{2}),$ (4.16)
$\displaystyle H_{\gamma}(z)$
$\displaystyle=H_{\beta}(zq)-(1-zq)(1-zq^{2})H_{\gamma}(zq^{3})+(1-zq)H_{\sigma}(zq^{2}),$
(4.17) $\displaystyle H_{\sigma}(z)$
$\displaystyle=H_{\alpha}(zq)-(1-zq)H_{\beta}(zq^{2})+H_{\gamma}(zq),$

where

$\alpha:=(1,1,1),\quad\beta:=(1,1,-1),\quad\gamma:=(1,-1,1),\quad\sigma:=(-1,1,1).$

and

(4.18)
$H_{\delta}(z):=(zq;q)_{\infty}\mathrm{SCP}_{(-\mathrm{rev}(\delta),\delta)}(z):=\sum_{n\geq
0}h_{\delta}(n)z^{n}$

for a profile $\delta$.

We can once again turn these $q$-difference equations to the equivalent
$q$-recurrence equations. Once these recurrences are uncoupled, we get that
these sequences satisfy the recurrences

$\displaystyle(1-q^{3n+9})(1+q^{3n+5}-$ $\displaystyle
q^{3n+6})h_{\alpha}(n+3)=-q^{3n+7}(1-q^{3}-q^{3n+11}-q^{6n+16}+q^{6n+17})h_{\alpha}(n+2)$
(4.19)
$\displaystyle-q^{3n+9}(1+q^{3n+5}+q^{3n+7}+q^{3n+8}-q^{3n+9}+q^{6n+12}-q^{6n+14})h_{\alpha}(n+1)$
$\displaystyle+q^{6n+12}(1+q^{3n+8}-q^{3n+9})h_{\alpha}(n),$
$\displaystyle(1-q^{3n+9})(1+q^{3n+5}-$ $\displaystyle
q^{3n+6})h_{\beta}(n+3)=-q^{3n+6}(1-q^{3}-q^{3n+11}-q^{6n+16}+q^{6n+17})h_{\beta}(n+2)$
(4.20)
$\displaystyle-q^{3n+7}(1+q^{3n+5}+q^{3n+7}+q^{3n+8}-q^{3n+9}+q^{6n+12}-q^{6n+14})h_{\beta}(n+1)$
$\displaystyle+q^{6n+9}(1+q^{3n+8}-q^{3n+9})h_{\beta}(n),$ (4.21)
$\displaystyle(1-q^{3n+9})h_{\gamma}(n+3)$
$\displaystyle=q^{6n+15}h_{\gamma}(n+2)-q^{3n+6}(1+q^{3n+4}+q^{3n+8})h_{\gamma}(n+1)+q^{6n+9}h_{\gamma}(n),$
(4.22) $\displaystyle(1-q^{3n+9})h_{\sigma}(n+3)$
$\displaystyle=q^{6n+13}h_{\sigma}(n+2)-q^{3n+8}(1+q^{3n+2}+q^{3n+4})h_{\sigma}(n+1)+q^{6n+9}h_{\sigma}(n).$

The initial conditions $h_{\delta}(0)=1$ and $h_{\delta}(m)=0$ for all
negative $m$ defines these sequences uniquely.

Looking at some initial values of the $h_{\gamma}(n)$ sequence, we can
identify that

(4.23) $\displaystyle h_{\gamma}(n)$ $\displaystyle=\sum_{m\geq
0}(-1)^{m}q^{3{n+1\choose
2}-3m(m+1)}\frac{(-q,-q^{5};q^{6})_{m}}{(q^{6};q^{6})_{m}(q^{3};q^{3})_{n-2m}}.$
Using $q$-Zeilberger algorithm (or the creative telescoping algorithm), we can
automatically prove that the right-hand side of (4.23) satisfies (4.21). In
the same way, using the relations (4.14)-(4.17), we can prove (4.24)
$\displaystyle h_{\beta}(n)$ $\displaystyle=\sum_{m\geq
0}(-1)^{m+1}q^{3{n-1\choose
2}+2n-3m(m+1)-1}\frac{(-q,-q^{5};q^{6})_{m}}{(q^{6};q^{6})_{m}(q^{3};q^{3})_{n-2m}}(1-q^{3n+1}+q^{3n-6m}),$
(4.25) $\displaystyle h_{\alpha}(n)$ $\displaystyle=\sum_{m\geq
0}(-1)^{m+1}q^{3{n+1\choose
2}-3m(m+1)-1}\frac{(-q,-q^{5};q^{6})_{m}}{(q^{6};q^{6})_{m}(q^{3};q^{3})_{n-2m}}(1-q^{3n+1}+q^{3n-6m}),$
and (4.26) $\displaystyle h_{\sigma}(n)$ $\displaystyle=\sum_{m\geq
0}(-1)^{m+1}q^{3{n+1\choose
2}-2n-3m(m+1)-1}\frac{(-q,-q^{5};q^{6})_{m}}{(q^{6};q^{6})_{m}(q^{3};q^{3})_{n-2m}}$
$\displaystyle\hskip
56.9055pt\times(1-q^{3n+1}-q^{3n-2}-q^{3n-6m}-q^{3n-6m-3}+q^{6n-6m-2}+q^{6n-12m-3})$

in this order.

Notice that the formulas in (4.23)-(4.26) are the summands of the left-hand
side sums of Theorem 1.2. Combining the definition of each $H_{\delta}(z)$
(4.18), and Proposition 2.9 (for $z=1$) for these profiles and weights
finishes the proof of Theorem 1.2.

###  4.3. New proofs of Little Göllnitz and Göllnitz–Gordon Theorems

We use the system of recurrences for standard weight, width 3 DSPPs. For
standard weight, reversing the order of the integer partitions in a DSPP leads
to the identity
$\mathrm{DSPP}_{\delta}(z;q)=\mathrm{DSPP}_{-\mathrm{rev}(\delta)}(z;q)$, thus
we need only write the system of recurrences using profiles

$\alpha:=(1,1,1),\quad\beta:=(1,1,-1),\quad\gamma:=(1,-1,1),\quad\sigma:=(-1,1,1).$

From equation 3.6, we have the system

$\displaystyle H_{\alpha}(z)$ $\displaystyle=H_{\beta}(zq),$ $\displaystyle
H_{\beta}(z)$ $\displaystyle=H_{\gamma}(zq^{2}),$ $\displaystyle
H_{\gamma}(z)$
$\displaystyle=H_{\beta}(zq)+H_{\delta}(zq)-(1-zq)H_{\gamma}(zq^{2}),$
$\displaystyle H_{\sigma}(z)$
$\displaystyle=H_{\alpha}(zq)+H_{\gamma}(zq)-(1-zq)H_{\beta}(zq^{2}),$

whence

$H_{\sigma}(z)=H_{\gamma}(zq)+zqH_{\gamma}(zq^{3}),$

and

$H_{\gamma}(z)=(1+zq)H_{\gamma}(zq^{2})+zq^{2}H_{\gamma}(zq^{4}).$

If we now write $H_{\gamma}(z)=\sum_{n\geq 0}h_{\gamma}(n)z^{n},$ where the
$h_{n}$ are rational functions in $q$ and $h_{\gamma}(0)=H_{\gamma}(0)=1,$
then the above recurrence gives

$h_{\gamma}(n)=q^{2n}h_{\gamma}(n)+(q^{2n-1}+q^{4n-2})h_{\gamma}(n-1).$

By iterating, we obtain

$h_{\gamma}(n)=\frac{(-q;q^{2})_{n}}{(q^{2};q^{2})_{n}}q^{n^{2}},$

therefore,

$\sum_{n\geq
0}\frac{(-q;q^{2})_{n}}{(q^{2};q^{2})_{n}}q^{n^{2}}=H_{(1,-1,1)}(1)=(q;q)_{\infty}\mathrm{DSPP}_{(1,-1,1)}(q)=\frac{(q;q)_{\infty}}{(q,q^{2},q^{3},q^{4};q^{4})_{\infty}(q^{2},q^{4},q^{7};q^{8})_{\infty}}.$

Upon cancelling common terms in the product, we obtain the third identity in
Theorem 1.3. The other identities can be proved using the expression for
$h_{\gamma}(n)$ and the relations above.

##  5\. Applications to Schmidt-type identities: proof of Theorem 1.5

In this section we prove Theorem 1.5, beginning with identity (1.18). It is
clear from the definitions that diamond partitions can be thought of as
weighted DSPPs. In particular,

$\sum_{\lambda\in\diamondsuit}z^{\lambda_{1}}q^{\lambda_{1}+\lambda_{4}+\lambda_{7}+\dots}=\mathrm{DSPP}_{(1,-1)}^{(0,1,0)}(z;q).$

Note that Proposition 2.9 then immediately implies (1.13), as

$\sum_{\lambda\in\diamondsuit}q^{\lambda_{1}+\lambda_{4}+\lambda_{7}+\dots}=\frac{1}{(q;q)_{\infty}^{3}(q;q^{2})_{\infty}}=\frac{(-q;q)_{\infty}}{(q;q)_{\infty}^{3}}.$

To prove our refinement, we apply the recurrences in Proposition 3.3 to obtain
the system,

$\displaystyle(1-z)\mathrm{DSPP}_{(1,1)}^{(0,1,0)}(z;q)=\mathrm{DSPP}_{(1,-1)}^{(0,1,0)}(z),$
$\displaystyle(1-z)\mathrm{DSPP}_{(-1,-1)}^{(0,1,0)}(z;q)=\mathrm{DSPP}_{(1,-1)}^{(0,1,0)}(z),$
$\displaystyle(1-zq)\mathrm{DSPP}_{(1,-1)}^{(0,1,0)}(z;q)=\mathrm{DSPP}_{(-1,1)}^{(0,1,0)}(zq),$
$\displaystyle(1-z)\mathrm{DSPP}_{(-1,1)}^{(0,1,0)}(z;q)=\mathrm{DSPP}_{(1,1)}^{(0,1,0)}(z)+\mathrm{DSPP}_{(-1,-1)}^{(0,1,0)}(z)-\mathrm{DSPP}_{(1,-1)}^{(0,1,0)}(z).$

It is then easy to show that

$\mathrm{DSPP}_{(1,-1)}^{(0,1,0)}(z;q)=\frac{1+zq}{(1-zq)^{3}}\mathrm{DSPP}_{(1,-1)}^{(0,1,0)}(zq;q)=\frac{(-zq;q)_{\infty}}{(zq;q)_{\infty}^{3}},$

proving (1.18).

Identities (1.16) and (1.17) are proved similarly by showing that

$\sum_{\lambda\in\mathcal{P}}z^{\lambda_{1}}q^{\lambda_{1}+\lambda_{3}+\dots}=\mathrm{CP}_{(-1,1)}^{(0,1)}(z;q),\quad\text{and}\quad\sum_{\lambda\in\mathcal{P}}z^{\lambda_{1}}q^{\lambda_{2}+\lambda_{4}+\dots}=\mathrm{CP}_{(1,-1)}^{(0,1)}(z;q),$

with the aid of the recurrence in Proposition 3.2.

For identities (1.14) and (1.15), we introduce cylindric partitions into
distinct parts, which are defined so that the inequalities along rows and
columns in a cylindric partition are strict. Define the generating function
$\mathrm{DCP}_{\delta}^{{\bf a}}(z;q)$ analogously. Then

$\sum_{\lambda\in\mathcal{D}}z^{\lambda_{1}}q^{\lambda_{1}+\lambda_{3}+\dots}=\mathrm{DCP}_{(-1,1)}^{(0,1)}(z;q),\qquad\sum_{\lambda\in\mathcal{D}}z^{\lambda_{1}}q^{\lambda_{2}+\lambda_{4}+\dots}=\mathrm{DCP}_{(1,-1)}^{(0,1)}(z;q).$

In this case, the recurrence is slightly different from Proposition 3.2 but is
proved as in [8].

######  Proposition 5.1.

With notation as in Proposition 3.2, we have

$\mathrm{DCP}_{\delta}^{{\bf a}}(z;q)=1+\sum_{\emptyset\subsetneq J\subseteq
I_{\delta}}(-1)^{|J|-1}zq^{\sum_{j\in J}a_{j}}\frac{\mathrm{DCP}^{{\bf
a}}_{\sigma_{J}(\delta)}\left(zq^{\sum_{j\in
J}a_{j}};q\right)}{1-zq^{\sum_{j\in J}a_{j}}}.$

Note that the $q$-difference equations above are inhomogeneous, in contrast to
those for cylindric partitions and DSPPs.

Thus, we have the system

$\displaystyle\mathrm{DCP}_{(-1,1)}^{(0,1)}(z;q)$
$\displaystyle=1+\frac{z}{1-z}\mathrm{DCP}_{(1,-1)}^{(0,1)}(z;q)$
$\displaystyle\mathrm{DCP}_{(1,-1)}^{(0,1)}(z;q)$
$\displaystyle=1+\frac{zq}{1-zq}\mathrm{DCP}_{(-1,1)}^{(0,1)}(zq;q),$

which implies

$\mathrm{DCP}_{(1,-1)}^{(0,1)}(z;q)=1+\frac{zq}{1-zq}+\frac{z^{2}q^{2}}{(1-zq)^{2}}\mathrm{DCP}_{(1,-1)}^{(0,1)}(zq;q)=\frac{1}{1-zq}+\frac{z^{2}q^{2}}{(1-zq)^{2}}\mathrm{DCP}_{(1,-1)}^{(0,1)}(zq;q),$

which one may iterate to get

(5.1) $\mathrm{DCP}_{(1,-1)}^{(0,1)}(z;q)=\sum_{n\geq
0}\frac{z^{2n}q^{n(n+1)}}{(zq;q)_{n}(zq;q)_{n+1}},$

which gives (1.14). Identity (1.15) can also be derived in a similar fashion
by studying $\mathrm{DCP}_{(-1,1)}^{(0,1)}(z;q)$.

##  6\. Conclusion and Outlook

Corteel and Welsh [8] developed the idea of using coupled systems of
$q$-difference equations for cylindric partitions to prove sum-product
identities. This paper expands their original idea in multiple ways. We apply
Corteel and Welsh’s argument to symmetric cylindric partitions and skew double
shifted plane partitions. The introduction of non-trivial weights then allows
for a bigger search space. We do not have a clear vision of which profile
families (those having the same width and same rank) or which group of weights
would yield an essentially solvable $q$-difference equation system.

New combinatorial questions arise from the correspondences noted in Remark
2.11. Can we define equivalence classes of cylindric partitions (such as for
the ones in (2.5) and (2.6)) or DSPPs? Are there direct bijections between
these sets of objects? Can finite versions of these sum-product identities be
proved?

In the alternate proof of Corollary 4.2, we showed that when $z=q$ both sides
of the identity in the Theorem 1.1 can be interpreted as a generating function
for a weighted count of partitions into distinct parts. It would be
interesting to see partition theoretic interpretations of Theorem 1.1 for
other specializations of $z$.

We should also note that Lemma 4.1 and our other experiments suggest that the
generating functions in many cases actually yield bivariate products in $q$
and $z$ rather than the univariate products coming from Propositions 2.2, 2.8
and 2.9. Similarly, in Section 5 we discussed cylindric partitions into
distinct parts and observed through Theorem 1.4 that for some width 2, rank 1
profiles the univariate generating functions (when $z=1$) turn into infinite
products. It is worth investigating whether we get bivariate products in other
cases as well.

The authors plan to return to some of these questions in follow-up work,
guided with the help of the new symbolic computation implementations tailor-
made for this purpose.

## Appendix A: modularity of $\mathrm{CP}_{\delta}(q)$

We prove that the multiset $W_{3}(\delta)$ in Proposition 2.2 contains every
$k<h$ as many times as it contains $h-k.$ We say that $W_{3}(\delta)$ is
$k$-balanced if the above property is true for $k$.

We begin by showing that $W_{3}(\delta)$ is 1-balanced. It is easy to see that
$W_{3}(-\delta)$ contains $h-k$ exactly as many times as $W_{3}(\delta)$
contains $k$, so $W_{3}(-\delta)$ is $k$-balanced if and only if
$W_{3}(\delta)$ is. Hence, we may assume without loss of generality that
$\delta_{1}=1.$ Thus, the multiplicity of 1 in $W_{3}(\delta)$ is

$\\#\\{(\delta_{j},\delta_{j+1})=(1,-1)\\},$

and the multiplicity of $h-1$ in $W_{3}(\delta)$ is

$\\#\\{(\delta_{j},\delta_{j+1})=(-1,1)\\}+1_{\delta_{h}=-1}.$

If the number of sign changes in the sequence $\delta$ is even, then
$\delta_{h}=1$ and the two multiplicities above are equal. If the number of
sign changes is odd, then $\delta_{h}=-1$ and furthermore

$\\#\\{(\delta_{j},\delta_{j+1})=(1,-1)\\}=1+\\#\\{(\delta_{j},\delta_{j+1})=(-1,1)\\},$

as required. Thus, $W_{3}(\delta)$ is 1-balanced for all profiles $\delta$.

Now let $1<k<h-1$ and partition $\delta$ into the following sub-profiles,

$\displaystyle(\delta_{1},\delta_{1+k},\dots,\delta_{1+q_{1}k})$
$\displaystyle(\delta_{2},\delta_{2+k},\dots,\delta_{2+q_{2}k})$
$\displaystyle\hskip 28.45274pt\vdots$
$\displaystyle(\delta_{k-1},\delta_{2k-1},\dots,\delta_{(q_{k-1}+1)k-1}),$

where each $q_{r}$ satisfies $q_{r}k+r\leq h<(q_{r}+1)k+r.$ The multiplicity
of $k$ in $W_{3}(\delta)$ is

$\\#\\{(\delta_{j},\delta_{j+k})=(1,-1)\\}+\\#\\{(\delta_{j},\delta_{j+h-k})=(-1,1)\\},$

and the multiplicity of $h-k$ in $W_{3}(\delta)$ is

$\\#\\{(\delta_{j},\delta_{j+k})=(-1,1)\\}+\\#\\{(\delta_{j},\delta_{j+h-k})=(1,-1)\\}.$

Here, the $k$ pairs $\\{(\delta_{j},\delta_{j+h-k})\\}_{1\leq j\leq k}$
occurring in $\delta$ are evenly distributed amongst the $k$ sub-profiles.
That $W_{3}(\delta)$ is $k$-balanced now follows from the fact that each sub-
profile is $1$-balanced.

## Appendix B

The second author is updating and adding new functionality to the qFunctions
package as a part of the Austrian Science Fund (FWF) sponsored project
P-34501N “Partition identities using the weighted word approach” he is
running. The most up to date released version of this package as well as the
up to date information can be found on the author’s website
https://akuncu.com/qfunctions/ and on the RISC website
https://risc.jku.at/software/.

One big difference between earlier work and this paper appears in the
description of the profiles for cylindric partitions. This paper follows the
profile definitions of Han and Xiong [13], however, in [8, 7, 20] the profiles
for cylindric partitions are given in an equivalent but different
representation. One can call SwitchCPProfileRepresentation and provide a
profile either in the form used in [8] or in [13] to get its equivalent
representation. This transformation is simple and, when needed, it is now
automatically detected and done by the qFunctions package.

The original $q$-difference relations (3.2) Corteel and Welsh noted for
cylindric partitions [8] is implemented in qFunctions. The functionality to
find all the $q$-difference relations in a coupled system through (3.3)-(3.4)
is also added to the package and it is tested meticulously during the
development of this work. These functions are called
WeightedCPFunctionalEquationSystem, WeightedDSPPFunctionalEquationSystem, and
WeightedDCPFunctionalEquationSystem similar to the original
CylindricFunctionalEquationCreator with one additional input of the weights at
the end. These functions and other related functionalities will be added to
the next release of the package. It should be noted that these functionality
is not necessary for the proofs of this paper. The implementation is being
done solely for assisting future users with problems involving larger coupled
systems of equations.

On top of the automated $q$-difference equation system setting functions. The
second author also implemented experimental tools where one can choose a
product as in the right-hand side of (2.3) or (2.4) and try any profile
against this product to find weights $a_{i}$’s which would give the product as
the generating function of the selected object. These are done by setting up
the linear systems induced by the $W^{a}_{i}$ sets presented in the
Propositions 2.8 and 2.9 against the selected residue classes and modulus.
These functionalities are called FitCPWeights and FitDSPPWeights.

## References

  * [1] J. Ablinger and A.K. Uncu, qFunctions \- A Mathematica package for $q$-series and partition theory applications, J. Symbolic Comput. 107, (2021), 145-166.
  * [2] G.E, Andrews, The Theory of Partitions, Cambridge University Press (1984).
  * [3] G.E. Andrews and R.J. Baxter, A motivated proof of the Rogers–Ramanujan identities, Am. Math. Mon. 96 (1989), 401–409.
  * [4] G.E. Andrews, A. Schilling, S.O. Warnaar, An $A_{2}$ Bailey lemma and Rogers-Ramanujan-type identities, J. Amer. Math. Soc., 12 (3), (1999) 677–702.
  * [5] G.E. Andrews and P. Paule, MacMahon’s partition analysis XIII: Schmidt type partitions and modular forms, J. Num. Theo. (2021). https://doi.org/10.1016/j.jnt.2021.09.008
  * [6] A. Borodin, Periodic Schur processes and cylindric partitions, Duke Math. J. 140 (2007), 391–468.
  * [7] S. Corteel, J. Dousse and A.K. Uncu, Cylindric partitions and some new $A_{2}$ Rogers–Ramanujan identies, Proc. Amer. Math. Soc. 150 (3), (February 2022), 481–497.
  * [8] S. Corteel and T. Welsh, The $A_{2}$ Rogers–Ramanujan identities revisited, Annals of Comb. 23 (2019), 683-–694.
  * [9] B. Feigin, O. Foda, and T. Welsh, Andrews-Gordon type identities from combinations of Virasoro characters, Ramanujan J. 17 (2008), no. 1, 33–52.
  * [10] A.M. Garsia and S.C. Milne, A Rogers–Ramanujan bijection, J. Comb. Theo. Ser. A 31 (1981), 289–339.
  * [11] I.M. Gessel and C. Krattenthaler, Cylindric partitions, Trans. Amer. Math. Soc. 349 (1997), 429–479.
  * [12] H. Göllnitz, Partitionen mit Differenzenbedingungen, Journal für die reine und angewandte Mathematik 225 (1967), 154–190.
  * [13] G.-N. Han and H. Xiong, Skew doubled shifted plane partitions: calculus and asymptotics, Amer. Inst. Math. Sci. 29 (2021), 1841–1857.
  * [14] C. Koutschan, Advanced Applications of the Holonomic Systems Approach, RISC, Johannes Kepler University, Linz. PhD Thesis. September 2009.
  * [15] J. Lepowsky and R.L. Wilson, The Rogers–Ramanujan identities: Lie theoretic interpretation and proof, Proc. Nat. Acad. Sci. 78 (1981), 699–701.
  * [16] P. Paule, On identities of the Rogers–Ramanujan type, J. Math. Anal. Appl. 107 (1985), 255–284.
  * [17] F. Schmidt, Interrupted partitions, Problem 10629, Am. Math. Mon. 104 (1999), 87–88.
  * [18] A. Sills, An Invitation to the Rogers–Ramanujan Identities, CRC Press (2017).
  * [19] A.K. Uncu, Weighted Rogers–Ramanujan partitions and Dyson crank, Ramanujan J. 46 (2018), 579–591.
  * [20] O. Warnaar, The $A_{2}$ Andrews–Gordon identities and cylindric partitions, https://arxiv.org/abs/2111.07550