#  $d$-fold partition diamonds

Dalen Dockery  Department of Mathematics, University of Tennessee, Knoxville,
TN 37996, USA<EMAIL_ADDRESS>,  Marie Jameson  Department of
Mathematics, University of Tennessee, Knoxville, TN 37996, USA
<EMAIL_ADDRESS>,  James A. Sellers  Department of Mathematics and
Statistics, University of Minnesota Duluth, Duluth, MN 55812, USA
<EMAIL_ADDRESS>and  Samuel Wilson  Department of Mathematics, University
of Tennessee, Knoxville, TN 37996, USA<EMAIL_ADDRESS>
###### Abstract.

In this work we introduce new combinatorial objects called $d$–fold partition
diamonds, which generalize both the classical partition function and the
partition diamonds of Andrews, Paule and Riese, and we set $r_{d}(n)$ to be
their counting function. We also consider the Schmidt type $d$–fold partition
diamonds, which have counting function $s_{d}(n).$ Using partition analysis,
we then find the generating function for both, and connect the generating
functions $\sum_{n=0}^{\infty}s_{d}(n)q^{n}$ to Eulerian polynomials. This
allows us to develop elementary proofs of infinitely many Ramanujan–like
congruences satisfied by $s_{d}(n)$ for various values of $d$, including the
following family: for all $d\geq 1$ and all $n\geq 0,$ $s_{d}(2n+1)\equiv
0\pmod{2^{d}}.$

##  1\. Introduction and statement of results

In 2001, George Andrews, Peter Paule, and Axel Riese [APR01b] defined plane
partition diamonds, which are partitions whose parts are non-negative integers
$a_{i}$ and $b_{i}$ which are placed at the nodes of the graph given in Figure
1, where each directed edge indicates $\geq.$ They found that the generating
function for such partitions is given by [APR01b, Corollary 2.1]

$\prod_{n=1}^{\infty}\frac{1+q^{3n-1}}{1-q^{n}}.$ (1)

$a_{0}$$b_{1}$$b_{2}$$a_{1}$$b_{3}$$b_{4}$$a_{2}$$b_{5}$$b_{6}$$a_{3}$$\cdots$
Figure 1. plane partition diamonds ($d=2$)

In this paper, we aim to generalize this construction. For any positive
integer $d$, we place $d$ nodes between any two nodes $a_{k}$ and $a_{k+1}$,
so that we have the classical partition function $p(n)$ when $d=1$ and we have
the plane partition diamonds of Andrews, Paule, and Riese when $d=2.$

More precisely, we define $d$-fold partition diamonds to be partitions whose
parts are non-negative integers $a_{i}$ and $b_{i,j}$ which are placed at the
nodes of the graph given in Figure 2, where each directed edge indicates
$\geq.$

$a_{0}$$a_{1}$$a_{2}$$a_{3}$$b_{1,1}$$b_{1,d}$$b_{2,1}$$b_{2,d}$$b_{3,1}$$b_{3,d}$$\vdots$$\vdots$$\vdots$$\cdots$
Figure 2. $d$-fold partition diamonds

Letting $r_{d}(n)$ denote the number of $d$-fold partition diamonds of $n,$ we
then have that

$\displaystyle\sum_{n=0}^{\infty}r_{1}(n)q^{n}$
$\displaystyle=\prod_{n=1}^{\infty}\frac{1}{1-q^{n}},$
$\displaystyle\sum_{n=0}^{\infty}r_{2}(n)q^{n}$
$\displaystyle=\prod_{n=1}^{\infty}\frac{1+q^{3n-1}}{1-q^{n}},$
$\displaystyle\sum_{n=0}^{\infty}r_{3}(n)q^{n}$
$\displaystyle=\prod_{n=1}^{\infty}\frac{1+2q^{4n-2}(1+q)+q^{8n-3}}{1-q^{n}}.$

Note that this corresponds to the classical partition function when $d=1$ and
to (1) when $d=2.$ More generally, we have the following result.

######  Theorem 1.1.

The generating function for the number of $d$-fold partition diamonds is given
by

$\sum_{n=0}^{\infty}r_{d}(n)q^{n}=\prod\limits_{n=1}^{\infty}\frac{F_{d}(q^{(n-1)(d+1)+1},q)}{1-q^{n}},$

where $F_{d}$ is a polynomial that is defined recursively in Lemma 2.5.

In fact, we can also prove an analogous result regarding Schmidt type $d$-fold
partition diamonds, which are a variant obtained by summing only the links
$a_{0}+a_{1}+\cdots+a_{k}$ (i.e., omitting the $b_{i}$’s when expressing $n$
as a sum) as in [AP22]. Letting $s_{d}(n)$ denote the number of Schmidt type
$d$-fold partition diamonds of $n$, we have that

$\displaystyle\sum_{n=0}^{\infty}s_{1}(n)q^{n}$
$\displaystyle=\prod_{n=1}^{\infty}\frac{1}{(1-q^{n})^{2}},$
$\displaystyle\sum_{n=0}^{\infty}s_{2}(n)q^{n}$
$\displaystyle=\prod_{n=1}^{\infty}\frac{1+q^{n}}{(1-q^{n})^{3}},$
$\displaystyle\sum_{n=0}^{\infty}s_{3}(n)q^{n}$
$\displaystyle=\prod_{n=1}^{\infty}\frac{1+4q^{n}+q^{2n}}{(1-q^{n})^{4}}.$

Note that the generating functions for $d=1$ and $d=2$ given above were known
previously [AP22, Theorems 2 and 4]. More generally, we have the following
result.

######  Theorem 1.2.

The generating function for the number of Schmidt type $d$-fold partition
diamonds is given by

$\sum_{n=0}^{\infty}s_{d}(n)q^{n}=\prod_{n=1}^{\infty}\frac{A_{d}(q^{n})}{(1-q^{n})^{d+1}}.$

Here, $A_{d}(q)$ denotes the $d$th Eulerian polynomial, which is defined by
$A_{0}(q)=1$ and, for all $d\geq 1,$

$A_{d}(q)=(1+(d-1)q)A_{d-1}(q)+q(1-q)A^{\prime}_{d-1}(q).$ (2)

Both Theorems 1.1 and 1.2 arise as special cases of a more technical theorem,
Theorem 2.3, which will be proved using MacMahon’s partition analysis.

In Section 2, we will describe the MacMahon operator and use it to understand
a generating function for $d$-fold partition diamonds; this will allow us to
state our main technical theorem, Theorem 2.3. In Section 2.4, we will prove
some preliminary results needed to prove Theorem 2.3, which we do in Section
2.5. Finally, in Section 3, we prove several infinite families of
Ramanujan–like congruences satisfied by the functions $s_{d}(n)$ in elementary
fashion. For example, we prove that for all $d\geq 1$ and all $n\geq 0,$

$s_{d}(2n+1)\equiv 0\pmod{2^{d}}.$

##  2\. Background and Notation

In order to describe the generating functions that arise in studying $d$-fold
partition diamonds, we start by introducing the MacMahon operator
$\Omega_{\geq},$ which is the primary tool for computing these generating
functions. The discussion below is inspired by the landmark series of papers
on the subject by Andrews, Paule, and others [APR01a, APR01b, AP22].

###  2.1. The MacMahon operator

First we define MacMahon’s Omega operator $\operatorname*{\Omega}_{\geq}.$

######  Definition 2.1.

The operator $\operatorname*{\Omega}_{\geq}$ is given by

$\operatorname*{\Omega}_{\geq}\sum_{s_{1}=-\infty}^{\infty}\cdots\sum_{s_{r}=-\infty}^{\infty}A_{s_{1},\ldots,s_{r}}\lambda_{1}^{s_{1}}\cdots\lambda_{r}^{s_{r}}\coloneqq\sum_{s_{1}=0}^{\infty}\cdots\sum_{s_{r}=0}^{\infty}A_{s_{1},\ldots,s_{r}},$

where the domain of the $A_{s_{1},\ldots,s_{r}}$ is the field of rational
functions over $\mathbb{C}$ in several complex variables and the $\lambda_{i}$
are restricted to a neighborhood of the circle $|\lambda_{i}|=1.$ In addition,
the $A_{s_{1},\ldots,s_{r}}$ are required to be such that any of the series
involved is absolutely convergent within the domain of the definition of
$A_{s_{1},\ldots,s_{r}}.$

We will always have $\operatorname*{\Omega}_{\geq}$ operate on variables
denoted by letters $\lambda,\mu$ from the Greek alphabet (so that letters from
the Latin alphabet will be unaffected by $\operatorname*{\Omega}_{\geq}$).

The benefit of using the MacMahon operator comes from our ability to compute
it quickly using elimination formulae. MacMahon [Mac04, pp. 102-103] gave a
list of some of these, e.g.,

$\operatorname*{\Omega}_{\geq}\frac{1}{(1-\lambda
x)(1-\lambda^{-1}y)}=\frac{1}{(1-x)(1-xy)}.$ (3)

This formula can be generalized as follows.

######  Proposition 2.2.

For $d$ a positive integer and $j$ an integer, we have that

$\operatorname*{\Omega}_{\geq}\frac{\lambda^{j}}{(1-\lambda
x_{1})\cdots(1-\lambda
x_{d})(1-\lambda^{-1}y)}=\frac{1}{(1-y)}\left[\frac{1}{(1-x_{1})(1-x_{2})\cdots(1-x_{d})}\right.\\\
\left.-\frac{y^{j+1}}{(1-x_{1}y)(1-x_{2}y)\cdots(1-x_{d}y)}\right].$

###### Proof.

Expanding the left hand side using geometric series and applying the
definition of $\operatorname*{\Omega}_{\geq}$, we obtain

$\displaystyle\operatorname*{\Omega}_{\geq}\frac{\lambda^{j}}{(1-\lambda
x_{1})\cdots(1-\lambda x_{d})(1-\lambda^{-1}y)}$
$\displaystyle=\operatorname*{\Omega}_{\geq}\sum_{a_{1},\ldots,a_{d+1}\geq
0}\lambda^{j+a_{1}+\cdots+a_{d}-a_{d+1}}x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}}y^{a_{d+1}}$
$\displaystyle=\operatorname*{\Omega}_{\geq}\sum_{\begin{subarray}{c}a_{1},\ldots,a_{d}\geq
0\\\
0\,\leq\,k\,\leq\,{a_{1}+\cdots+a_{d}+j}\end{subarray}}\lambda^{k}x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}}y^{a_{1}+\cdots+a_{d}+j-k}$
$\displaystyle=\sum_{a_{1},\ldots,a_{d}\geq 0}x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}}\sum_{k=0}^{a_{1}+\cdots+a_{d}+j}y^{k}$
$\displaystyle=\sum_{a_{1},\ldots,a_{d}\geq 0}x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}}\left[\frac{1-y^{a_{1}+\cdots+a_{d}+j+1}}{1-y}\right],$

and the result follows by simplifying the geometric series. ∎

###  2.2. Generating functions for $d$-fold partition diamonds

Following the work of Andrews, Paule, and Riese in [APR01b], for integers
$d,n\geq 1$ we set

$D_{d,n}\coloneqq D_{d,n}(q_{0},q_{1},\ldots,q_{n};w)\coloneqq\sum
q_{0}^{a_{0}}\cdots q_{n}^{a_{n}}w^{\sum_{j,k}b_{j,k}}$

where the (outer) sum for $D_{d,n}$ ranges over all non-negative integers
$a_{i}$ and $b_{j,k}$ which satisfy the inequalities encoded by Figure 3.
Thus, $D_{d,n}$ is the generating function for all $d$-fold partition diamonds
of fixed length $n,$ and each part (i.e., vertex of the graph) corresponds to
some $q_{i}$ or $w$.

$b_{1,1}$$b_{1,d}$$b_{2,1}$$b_{2,d}$$b_{n,1}$$b_{n,d}$$a_{0}$$a_{1}$$a_{2}$$a_{n}$$a_{n-1}$$\vdots$$\vdots$$\vdots$$\cdots$
Figure 3. $d$-fold partition diamond of length n.

Using this notation, we are now ready to state our main technical theorem.

######  Theorem 2.3.

For $d\geq 1$ and $n\geq 1,$

$D_{d,n}(q_{0},\ldots,q_{n};w)=\left(\prod_{k=0}^{n-1}\frac{F_{d}(Q_{k}w^{dk},w)}{(1-Q_{k}w^{dk})(1-Q_{k}w^{dk+1})\cdots(1-Q_{k}w^{dk+d})}\right)\frac{1}{1-Q_{n}w^{dn}},$

where $F_{d}$ is the polynomial that is defined recursively in Lemma 2.5, and
$Q_{k}\coloneqq q_{0}q_{1}\dots q_{k}$ for all $k\geq 0.$

###  2.3. The crude form of the generating function

Here, we continue to generalize the objects described in [APR01b] as we fix
some more notation to describe the crude form of the generating function
$D_{d,n}.$ For integers $d,n\geq 1$ we set

$\Lambda_{n}\coloneqq\Lambda_{d,n}\coloneqq\left(\lambda_{n,1}^{a_{n-1}-b_{n,1}}\lambda_{n,2}^{a_{n-1}-b_{n,2}}\cdots\lambda_{n,d}^{a_{n-1}-b_{n,d}}\right)\left(\mu_{n,1}^{b_{n,1}-a_{n}}\mu_{n,2}^{b_{n,2}-a_{n}}\cdots\mu_{n,d}^{b_{n,d}-a_{n}}\right).$

Note here that $\Lambda_{d,1}\Lambda_{d,2}\cdots\Lambda_{d,n}$ encodes all of
the inequalities that must be satisfied in a $d$-fold diamond of length $n$.
That is, each inequality (i.e., edge of the graph) corresponds to some
$\lambda_{i,j}$ or $\mu_{i,j}.$ In other words, we immediately have that

$D_{d,n}=\operatorname*{\Omega}_{\geq}\sum_{a_{i}\geq 0,b_{j,k}\geq
0}\Lambda_{d,1}\Lambda_{d,2}\cdots\Lambda_{d,n}q_{0}^{a_{0}}\cdots
q_{n}^{a_{n}}w^{\sum_{j,k}b_{j,k}}.$ (4)

In the proof of Theorem 2.3, we will also need (for integers $d,n\geq 1,$ and
$\rho\geq 0$)

$D_{d,n}^{(\rho)}\coloneqq
D_{d,n}^{(\rho)}(q_{0},q_{1},\ldots,q_{n};w)\coloneqq\sum q_{0}^{a_{0}}\cdots
q_{n}^{a_{n}}w^{\sum_{j,k}b_{j,k}}$

where the sum for $D_{d,n}^{(\rho)}$ ranges over all non-negative integers
$a_{i}$ and $b_{j,k}$ which satisfy the inequalities encoded by
$\Lambda_{d,1}\Lambda_{d,2}\cdots\Lambda_{d,n}$ and where $q_{n}\geq\rho.$

Now we set (for $d,n\geq 1$ and $1\leq k\leq n$)

$\displaystyle h\coloneqq h_{d}$
$\displaystyle\coloneqq\frac{1}{1-\lambda_{1,1}\cdots\lambda_{1,d}\,q_{0}}$
$\displaystyle f_{k}\coloneqq f_{k,d}$
$\displaystyle\coloneqq\frac{1}{\left(1-\frac{\mu_{k,1}}{\lambda_{k,1}}w\right)\cdots\left(1-\frac{\mu_{k,d}}{\lambda_{k,d}}w\right)\cdot\left(1-\frac{\lambda_{k+1,1}\cdots\lambda_{k+1,d}}{\mu_{k,1}\cdots\mu_{k,d}}q_{k}\right)}$
$\displaystyle g_{n}\coloneqq g_{n,d}$
$\displaystyle\coloneqq\frac{1-\frac{\lambda_{n+1,1}\cdots\lambda_{n+1,d}}{\mu_{n,1}\cdots\mu_{n,d}}q_{n+1}}{1-\frac{q_{n+1}}{\mu_{n,1}\cdots\mu_{n,d}}}$

and note that one has the following crude form of the generating functions.

######  Lemma 2.4.

For $d,n\geq 1,$ we have that

$\displaystyle D_{d,n}$ $\displaystyle=\operatorname*{\Omega}_{\geq}h\cdot
f_{1}\cdots f_{n}\cdot g_{n}$ $\displaystyle D_{d,n}^{(\rho)}$
$\displaystyle=\operatorname*{\Omega}_{\geq}h\cdot f_{1}\cdots f_{n}\cdot
g_{n}\left(\frac{q_{n}}{\mu_{n,1}\cdots\mu_{n,d}}\right)^{\rho}.$

###### Proof.

The proof follows, mutatis mutandis, as in the proofs of Propositions 2.1 and
2.2 in [APR01b]. ∎

###  2.4. Preliminary Results

In order to prove Theorem 2.3, we must be able to apply some elimination
formulae in order to simplify the expression for $D_{d,n}$ given in Lemma 2.4.
We’ll first do this in the case where $n=1.$

######  Lemma 2.5.

For $d\geq 1,$ we have that

$D_{d,1}(q_{0},q_{1};w)=\frac{F_{d}(q_{0},w)}{(1-q_{0})(1-q_{0}w)\cdots(1-q_{0}w^{d})\cdot(1-q_{0}q_{1}w^{d})},$

where $F_{d}(q_{0},w)\in\mathbb{Z}[q_{0},w]$ is a polynomial of degree $d-1$
in $q_{0}$ that is given by $F_{1}(q_{0},w)=1$ and

$F_{d}(q_{0},w)=\frac{(1-q_{0}w^{d})F_{d-1}(q_{0},w)-w(1-q_{0})F_{d-1}(q_{0}w,w)}{1-w}.$

###### Proof.

Proceed by induction on $d$. The case $d=1$ corresponds to classical
partitions with at most three parts. We may compute the generating function
for such partitions using MacMahon’s partition analysis: we have

$\displaystyle D_{1,1}(q_{0},q_{1};w)$
$\displaystyle=\operatorname*{\Omega}_{\geq}\sum\limits_{a_{0},a_{1},b_{1}\geq
0}\lambda_{1}^{a_{0}-b_{1}}\mu_{1}^{b_{1}-a_{1}}q_{0}^{a_{0}}w^{b_{1}}q_{1}^{a_{1}}=\operatorname*{\Omega}_{\geq}\sum\limits_{a_{0},a_{1},b_{1}\geq
0}(\lambda_{1}q_{0})^{a_{0}}\left(\frac{\mu_{1}}{\lambda_{1}}w\right)^{b_{1}}\left(\frac{q_{1}}{\mu_{1}}\right)^{a_{1}}$
$\displaystyle=\operatorname*{\Omega}_{\geq}\frac{1}{(1-\lambda_{1}q_{0})(1-\lambda_{1}^{-1}\mu_{1}w)(1-\mu_{1}^{-1}q_{1})}=\operatorname*{\Omega}_{\geq}\frac{1}{(1-q_{0})(1-\mu_{1}q_{0}w)(1-\mu_{1}^{-1}q_{1})}$
$\displaystyle=\frac{1}{(1-q_{0})(1-q_{0}w)(1-q_{0}q_{1}w)},$

by applying (3) to successively eliminate $\lambda_{1}$ and $\mu_{1}.$ Hence,
$F_{1}=1.$

Now suppose that the conclusion holds for $d-1.$ Then by Lemma 2.4 (and the
induction hypothesis) we have

$\displaystyle D_{d,1}(q_{0},q_{1};w)$
$\displaystyle=\operatorname*{\Omega}_{\geq}\frac{1}{(1-\lambda_{1}\cdots\lambda_{d}\,q_{0})\cdot\left(1-\lambda_{1}^{-1}\mu_{1}w\right)\cdots\left(1-\lambda_{d}^{-1}\mu_{d}w\right)\cdot\left(1-\mu_{1}^{-1}\cdots\mu_{d}^{-1}q_{1}\right)}$
$\displaystyle=\operatorname*{\Omega}_{\geq}\frac{D_{d-1,1}(\lambda_{d}q_{0},\mu_{d}^{-1}q_{1};w)}{1-\lambda_{d}^{-1}\mu_{d}w}$
$\displaystyle=\operatorname*{\Omega}_{\geq}\frac{F_{d-1}(\lambda_{d}q_{0},w)}{(1-\lambda_{d}q_{0})\cdots(1-\lambda_{d}q_{0}w^{d-1})(1-\lambda_{d}\mu_{d}^{-1}q_{0}q_{1}w^{d-1})(1-\lambda_{d}^{-1}\mu_{d}w)}.$

Write $F_{d-1}$ as a polynomial in $q_{0}$ with coefficients in
$\mathbb{Z}[w]$ as

$F_{d-1}(q_{0},w)=\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}.$

Then

$\displaystyle D_{d,1}(q_{0},q_{1};w)$
$\displaystyle=\operatorname*{\Omega}_{\geq}\frac{\sum\limits_{i=0}^{d-2}a_{i}(w)(\lambda_{d}q_{0})^{i}}{(1-\lambda_{d}q_{0})\cdots(1-\lambda_{d}q_{0}w^{d-1})(1-\frac{\lambda_{d}}{\mu_{d}}q_{0}q_{1}w^{d-1})(1-\frac{\mu_{d}}{\lambda_{d}}w)}$
$\displaystyle=\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}\operatorname*{\Omega}_{\geq}\frac{\lambda_{d}^{i}}{(1-\lambda_{d}q_{0})\cdots(1-\lambda_{d}q_{0}w^{d-1})(1-\frac{\lambda_{d}}{\mu_{d}}q_{0}q_{1}w^{d-1})(1-\frac{\mu_{d}}{\lambda_{d}}w)}$
$\displaystyle=\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}\operatorname*{\Omega}_{\geq}\frac{\lambda_{d}^{i}}{(1-\lambda_{d}q_{0})\cdots(1-\lambda_{d}q_{0}w^{d-1})(1-\lambda_{d}^{-1}w)(1-q_{0}q_{1}w^{d})},$

by (3). Rearranging,

$\displaystyle D_{d,1}(q_{0},q_{1};w)$
$\displaystyle=\frac{\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}}{1-q_{0}q_{1}w^{d}}\operatorname*{\Omega}_{\geq}\frac{\lambda_{d}^{i}}{(1-\lambda_{d}q_{0})\cdots(1-\lambda_{d}q_{0}w^{d-1})(1-\lambda_{d}^{-1}w)}$
$\displaystyle=\frac{\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}}{(1-q_{0}q_{1}w^{d})(1-w)}\left[\frac{1}{(1-q_{0})\cdots(1-q_{0}w^{d-1})}-\frac{w^{i+1}}{(1-q_{0}w)\cdots(1-q_{0}w^{d})}\right]$
$\displaystyle=\frac{1}{(1-q_{0})\cdots(1-q_{0}w^{d})(1-q_{0}q_{1}w^{d})}\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}\left[\frac{1-q_{0}w^{d}}{1-w}-\frac{w^{i+1}(1-q_{0})}{1-w}\right]$

by applying Proposition 2.2. Thus, we have shown that

$\displaystyle F_{d}(q_{0},w)$
$\displaystyle=\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}\left[\frac{1-q_{0}w^{d}}{1-w}-\frac{w^{i+1}(1-q_{0})}{1-w}\right]$
$\displaystyle=\frac{1-q_{0}w^{d}}{1-w}\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}-\frac{w}{1-w}(1-q_{0})\sum\limits_{i=0}^{d-2}a_{i}(w)q_{0}^{i}w^{i}$
$\displaystyle=\frac{(1-q_{0}w^{d})F_{d-1}(q_{0},w)-w(1-q_{0})F_{d-1}(q_{0}w,w)}{1-w}.$

To complete the proof, note that $F_{d}(q_{0},w)$ is indeed a polynomial since
$w=1$ is a root of the numerator given above. ∎

Lemma 2.5 will serve as the base case of an induction argument in the proof of
Theorem 2.3. We will also need the following results in order to complete that
proof.

######  Lemma 2.6.

For $k\geq 1,$ and $y_{1},\ldots,y_{k},z\neq 0$ distinct elements from an
appropriate field, define

$p(\mathbf{y};z)\coloneqq\prod_{i=1}^{k}\left(1-\frac{y_{i}}{z}\right),$

where $\mathbf{y}=(y_{1},\ldots,y_{k}).$ For $i\leq j\leq k$ define

$p_{j}(\mathbf{y};z)\coloneqq\prod_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{k}\left(1-\frac{y_{i}}{z}\right)^{-1}.$

Then we have

$\frac{1}{p(\mathbf{y};z)}=\sum_{j=1}^{k}\frac{p_{j}(\mathbf{y};y_{j})}{1-(y_{j}/z)}.$

###### Proof.

See [APR01b, Lemma 2.1]. ∎

######  Lemma 2.7.

Let $d,n\geq 1$ and $\rho\geq 0.$ Then

$D_{d,n}^{(\rho)}(q_{0},\ldots,q_{n};w)=(q_{0}\cdots
q_{n})^{\rho}w^{dn\rho}D_{d,n}(q_{0},\ldots,q_{n};w).$

###### Proof.

The above equality is given by the following bijection. Given an arbitrary
$d$-fold partition diamond, adding $\rho$ to each part assigns the partition a
unique $d$-fold partition diamond with smallest part at least $\rho$.
Conversely, given an arbitrary $d$-fold partition diamond with smallest part
at least $\rho$, subtracting $\rho$ from each part assigns the partition a
unique $d$-fold partition partition. Thus set of $d$-fold partition diamonds
and the set of $d$-fold partition diamonds with smallest part at least $\rho$
are in bijection, which is encoded by the above equality on their generating
functions; this proves the result. ∎

###  2.5. Proof of Theorem 2.3

To prove Theorem 2.3, we follow the same approach as in the proof of Theorem
2.1 of [APR01b].

###### Proof.

We proceed by induction on $n.$ The base case (for $n=1$) has already been
discussed in Lemma 2.5. Now, suppose the theorem holds for $n$ and note that
by Lemma 2.4 we have

$\displaystyle D_{n+1}=$ $\displaystyle
D_{d,n+1}(q_{0},\ldots,q_{n};w)=\operatorname*{\Omega}_{\geq}h\cdot
f_{1}\cdots f_{n+1}\cdot g_{n+1}$ $\displaystyle=$
$\displaystyle\operatorname*{\Omega}_{\geq}h\cdot f_{1}\cdots
f_{n-1}\frac{1}{\left(1-\frac{\mu_{n,1}}{\lambda_{n,1}}w\right)\cdots\left(1-\frac{\mu_{n,d}}{\lambda_{n,d}}w\right)}\cdot\frac{1}{\left(1-\frac{\lambda_{n+1,1}\cdots\lambda_{n+1,d}}{\mu_{n,1}\cdots\mu_{n,d}}q_{n}\right)}$
$\displaystyle\cdot\frac{1}{\left(1-\frac{\mu_{n+1,1}}{\lambda_{n+1,1}}w\right)\cdots\left(1-\frac{\mu_{n+1,d}}{\lambda_{n+1,d}}w\right)}\cdot\frac{1}{\left(1-\frac{1}{\mu_{n+1,1}\cdots\mu_{n+1,d}}q_{n+1}\right)}.$

Now, we apply Lemma 2.5 to the last $d+2$ factors to eliminate
$\lambda_{n+1,1},\ldots\lambda_{n+1,d}$ and $\mu_{n+1,1},\ldots\mu_{n+1,d}$
and find that

$\displaystyle D_{n+1}=$ $\displaystyle\operatorname*{\Omega}_{\geq}h\cdot
f_{1}\cdots
f_{n-1}\frac{1}{\left(1-\frac{\mu_{n,1}}{\lambda_{n,1}}w\right)\cdots\left(1-\frac{\mu_{n,d}}{\lambda_{n,d}}w\right)}D_{1}\left(\frac{q_{n}}{\mu_{n,1}\dots\mu_{n,d}},q_{n+1};w\right)$
$\displaystyle=$ $\displaystyle\operatorname*{\Omega}_{\geq}h\cdot f_{1}\cdots
f_{n-1}\frac{1}{\left(1-\frac{\mu_{n,1}}{\lambda_{n,1}}w\right)\cdots\left(1-\frac{\mu_{n,d}}{\lambda_{n,d}}w\right)}\cdot\frac{F_{d}\left(\frac{q_{n}}{\mu_{n,1}\cdots\mu_{n,d}},w\right)}{p(\mathbf{y};\mu_{n,1}\cdots\mu_{n,d})},$

where we set
$\mathbf{y}=(y_{1},\ldots,y_{d+2})\coloneqq(q_{n},q_{n}w,\dots,q_{n}w^{d},q_{n}q_{n+1}w^{d}).$
Then, applying Lemma 2.6 and substituting
$F_{d}\left(\frac{q_{n}}{\mu_{n,1}\cdots\mu_{n,d}},w\right)=\sum_{i=0}^{d-1}a_{i}(w)\frac{q_{n}^{i}}{(\mu_{n,1}\cdots\mu_{n,d})^{i}}$
(where $a_{i}(w)\in\mathbb{Z}[w]$ depends on $d$) gives

$\displaystyle D_{n+1}$ $\displaystyle=\operatorname*{\Omega}_{\geq}h\cdot
f_{1}\cdots
f_{n-1}\frac{F_{d}\left(\frac{q_{n}}{\mu_{n,1}\cdots\mu_{n,d}},w\right)}{\left(1-\frac{\mu_{n,1}}{\lambda_{n,1}}w\right)\cdots\left(1-\frac{\mu_{n,d}}{\lambda_{n,d}}w\right)}\cdot\sum_{j=1}^{d+2}\frac{p_{j}(\mathbf{y};y_{j})}{1-\frac{y_{j}}{\mu_{n,1}\cdots\mu_{n,d}}}$
$\displaystyle=\sum_{i=0}^{d-1}a_{i}(w)q_{n}^{i}\sum_{j=1}^{d+2}p_{j}(\mathbf{y};y_{j})\frac{1}{y_{j}^{i}}\left[\operatorname*{\Omega}_{\geq}\frac{h\cdot
f_{1}\cdots
f_{n-1}}{\left(1-\frac{\mu_{n,1}}{\lambda_{n,1}}w\right)\cdots\left(1-\frac{\mu_{n,d}}{\lambda_{n,d}}w\right)}\cdot\frac{1}{1-\frac{y_{j}}{\mu_{n,1}\cdots\mu_{n,d}}}\cdot\left(\frac{y_{j}}{\mu_{n,1}\cdots\mu_{n,d}}\right)^{i}\right]$
$\displaystyle=\sum_{i=0}^{d-1}a_{i}(w)q_{n}^{i}\sum_{j=1}^{d+2}p_{j}(\mathbf{y};y_{j})\frac{1}{y_{j}^{i}}D_{n}^{(i)}(q_{0},q_{1},\dots,q_{n-1},y_{j};w)$
$\displaystyle=\sum_{i=0}^{d-1}a_{i}(w)Q_{n}^{i}w^{din}\sum_{j=1}^{d+2}p_{j}(\mathbf{y};y_{j})D_{n}(q_{0},q_{1},\dots,q_{n-1},y_{j};w)$
$\displaystyle=F_{d}(Q_{n}w^{dn},w)\sum_{j=1}^{d+2}p_{j}(\mathbf{y};y_{j})D_{n}(q_{0},q_{1},\dots,q_{n-1},y_{j};w),$

where we have used Lemma 2.4 and Lemma 2.7. By the induction hypothesis, we
may observe that

$D_{n}(q_{0},q_{1},\dots,q_{n-1},y_{j};w)=\frac{1-Q_{n}w^{dn}}{1-Q_{n-1}y_{j}w^{dn}}D_{n}(q_{0},q_{1},\dots,q_{n-1},q_{n};w),$

and thus we have

$\displaystyle D_{n+1}=$ $\displaystyle
F_{d}(Q_{n}w^{dn},w)(1-Q_{n}w^{dn})D_{n}(q_{0},q_{1},\dots,q_{n-1},q_{n};w)\sum_{j=1}^{d+2}\frac{p_{j}(\mathbf{y};y_{j})}{1-(q_{0}\dots
q_{n-1})y_{j}w^{dn}}$ $\displaystyle=$ $\displaystyle
F_{d}(Q_{n}w^{dn},w)(1-Q_{n}w^{dn})D_{n}(q_{0},q_{1},\dots,q_{n-1},q_{n};w)\frac{1}{p(\mathbf{y};\frac{q_{n}}{Q_{n}w^{dn}})}$
$\displaystyle=$ $\displaystyle
F_{d}(Q_{n}w^{dn},w)(1-Q_{n}w^{dn})D_{n}(q_{0},q_{1},\dots,q_{n-1},q_{n};w)$
$\displaystyle\cdot\frac{1}{(1-Q_{n}w^{dn})(1-Q_{n}w^{dn+1})\cdots(1-Q_{n}w^{dn+d})(1-Q_{n+1}w^{dn+d})}$

by Lemma 2.6 with $z=(Q_{n-1}w^{dn})^{-1}.$ Thus we have shown that

$D_{n+1}=\frac{F_{d}(Q_{n}w^{dn},w)}{(1-Q_{n}w^{dn+1})\cdots(1-Q_{n}w^{dn+d})(1-Q_{n+1}w^{dn+d})}D_{n},$

which gives the desired result. ∎

###  2.6. Proof of Theorems 1.1 and 1.2

Now, Theorems 1.1 and 1.2 can be obtained as corollaries of Theorem 2.3.

###### Proof of Theorems 1.1 and 1.2.

Theorem 1.1 comes from applying Theorem 2.3 to

$\sum_{n=0}^{\infty}r_{d}(n)q^{n}=\lim_{n\rightarrow\infty}D_{d,n}(q,q,\ldots,q;q).$

To prove Theorem 1.2, we first apply Theorem 2.3 to obtain

$\sum_{n=0}^{\infty}s_{d}(n)q^{n}=\lim_{n\rightarrow\infty}D_{d,n}(q,q,\ldots,q;1)=\prod_{n=1}^{\infty}\frac{F_{d}(q^{n},1)}{(1-q^{n})^{d+1}},$

so we must show that $F_{d}(q,1)=A_{d}(q).$ Since $F_{1}(q,1)=A_{1}(q)=1,$ it
suffices to show that $F_{d}(q,1)$ and $A_{d}(q)$ satisfy the same recurrence
(2).

Recalling the recurrence from Lemma 2.5 and writing
$F_{d-1}(q,w)=\sum_{i=0}^{d-2}a_{i}(w)q^{i}$ gives

$\displaystyle F_{d}(q,w)(1-w)$
$\displaystyle=(1-qw^{d})F_{d-1}(q,w)-w(1-q)F_{d-1}(qw,w)$
$\displaystyle=(1-qw^{d})\sum_{i=0}^{d-2}a_{i}(w)q^{i}-w(1-q)\sum_{i=0}^{d-2}a_{i}(w)q^{i}w^{i}$
$\displaystyle=\sum_{i=0}^{d-2}a_{i}(w)q^{i}\left[1-qw^{d}-w^{i+1}+qw^{i+1}\right]$
$\displaystyle=(1-w)\sum_{i=0}^{d-2}a_{i}(w)q^{i}\left[(1+w+\cdots+w^{i})+q(w^{i+1}+\cdots+w^{d-1}))\right].$

Finally, dividing by $w-1$ and then setting $w=1$ gives

$\displaystyle F_{d}(q,1)$
$\displaystyle=\sum_{i=0}^{d-2}a_{i}(1)q^{i}\left[(i+1)+q(d-i-1))\right]$
$\displaystyle=\sum_{i=0}^{d-2}a_{i}(1)q^{i}\left[1+(d-1)q+i(1-q)\right]$
$\displaystyle=(1+(d-1)q)F_{d-1}(q,1)+q(1-q)F^{\prime}_{d-1}(q,1),$

as desired. ∎

##  3\. Ramanujan–Like Congruences Satisfied by $s_{d}(n)$

We begin this section by noting the following corollary of Theorem 1.2, which
gives an alternate form of the generating function for $s_{d}(n)$.

######  Corollary 3.1.

We have

$\sum_{n=0}^{\infty}s_{d}(n)q^{n}=\prod_{n=1}^{\infty}\left(\sum_{j=0}^{\infty}(j+1)^{d}q^{jn}\right).$

###### Proof.

This follows immediately from Theorem 1.2, together with the following
identity proved by Euler in [Eul68].

$\sum_{j=0}^{\infty}(j+1)^{d}q^{jn}=\frac{A_{d}(q^{n})}{(1-q^{n})^{d+1}}.$

∎

######  Remark.

One can view Corollary 3.1 combinatorially as follows. In order to build an
arbitrary Schmidt type $d$-fold partition diamond, one must pick the
difference, $j,$ between $a_{n-1}$ and $a_{n}$ for each positive integer $n.$
This completely determines all of the linking nodes $a_{0},a_{1},\ldots,$ and
the $q^{jn}$ factors keep track of their contribution. There are then
$(j+1)^{d}$ options for the nodes $b_{n,1},\ldots,b_{n,d}$ (since there are
$j+1$ possibilities for each of those nodes).

We utilize this new view of the generating function for $s_{d}(n)$ to easily
prove a variety of arithmetic properties satisfied by these functions. Prior
to doing so, we remind the reader of two well–known results which will also be
helpful below:

######  Lemma 3.2.

(Euler’s Pentagonal Number Theorem) We have

$\prod_{m=1}^{\infty}(1-q^{m})=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n+1)/2}.$

###### Proof.

See [Hir17, Eq. (1.6.1)]. ∎

######  Lemma 3.3.

(Jacobi) We have

$\prod_{m=1}^{\infty}(1-q^{m})^{3}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1)q^{n(n+1)/2}.$

###### Proof.

See [Hir17, Eq. (1.7.1)]. ∎

We now prove the following theorem which yields an infinite family of
Ramanujan–like congruences modulo arbitrarily large powers of 2.

######  Theorem 3.4.

For all $d\geq 1$ and all $n\geq 0,$ $s_{d}(2n+1)\equiv 0\pmod{2^{d}}.$

###### Proof.

For fixed $d\geq 1,$

$\displaystyle\sum_{n=0}^{\infty}s_{d}(n)q^{n}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\left(1+2^{d}q^{n}+3^{d}q^{2n}+4^{d}q^{3n}+5^{d}q^{4n}+6^{d}q^{5n}+\dots\right)$
$\displaystyle\equiv$
$\displaystyle\prod_{n=1}^{\infty}\left(1+0q^{n}+3^{d}q^{2n}+0q^{3n}+5^{d}q^{4n}+0q^{5n}+\dots\right)\pmod{2^{d}}$
$\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\left(1+3^{d}q^{2n}+5^{d}q^{4n}+\dots\right).$

In the penultimate line above, we have used the fact that $2^{d}\,|\,(2j)^{d}$
for all $j\geq 1.$ The last expression above is a function of $q^{2},$ and
this immediately implies the result. ∎

We note, in passing, that the $d=2$ case of Theorem 3.4 was proven by Andrews
and Paule [AP22, Corollary 2].

We next prove the following overarching lemma that will provide us with the
machinery needed to prove infinite families of divisibility properties
satisfied by $s_{d}(n).$

######  Lemma 3.5.

Let $k$ and $r$ be nonnegative integers and let $m\geq 2$. For all $n\geq 0,$
$s_{\phi(m)k+r}(n)\equiv s_{r}(n)\pmod{m}$ where $\phi(m)$ is Euler’s totient
function.

###### Proof.

Note that

$\displaystyle\sum_{n=0}^{\infty}s_{\phi(m)k+r}(n)q^{n}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\left(1+2^{\phi(m)k+r}q^{n}+3^{\phi(m)k+r}q^{2n}+4^{\phi(m)k+r}q^{3n}\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \
\left.{}+5^{\phi(m)k+r}q^{4n}+6^{\phi(m)k+r}q^{5n}+\dots\right)$
$\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\left(1+(2^{\phi(m)})^{k}\cdot
2^{r}q^{n}+(3^{\phi(m)})^{k}\cdot 3^{r}q^{2n}+(4^{\phi(m)})^{k}\cdot
4^{r}q^{3n}\right.$ $\displaystyle\ \ \ \ \ \ \ \ \ \
\left.{}+(5^{\phi(m)})^{k}\cdot 5^{r}q^{4n}+(6^{\phi(m)})^{k}\cdot
6^{r}q^{5n}+\dots\right)$ $\displaystyle\equiv$
$\displaystyle\prod_{n=1}^{\infty}\left(1+2^{r}q^{n}+3^{r}q^{2n}+4^{r}q^{3n}+5^{r}q^{4n}+6^{r}q^{5n}+\dots\right)\pmod{m}$
$\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}s_{r}(n)q^{n}.$

The penultimate line above follows from Euler’s generalization of Fermat’s
Little Theorem. ∎

We now transition to a consideration of various families of congruences modulo
5. We first prove an infinite family of such congruences thanks to Lemma 3.5
which uses the function $s_{1}(n)$ as its starting point.

######  Theorem 3.6.

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

$s_{4k+1}(5n+2)\equiv s_{4k+1}(5n+3)\equiv s_{4k+1}(5n+4)\equiv 0\pmod{5}.$

###### Proof.

Note that

$\displaystyle\sum_{n=0}^{\infty}s_{1}(n)q^{n}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{1}{(1-q^{n})^{2}}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{(1-q^{n})^{3}}{(1-q^{n})^{5}}$
$\displaystyle\equiv$
$\displaystyle\prod_{n=1}^{\infty}\frac{(1-q^{n})^{3}}{(1-q^{5n})}\pmod{5}$
$\displaystyle=$
$\displaystyle\left(\sum_{j=0}^{\infty}(-1)^{j}(2j+1)q^{j(j+1)/2}\right)\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n})}$

thanks to Lemma 3.3. Since $\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n})}$ is a
function of $q^{5}$, the above product will satisfy a congruence modulo 5 in
an arithmetic progression of the form $5n+r$, for $0\leq r\leq 4,$ if and only
if such a congruence is satisfied by

$\left(\sum_{j=0}^{\infty}(-1)^{j}(2j+1)q^{j(j+1)/2}\right).$

Since $5n+2$ and $5n+4$ are never triangular numbers, this immediately tells
us that, for all $n\geq 0,$

$s_{1}(5n+2)\equiv s_{1}(5n+4)\equiv 0\pmod{5}.$

Moreover, we know that $5n+3=j(j+1)/2$ if and only if $j\equiv 2\pmod{5},$ and
in such cases, $2j+1\equiv 0\pmod{5}.$ Because of the presence of the factor
$2j+1$ in the sum above, we then see that, for all $n\geq 0,$

$s_{1}(5n+3)\equiv 0\pmod{5}$

since $2\cdot 2+1=5\equiv 0\pmod{5}.$ The remainder of the proof immediately
follows from Lemma 3.5. ∎

We note, in passing, that Baruah and Sarmah [BS13, Theorem 5.1, equation
(5.4)] also provided a proof of the $k=0$ case of Theorem 3.6.

We next consider the following modulo 5 congruences, with a focus on the case
$d=2.$

######  Theorem 3.7.

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

$s_{4k+2}(25n+23)\equiv 0\pmod{5}.$

###### Proof.

We begin by noting that

$\displaystyle\sum_{n=0}^{\infty}s_{2}(n)q^{n}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{1+q^{n}}{(1-q^{n})^{3}}$
$\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{1-q^{2n}}{(1-q^{n})^{4}}$
$\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{(1-q^{2n})(1-q^{n})}{(1-q^{n})^{5}}$
$\displaystyle\equiv$
$\displaystyle\prod_{n=1}^{\infty}\frac{(1-q^{2n})(1-q^{n})}{(1-q^{5n})}\pmod{5}$
$\displaystyle=$
$\displaystyle\left(\sum_{j,k=-\infty}^{\infty}q^{2(j(3j+1)/2)+k(3k+1)/2}\right)\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n})}$

thanks to Lemma 3.2. At this point, we consider those terms of the form
$q^{25n+23}$ in the power series representation of the last expression above.
There are no terms of the form $q^{25n+8},q^{25n+13},q^{25n+18},$ or
$q^{25n+23}$ in the double sum above, although there are terms of the form
$q^{25n+3}.$ Thus, once we multiply the double sum above by

$\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n})}=\sum_{n=0}^{\infty}p(n)q^{5n},$ (5)

we see that the only way to obtain a term of the form $q^{25n+23}$ which can
contribute to $s_{2}(25n+23)$ modulo 5 is to multiply by a term of the form
$q^{5(5m+4)}$ from the power series representation of (5). This will then
contribute, modulo 5, to the value of $s_{2}(25n+23)$ by multiplying by the
value $p(5m+4).$ Thanks to Ramanujan’s well–known result that, for all $m\geq
0,$ $p(5m+4)\equiv 0\pmod{5},$ we then know that, for all $n\geq 0,$
$s_{2}(25n+23)\equiv 0\pmod{5}.$ The theorem then follows thanks to Lemma 3.5.
∎

We can also show that the functions $s_{4k+3}$ satisfy a rich set of
congruences modulo 5 via the following:

######  Theorem 3.8.

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

$s_{4k+3}(5n+2)\equiv s_{4k+3}(5n+4)\equiv 0\pmod{5}.$

###### Proof.

We begin with the following generating function manipulations:

$\displaystyle\sum_{n=0}^{\infty}s_{3}(n)q^{n}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{1+4q^{n}+q^{2n}}{(1-q^{n})^{4}}$
$\displaystyle\equiv$
$\displaystyle\prod_{n=1}^{\infty}\frac{1-q^{n}+q^{2n}}{(1-q^{n})^{4}}\pmod{5}$
$\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{1+(-q^{n})+(-q^{n})^{2}}{(1-q^{n})^{4}}$
$\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{1-(-q^{n})^{3}}{(1-(-q^{n}))(1-q^{n})^{4}}$
$\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{1+q^{3n}}{(1+q^{n})(1-q^{n})^{4}}$
$\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\frac{(1-q^{6n})(1-q^{n})^{2}}{(1-q^{3n})(1-q^{2n})(1-q^{n})^{5}}$
$\displaystyle\equiv$
$\displaystyle\prod_{n=1}^{\infty}\frac{(1-q^{6n})(1-q^{n})^{2}}{(1-q^{3n})(1-q^{2n})}\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n})}\pmod{5}$

We now consider the power series representation of

$F(q):=\prod_{n=1}^{\infty}\frac{(1-q^{6n})(1-q^{n})^{2}}{(1-q^{3n})(1-q^{2n})},$
(6)

which turns out to be a well–known modular form; indeed, it appears as the
seventh modular form in Mersmann’s list of the 14 primitive eta–products which
are holomorphic modular forms of weight 1/2. See [BvdGHZ08, page 30] and
[OEI23, Entry A089812] for additional details. After some elementary
calculations (for example, by noting that $qF(q^{8})$ appears in [LO13,
Theorem 1.2]), we find that

$F(q)=\sum_{t=0}^{\infty}q^{t(t+1)/2}-3\sum_{t=0}^{\infty}q^{(3t+1)(3t+2)/2}.$

As noted in the proof of Theorem 3.8, $5n+2$ and $5n+4$ can never be
triangular numbers. Since $\frac{1}{2}(3t+1)(3t+2)$ is always triangular, we
conclude that $s_{3}(5n+2)\equiv 0\pmod{5}$ and $s_{3}(5n+4)\equiv 0\pmod{5}$.
The case where $k>0$ follows from Lemma 3.5. ∎

Related to the above, we can also prove the following additional congruence
family modulo 5.

######  Theorem 3.9.

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

$s_{4k+3}(25n+23)\equiv 0\pmod{5}.$

###### Proof.

As with the proof of Theorem 3.8 above, we begin by considering $s_{3}(n)$.
Using the notation from above, recall that

$\displaystyle\sum_{n=0}^{\infty}s_{3}(n)q^{n}$ $\displaystyle\equiv$
$\displaystyle\left(\sum_{t=0}^{\infty}q^{t(t+1)/2}-3\sum_{t=0}^{\infty}q^{(3t+1)(3t+2)/2}\right)\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n})}\pmod{5}$
(7) $\displaystyle=$
$\displaystyle\left(\sum_{t=0}^{\infty}q^{t(t+1)/2}-3\sum_{t=0}^{\infty}q^{(3t+1)(3t+2)/2}\right)\left(\sum_{n=0}^{\infty}p(n)q^{5n}\right).$

We then consider ways that $q^{25n+23}$ can arise when (7) is expanded. We
note that no numbers of the form $25n+8,25n+13,25n+18,$ or $25n+23$ can be
represented as a triangular number. Thus, the only way to obtain a term of the
form $q^{25n+23}$ in (7) is to multiply by a term of the form
$p(5m+4)q^{5(5m+4)}$ (in the same way as was discussed in the proof of Theorem
3.7). This implies that, for all $n\geq 0,$ $s_{3}(25n+23)\equiv 0\pmod{5}.$
The full result follows from Lemma 3.5. ∎

We close this section with one last infinite family of congruences, this time
modulo 11.

######  Theorem 3.10.

For all $n\geq 0,$

$s_{10k+1}(121n+111)\equiv 0\pmod{11}.$

###### Proof.

In the work of Gordon [Gor83, Theorem 2], we set $k=2$ and $r=2,$ so that
$\alpha=1.$ We then see that, if $24n\equiv 2\pmod{11^{2}},$ then $12n\equiv
1\pmod{121}$ or $n\equiv 111\pmod{121}$. Gordon’s work then implies that, for
all $n\geq 0,$ $s_{1}(121n+111)\pmod{11}$. The full result then follows from
Lemma 3.5.

∎

##  4\. Closing Thoughts

We close this work with the following set of conjectured congruence families
modulo 7:

######  Conjecture 4.1.

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

$\displaystyle s_{6k+1}(49n+17)$ $\displaystyle\equiv$ $\displaystyle
s_{6k+2}(49n+17)\ \equiv\ 0\pmod{7},$ $\displaystyle s_{6k+1}(49n+31)$
$\displaystyle\equiv$ $\displaystyle s_{6k+2}(49n+31)\ \equiv\ 0\pmod{7},$
$\displaystyle s_{6k+1}(49n+38)$ $\displaystyle\equiv$ $\displaystyle
s_{6k+2}(49n+38)\ \equiv\ 0\pmod{7},$ $\displaystyle s_{6k+1}(49n+45)$
$\displaystyle\equiv$ $\displaystyle s_{6k+2}(49n+45)\ \equiv\ 0\pmod{7}.$

We would be very interested to see elementary proofs of the above results.

In addition, Theorems 3.4 and 3.8 provide three Ramanujan congruences for
$s_{3}(n),$ namely

$\displaystyle s_{3}(2n+1)$ $\displaystyle\equiv 0\pmod{2}$ $\displaystyle
s_{3}(5n+2)$ $\displaystyle\equiv 0\pmod{5}$ $\displaystyle s_{3}(5n+4)$
$\displaystyle\equiv 0\pmod{5},$

by taking $d=3$ and $k=0,$ respectively. We would like to know if there are
any other Ramanujan congruences for $s_{3}(n)$ and if any exist for $d\geq 4.$
We are especially interested in knowing if $s_{d}(n)$ satisfies only finitely
many Ramanujan congruences for fixed $d\geq 3.$

## References

  * [AP22] George E. Andrews and Peter Paule.  MacMahon’s partition analysis XIII: Schmidt type partitions and modular forms.  J. Number Theory, 234:95–119, 2022.
  * [APR01a] George E. Andrews, Peter Paule, and Axel Riese.  MacMahon’s partition analysis: the Omega package.  European J. Combin., 22(7):887–904, 2001.
  * [APR01b] George E. Andrews, Peter Paule, and Axel Riese.  MacMahon’s partition analysis. VIII. Plane partition diamonds.  Adv. in Appl. Math., 27(2-3):231–242, 2001.  Special issue in honor of Dominique Foata’s 65th birthday (Philadelphia, PA, 2000).
  * [BS13] Nayandeep Deka Baruah and Bipul Kumar Sarmah.  Identities and congruences for the general partition and Ramanujan’s tau functions.  Indian J. Pure Appl. Math., 44(5):643–671, 2013.
  * [BvdGHZ08] Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, and Don Zagier.  The 1-2-3 of modular forms.  Universitext. Springer-Verlag, Berlin, 2008.  Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, June 2004.
  * [Eul68] L. Euler.  Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques.  Mémoires de l’académie des sciences de Berlin, pages 83–106, 1768.
  * [Gor83] Basil Gordon.  Ramanujan congruences for $p_{-k}\ ({\rm mod}\,11^{r})$.  Glasgow Math. J., 24(2):107–123, 1983.
  * [Hir17] Michael D. Hirschhorn.  The power of $q$, volume 49 of Developments in Mathematics.  Springer, Cham, 2017.  A personal journey, With a foreword by George E. Andrews.
  * [LO13] Robert J. Lemke Oliver.  Eta-quotients and theta functions.  Adv. Math., 241:1–17, 2013.
  * [Mac04] Percy A. MacMahon.  Combinatory analysis. Vol. I, II (bound in one volume).  Dover Phoenix Editions. Dover Publications, Inc., Mineola, NY, 2004.  Reprint of ıt An introduction to combinatory analysis (1920) and ıt Combinatory analysis. Vol. I, II (1915, 1916).
  * [OEI23] OEIS Foundation Inc.  The On-Line Encyclopedia of Integer Sequences, 2023.  Published electronically at http://oeis.org.


# Sparse Sensing, Communication, and Actuation via Self-Triggered Control
Algorithms

MirSaleh Bahavarnia<EMAIL_ADDRESS>Hossein K. Mousavi<EMAIL_ADDRESS>Nader Motee<EMAIL_ADDRESS>Department of Electrical and Computer Engineering
and Institute for Systems Research, University of Maryland at College Park,
USA  Department of Mechanical Engineering and Mechanics, Lehigh University,
Bethlehem, USA

###### Abstract

We propose a self-triggered control algorithm to reduce onboard processor
usage, communication bandwidth, and energy consumption across a linear time-
invariant networked control system. We formulate an optimal control problem by
penalizing the $\ell_{0}$–measures of the feedback gain and the vector of
control inputs and maximizing the dwell time between the consecutive
triggering times. It is shown that the corresponding $\ell_{1}$–relaxation of
the optimal control problem is feasible and results in a stabilizing feedback
control law with guaranteed performance bounds, while providing a sparse
schedule for collecting samples from sensors, communication with other
subsystems, and activating the input actuators.

††thanks: The corresponding author is M. Bahavarnia.

, ,

##  1 Introduction

The design of feedback control strategies with sparse communication topologies
has been one of the active research areas in networked control systems in
recent years [1, 3, 8, 11], where the control objective is to find a sparse
state feedback gain by minimizing a $\ell_{1}$-regularized, which is usually
quadratic, cost functional. Promoting sparsity in the state feedback gain
results in a reduced communication requirement among the subsystems, which is
practically plausible in networked systems with limited onboard resources
(e.g., short battery life on quadcopters). In this paper, we take further
steps and enhance sparsity by using a self-triggered state feedback control
mechanism and allowing subsystems to sense less frequently, communicate
whenever performance deteriorates below some pre-specified threshold, and
activate their actuators if necessary.

The fundamental idea of the self-triggered control is to reduce the need for
continuous sensing and actuation by incurring some performance loss in a
controlled manner. A performance-preserving condition or Lyapunov-based
stability condition, which is known as the self-triggering condition, is
usually verified to determine whether new updates are necessary or not. In
this approach, the next update time is solely computed based on the current
state information [6, 4]. In [14], the problem of linear time-invariant
networked systems with limited bandwidth on their communication channels is
considered, where the authors develop a self-triggered method to schedule
sampling instances as well as switching times of feedback gains with
$\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ performance guarantees. In [13],
the authors employ ideas from the self-triggered control to develop an
algorithm for a network of multiple agents on how to collect new samples and
compute localized control inputs in order to achieve optimal static deployment
in a given convex region. The authors of [12] show that if the sparsest
control signal for a finite-horizon optimal control is unique, then the
corresponding $\ell_{1}$-regularized optimal control problem recovers the
sparsest solution. They leverage this idea and design a sparse self-triggered
control law for linear time-invariant systems. In [7], the $L^{p}$ control
problems with $0<p<1$ is considered, where the authors obtain a maximum
principle and conditions for the existence of solution for optimal control
problems with $L^{0}$-regularized quadratic costs. In [4], the triggering
times and feedback gains for the quadratic optimal control of linear time-
invariant systems are designed with gauranteed performance bounds.

In this paper, we propose a control algorithm that is based on ideas from
sparse feedback control and self-triggered control design techniques to
achieve higher levels of sparsity, both in time and space, by reducing
sensing, communication, and input actuation requirements with a prescribed
performance bound. To achieve these goals, in Sections 2 and 3, we formulate
an optimal control problem in which subsystems measure their own state
variables and then share them over a (time-varying) communication topology.
Each subsystem receives the state information of other neighboring subsystems
once (at triggering times) and forms its optimal control input using these
samples and apply it for some period of time without updating it. As soon as
the performance deteriorates below some threshold, subsystems trigger and
collect new samples and repeat the process. The control inputs are formed
using state feedback gains that are computed by minimizing a cost function
that is penalized by the $\ell_{0}$-measures of the feedback gain and the
vector of control inputs. In Section 4, we show that this optimal control
problem and its $\ell_{1}$-relaxation are feasible. To prolong the triggering
intervals, where over which new samples will not be collected and feedback
gains stay fixed, our algorithm in Section 5 solves a nonlinear optimization
problem for the next triggering time. Then, using this value, a
$\ell_{1}$-relaxation of the optimal control problem is solved to find a
sparse feedback gain. In the last step, the value of the next triggering time
is corrected using this feedback gain. It is shown that the resulting feedback
control strategy is stabilizing and provides guaranteed performance bounds.
The usefulness of our theoretical findings is verified through extensive
simulations in Section 6. This paper is an outgrowth of [2] with several
updated and new results and simulations.

Notations: The set of real numbers, positive real numbers, positive integer
numbers, and non-negative integer numbers are denoted by $\mathbb{R}$,
$\mathbb{R}_{++}$, $\mathbb{N}$, and $\mathbb{Z}_{+}$, respectively. The
supremum of a subset of $\mathbb{R}$ is denoted by $\sup$. For a function of
time $f(t)$, $f^{\prime}(t)=\partial f(t)/\partial t$. The positive semi-
definiteness and positive-definiteness are denoted by $\succeq 0$ and $\succ
0$, respectively. The identity matrix is represented by $I$. The Euclidean
norm of vector $v$ is denoted by $\|v\|_{2}$. The cardinality ($\ell_{0}$
sparsity measure), $\ell_{1}$ norm, and largest singular value of a matrix $M$
are represented by $\|M\|_{0}$, $\|M\|_{1}$, and $\|M\|$, respectively, where
$\|M\|_{0}$ is identical to its number of nonzero elements and $\|M\|_{1}$ is
the sum of absolute values of its elements. A matrix is called Hurwitz if all
of its eigenvalues have negative real parts. The vectorization and determinant
of a matrix $M$ are denoted by $\mathrm{vec}(M)$ and $\det(M)$, respectively.
The Kronecker product is denoted by $\otimes$.

##  2 Problem Statement

We consider the linear time-invariant (LTI) control systems described by

$\dot{x}(t)=Ax(t)+Bu(t),~{}~{}x(0)=x_{0},$ (1)

where $x\in\mathbb{R}^{n}$ denotes the state vector, $u\in\mathbb{R}^{m}$
represents the control input, and $x_{0}$ is the initial state. To stabilize
system (1) and regularize $x(t)$ toward the origin, we use the sample-and-hold
feedback control laws

$u(t)=u_{k}:=F_{k}\,x(t_{k})=F_{k}\,x_{k},$ (2)

for all $t\in[t_{k},t_{k+1})$ , in which the sequence of time instants
$\\{t_{k}\\}_{k=0}^{\infty}\subset\mathbb{R}_{+}$ are called triggering times,
and $[t_{k},t_{k+1})$ is the $k$’th time interval. The resulting sequence of
inter-execution times is $\\{\delta_{k}\\}_{k=0}^{\infty}$, wherein

$\delta_{k}:=t_{k+1}-t_{k}.$ (3)

The sequence of feedback gains is also denoted by
$\\{F_{k}\\}_{k=1}^{\infty}\subset\mathbb{R}^{m\times n}$. The cost-to-go
$J_{k}(F_{k},\xi;x_{k})$ corresponding to time interval $[t_{k},t_{k}+\xi)$ is

$J_{k}(F_{k},\xi;x_{k}):=\displaystyle\int_{t_{k}}^{t_{k}+\xi}\left({x(t)}^{T}Qx(t)+{u(t)}^{T}Ru(t)\right)~{}dt$
(4)

for all $\xi\in[0,\delta_{k})$ and weight matrices $Q\succ 0$ and $R\succ 0$.

######  Assumption 1.

The pair $(A,B)$ is stabilizable.

Since $Q\succ 0$ and Assumption 1 holds, the LQR problem corresponding to cost
functional (4) has a unique stabilizing solution. For a pre-designed and
stabilizing feedback gain $\tilde{F}$, the infinite-horizon cost value is
given by

$\displaystyle\tilde{J}(x_{0}):=x_{0}^{T}\,\tilde{P}\,x_{0},$ (5)

where $\tilde{P}\succ 0$ is the unique solution to Lyapunov equation

$\left(A+B\tilde{F}\right)^{T}\tilde{P}+\tilde{P}\left(A+B\tilde{F}\right)+Q+\tilde{F}^{T}R\tilde{F}=0.$
(6)

This solution exists because $A+B\tilde{F}$ is Hurwitz and
$Q+\tilde{F}^{T}R\tilde{F}\succ 0$. A popular choice for $\tilde{F}$ is the
standard linear quadratic regulator (LQR). In the traditional controller
design methods, $\tilde{F}$ is usually non-sparse; i.e., most of its elements
are nonzero.

The problem is to design the sequences of inter-execution times
$\left\\{\delta_{k}\right\\}_{k=0}^{\infty}$ and feedback gains
$\left\\{F_{k}\right\\}_{k=0}^{\infty}$ in order to ensure the following
control objectives: (i) sparse sensing both in space and time, (ii) sparse
scheduling of actuators, and (iii) guaranteed stability and closed-loop
performance compared to a well-performing feedback gain $\tilde{F}$.

##  3 Initial Problem Developments

We show how the objectives of the paper can be translated to an optimization
problem. To achieve objective (i), one can maximize the inter-execution time
$\delta_{k}$ and minimize $\|F_{k}\|_{0}$. To realize objective (ii), we may
minimize $\|u_{k}\|_{0}$, which is the number of actuators that is used in
interval $[t_{k},t_{k+1})$. To achieve objective (iii), we consider a
Lyapunov-based performance and stability analysis, which is a well-
investigated methodology in model predictive control (MPC) (e.g., see [4]).
First, the total cost value is given by

$\displaystyle J(x_{0}):=\sum_{k=0}^{\infty}J_{k}(F_{k},\delta_{k};x_{k}).$
(7)

The relative performance loss with respect to the benchmark cost value (5) is

$\displaystyle\nu(x_{0}):=\frac{J(x_{0})-\tilde{J}(x_{0})}{\tilde{J}(x_{0})}.$
(8)

######  Theorem 1.

Suppose that for some given $\alpha>1$, the sequences of triggering times,
$\\{t_{k}\\}_{k=0}^{\infty}$, inter-execution times
$\left\\{\delta_{k}\right\\}_{k=0}^{\infty}$, and feedback gains
$\left\\{F_{k}\right\\}_{k=0}^{\infty}$, inequality

$\displaystyle
J_{k}(F_{k},\xi;x_{k})\leq\alpha\left(V\big{(}x(t_{k})\big{)}-V\big{(}x(t_{k}+\xi)\big{)}\right),$
(9)

holds for every $\xi\in[0,\delta_{k}]$, in which the Lyapunov function
$V:~{}\mathbb{R}^{n}\rightarrow\mathbb{R}$ is defined as

$V(x(t)):=x(t)^{T}\,\tilde{P}\,x(t).$

Then, control law (2) is stabilizing. Moreover, for all
$x_{0}\in\mathbb{R}^{n}$, the relative performance loss $\nu(x_{0})$ satisfies

$\displaystyle\nu(x_{0})\,\leq\,\alpha-1.$ (10)

###### Proof.

Let us define
$S_{N}:=\displaystyle\sum_{k=0}^{N}J_{k}(F_{k},\delta_{k};x_{k}).$ We can use
(9) and the limit of $\xi\rightarrow\delta_{k}$ for each term to get

$\displaystyle S_{N}\,$
$\displaystyle\leq\,\sum_{k=0}^{N}\alpha\left(V(x_{k})-V(x_{k+1})\right)$
$\displaystyle=\alpha(V(x_{0})-V(x_{N}))<\alpha
V(x_{0})=\alpha\tilde{J}(x_{0}).$

Since $\\{S_{N}\\}_{N=0}^{\infty}$ is a nondecreasing sequence that is bounded
from above, it has a limit, that is actually $J(x_{0})$. Moreover,

$\displaystyle
J(x_{0})=\lim_{N\rightarrow\infty}S_{N}\leq\alpha\tilde{J}(x_{0}).$ (11)

A simple reorganization of (11) reveals that (10) holds. ∎

We combine these objectives and formulate the self-triggered sparse optimal
control problem as

$\displaystyle\underset{F_{k},~{}\delta_{k}}{\mathrm{minimize}}~{}~{}~{}~{}~{}~{}-\delta_{k}+\gamma\|F_{k}\|_{0}+\eta\|u_{k}\|_{0}$
(P1)
$\displaystyle\mathrm{subject~{}to}:~{}~{}~{}(\ref{STLQR})~{}~{}\mathrm{and}~{}(\ref{KTLQR}),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathrm{~{}\eqref{PerfoC}}\mathrm{~{}for~{}all~{}}\xi\in[0,\delta_{k}]$

The nonnegative parameters $\gamma$ and $\eta$ adjust the balance between the
levels of spatial and temporal sparsity in terms of sensing and actuation,
respectively.

######  Theorem 2.

Every feasible solution of problem (P1) is a stabilizing control law with
performance guarantee (10).

######  Remark 1.

The method proposed by [6] solely takes advantage of the Lyapunov function
evolutions. However, our proposed problem setup resembles the method proposed
by [4], in which in addition to the Lyapunov functions, a performance-based
condition is utilized.

##  4 Equivalent Formulation, Relaxation, and Feasibility

We show that (P1) can be reformulated as a regularized quadratically-
constrained quadratic program (QCQP) when $\delta_{k}$ is kept fixed. Then, we
suggest appropriate relaxations that make the problem more tractable.

######  Lemma 1.

For all $k\in\mathbb{N}$ and $j\in\mathbb{Z}_{+}$, feedback control law (2)
can be decomposed as

$u(t)=F_{k}N_{k}x_{0}$

for all $t\in[t_{k},t_{k+1})$, where $N_{0}:=I$ and $N_{k}$ is recursively
given by

$\displaystyle N_{k}$ $\displaystyle:=M_{k-1}({\delta}_{k-1})N_{k-1}$

and the matrices $M_{j}$’s are given by

$M_{j}(\xi):=\e^{A\xi}\big{(}I+Z(\xi)BF_{j}\big{)},~{}Z(\xi):=\int_{0}^{\xi}\e^{-A\tau}d\tau.$

###### Proof.

Solving system (1) together with (2) for time interval $[t_{k},t_{k+1})$ gives
us

$\displaystyle x(t)$
$\displaystyle=\e^{A(t-t_{k})}x(t_{k})+\int_{t_{k}}^{t}\e^{A(t-\phi)}BF_{k}x(t_{k})d\phi$
$\displaystyle=\e^{A(t-t_{k})}\left(I+\int_{0}^{t-t_{k}}\e^{A(-\tau)}BF_{k}~{}d\tau\right)x(t_{k})$
$\displaystyle=\e^{A(t-t_{k})}\big{(}I+Z(t-t_{k})BF_{k}\big{)}x(t_{k})$
$\displaystyle=M_{k}(t-t_{k})x(t_{k}).$

Since $x(t)$ is continuous at $t=t_{k+1}$, thus

$x(t_{k+1})=\e^{A\delta_{k}}\big{(}I+Z(\delta_{k})BF_{k}\big{)}x(t_{k})=M_{k}(\delta_{k})x(t_{k}).$

By induction, $x(t_{k})=N_{k}x_{0}$. ∎

We derive a quadratic form for the cost-to-go.

######  Lemma 2.

The $k$’th time interval cost $J_{k}(F_{k},\xi;x_{k})$ can be expressed as the
quadratic term

$J_{k}(F_{k},\xi;x_{k})=x_{k}^{T}Y_{k}(F_{k},\xi)x_{k},$

where the matrix $Y_{k}$ is given by

$Y_{k}(F_{k},\xi):=H_{0}(\xi)+F_{k}^{T}H_{1}(\xi)^{T}+H_{1}(\xi)F_{k}+F_{k}^{T}H_{2}(\xi)F_{k},$

for all $k\in\mathbb{Z}_{+}$ and

$\displaystyle H_{0}(\xi)$
$\displaystyle:=\int_{0}^{\xi}\e^{A^{T}\tau}Q\e^{A\tau}d\tau,$ $\displaystyle
H_{1}(\xi)$
$\displaystyle:=\int_{0}^{\xi}\e^{A^{T}\tau}Q\e^{A\tau}Z(\tau)Bd\tau,$
$\displaystyle H_{2}(\xi)$
$\displaystyle:=\int_{0}^{\xi}\big{(}\e^{A\tau}Z(\tau)B\big{)}^{T}Q\big{(}\e^{A\tau}Z(\tau)B\big{)}d\tau+\xi
R.$

###### Proof.

We can write

$J_{k}(F_{k},\xi;x_{k})=J_{k}^{x}(F_{k},\xi;x_{k})+J_{k}^{u}(F_{k},\xi;x_{k}),$

wherein

$\displaystyle J_{k}^{x}(F_{k},\xi;x_{k})$
$\displaystyle=\int_{t_{k}}^{t_{k}+\xi}x_{k}^{T}M_{k}(t-t_{k})^{T}QM_{k}(t-t_{k})x_{k}dt$
$\displaystyle=x_{k}^{T}\int_{0}^{\xi}M_{k}(\tau)^{T}QM_{k}(\tau)d\tau x_{k}$

and

$J_{k}^{u}(F_{k},\xi;x_{k})=\displaystyle\int_{t_{k}}^{t_{k}+\xi}x_{k}^{T}F_{k}^{T}RF_{k}x_{k}dt=x_{k}^{T}(\xi
F_{k}^{T}RF_{k})x_{k}.$

Then, we get $J_{k}(F_{k},\xi;x_{k})=x_{k}^{T}Y_{k}(F_{k},\xi)x_{k}$, wherein

$Y_{k}(F_{k},\xi)=\int_{0}^{\xi}M_{k}(\tau)^{T}QM_{k}(\tau)d\tau+\xi_{k}F_{k}^{T}RF_{k}.$
(12)

The claim follows by substituting
$M_{k}(\tau)=\e^{A\tau}\big{(}I+Z(\tau)BF_{k}\big{)}$ in the right side of
(12). ∎

When $\delta_{k}$ is kept fixed, the following proposition states an important
property about matrix $H_{2}(\xi)$ which leads to the convex reformulation of
(P1) (in terms of variable $F_{k}$) .

######  Proposition 1.

For all $\xi>0$, $H_{2}(\xi)$ is positive-definite.

###### Proof.

For arbitrary nonzero vector $v\in\mathbb{R}^{m}$, we get

$\displaystyle v^{T}H_{2}(\xi)v$
$\displaystyle=v^{T}\left(\int_{0}^{\xi}\big{(}e^{A\tau}Z(\tau)B\big{)}^{T}Q\big{(}e^{A\tau}Z(\tau)B\big{)}d\tau+\xi
R\right)v$
$\displaystyle=\int_{0}^{\xi}\big{(}e^{A\tau}Z(\tau)Bv\big{)}^{T}Q\big{(}e^{A\tau}Z(\tau)Bv\big{)}d\tau+\xi
v^{T}Rv.$

Since $Q\succ 0$, the last integrand is non-negative. Therefore, the integral
is also non-negative. Since $R\succ 0$, $\xi v^{T}Rv>0$. Thus,
$v^{T}H_{2}(\xi)v>0$ and the conclusion follows. ∎

Now, we can state the following equivalent formulation.

######  Theorem 3.

The optimization problem (P1) is equivalent to

$\displaystyle\underset{f_{k},\delta_{k}}{\mathrm{minimize}}~{}~{}-\delta_{k}+\gamma\|f_{k}\|_{0}+\eta\|(x_{k}^{T}\otimes
I)f_{k}\|_{0}$ (P2)
$\displaystyle\mathrm{subject~{}to}:\mathrm{~{}for~{}all~{}}\xi\in[0,\delta_{k}):$
(13)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{2}f_{k}^{T}P_{1}(\xi)f_{k}+q_{1}(\xi)^{T}f_{k}+r_{1}(\xi)\leq
0,$

where $f_{k}:=\mathrm{vec}(F_{k})$ and

$P_{1}(\xi):=2(x_{k}x_{k}^{T})\otimes\big{(}H_{2}(\xi)+\alpha
B^{T}Z(\xi)^{T}e^{A^{T}\xi}\tilde{P}e^{A\xi}Z(\xi)B\big{)},$
$q_{1}(\xi):=2\mathrm{vec}\bigg{(}\big{(}H_{1}(\xi)^{T}+\alpha
B^{T}Z(\xi)^{T}e^{A^{T}\xi}\tilde{P}e^{A\xi}\big{)}x_{k}x_{k}^{T}\bigg{)},$
$r_{1}(\xi):=x_{k}^{T}\big{(}H_{0}(\xi)+\alpha(e^{A^{T}\xi}\tilde{P}e^{A\xi}-\tilde{P})\big{)}x_{k}.$

###### Proof.

The building block of the proof is the identity

$\mathrm{vec}(UVW)=(W^{T}\otimes U)\mathrm{vec}(V)$ (14)

for any triplet $(U,V,W)$. Using (14) for $U=I$, $V=F_{k}$, and $W=x_{k}$, the
objective function of (P1) takes the form of objective function in (P2).
Likewise, by repeated application of (14) on (9), one can verify (15).∎

If we combine Theorem 3 and of Proposition 1, we observe that constraint (15)
involves a quadratic convex function of variable $f_{k}=\mathrm{vec}(F_{k})$.
Thus, when $\delta_{k}$ is kept fixed, (P2) takes the form of a regularized
QCQP. Next, we relax (P2) by replacing the $\ell_{0}$-norm with its convex
surrogate the $\ell_{1}$-norm to get the following form.

$\displaystyle\underset{f_{k},\delta_{k}}{\mathrm{minimize}}~{}~{}-\delta_{k}+\gamma\|f_{k}\|_{1}+\eta\|(x_{k}^{T}\otimes
I)f_{k}\|_{1}$ (P3)
$\displaystyle\mathrm{subject~{}to}:\mathrm{~{}for~{}all~{}}\xi\in[0,\delta_{k}]:$
(15)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{2}f_{k}^{T}P_{1}(\xi)f_{k}+q_{1}(\xi)^{T}f_{k}+r_{1}(\xi)\leq
0,$

Although the $\ell_{1}$-relaxation makes (P2) more tractable, the left hand
side of constraint (P3) may be still a nonconvex function of $\xi$. Thus,
solving (P3) for optimal solutions generally remains a difficult non-convex
task. In order to sub-optimally solve (P3) we propose a two-stage method that
is discussed in Section 5. The constraints of relaxed problem (P3) and the
original problem (P1) are identical. Moreover, stability and performance
guarantees of (P1) originate from its constraint. Hence, their feasibility,
stability, and performance guarantee are equivalent.

######  Theorem 4.

The feasible solutions to (P3) are feasible solutions to (P1). Moreover, any
solution to (P3) results in a stabilizing control (2) law with performance
guarantee (10).

We prove the feasibility of these optimization problems.

######  Theorem 5.

For every $\alpha>1$, problem (P1) and its relaxation (P3) are feasible.

###### Proof.

(9) is equivalent to $g(\xi;F_{k})\leq 0$, where

$g(\xi;F_{k}):=x_{k}^{T}\left(Y_{k}(F_{k},\xi)+\alpha\big{(}-\tilde{P}+M_{k}(\xi)^{T}\tilde{P}M_{k}(\xi)\big{)}\right)x_{k}.$

We can show that $g^{\prime}(0;F_{k})=x_{k}^{T}W_{k}x_{k}$, in which

$W_{k}=Q+F_{k}^{T}RF_{k}+\alpha\big{(}(A+BF_{k})^{T}\tilde{P}+\tilde{P}(A+BF_{k})\big{)}.$

According to (6), we get

$g^{\prime}(0;\tilde{F})=(1-\alpha)x_{k}^{T}\left(Q+\tilde{F}^{T}R\tilde{F}\right)x_{k}.$

Since $\alpha>1$ and $Q+\tilde{F}^{T}R\tilde{F}\succ 0$ , we get
$g^{\prime}(0;\tilde{F})<0$. In addition, $g(0;\tilde{F})=0$. Therefore, there
exists a positive $\theta_{k}$ at $F_{k}=\tilde{F}$ such that for all
$\xi\in(0,\theta_{k})$ we have $g(\xi;\tilde{F})<0$. This proves the
feasibility of (P1). Using Theorem 4 the feasibility of (P3) is followed as
well.∎

##  5 Self-Triggered Sparse Optimal Control

We find a sequence of stabilizing feedback gains and inter-execution times
that satisfy desired performance guarantees.

###  5.1 Solving (P3) for $\delta_{k}$ while $F_{k}$ is Fixed

This concerns a subproblem that is a component of our algorithm. Inequality
(15) is equivalent to the constraint

$\frac{1}{2}u_{k}^{T}P_{2}(\xi)u_{k}+q_{2}(\xi)^{T}u_{k}+r_{1}(\xi)\leq 0,$
(16)

where $u_{k}:=F_{k}x_{k}$ and

$\displaystyle P_{2}(\xi)$ $\displaystyle:=2H_{2}(\xi)+2\alpha
B^{T}Z(\xi)^{T}\e^{A^{T}\xi}\tilde{P}e^{A\xi}Z(\xi)B,$ $\displaystyle
q_{2}(\xi)$ $\displaystyle:=\left(2H_{1}(\xi)^{T}+2\alpha
B^{T}Z(\xi)^{T}\e^{A^{T}\xi}\tilde{P}e^{A\xi}\right)x_{k}.$

When we fix feedback gain $F_{k}$, (P3) becomes equivalent to the nonlinear
optimization problem

$\displaystyle\underset{\delta_{k}}{\mathrm{maximize}}~{}~{}\delta_{k}$ (P4)
$\displaystyle\mathrm{subject~{}to:}~{}\eqref{equpper}\mathrm{~{}for~{}every~{}}\xi\in[0,\delta_{k}].$

The optimal solution of (P4) can be characterized by

$\displaystyle\delta_{k}^{*}=\sup\left.\Big{\\{}\theta_{k}>0~{}\right|(\ref{equpper})~{}\mathrm{holds~{}for~{}all~{}}\xi\in[0,\theta_{k})\Big{\\}}.$
(17)

In order to solve (17), we utilize the simple and commonly-used discretization
rule with fixed step-size, e.g., see [9, 6]. The general nonlinear
optimization methods can be effective to solve (17) as well. In the rest of
this paper, we suppose that there exists a procedure
$\mathrm{InterExec}(F_{k})$ that solves (P4) for a fixed gain $F_{k}$ and
returns $\delta_{k}^{*}$.

###  5.2 Solving for $F_{k}$ when $\delta_{k}$ is Fixed

When $\delta_{k}$ is kept fixed, solving (P3) for $F_{k}$ is still an
infinite-dimensional optimization. Therefore, we relax the problem by
evaluating (16) only at $\xi=\delta_{k}$ and ensuring the feasibility of (P4)
after updating $F_{k}$ by imposing an additional constraint. Applying the
Schur complement to (16) and setting the argument $\xi$ to $\delta_{k}$, we
arrive at the LMI constraint

$\begin{bmatrix}2P_{2}(\delta_{k})^{-1}&F_{k}x_{k}\\\
x_{k}^{T}F_{k}^{T}&-q_{2}(\delta_{k})^{T}F_{k}x_{k}-r_{1}(\delta_{k})\end{bmatrix}\succeq
0.$ (18)

######  Proposition 2.

Suppose that $F_{k}$ satisfies

$\begin{bmatrix}R^{-1}&F_{k}\\\
F_{k}^{T}&-\alpha\big{(}(A+BF_{k})^{T}\tilde{P}+\tilde{P}(A+BF_{k})\big{)}-Q\end{bmatrix}\succ
0.$ (19)

Then, (P4) for feedback gain $F_{k}$ is feasible.

###### Proof.

If $g^{\prime}(0;F_{k})<0$, then using the same argument given in the proof of
Theorem 5, (P4) would be feasible. One inspects that inequality
$g^{\prime}(0;F_{k})<0$ is equivalent to

$x_{k}^{T}\big{(}Q+F_{k}^{T}RF_{k}+\alpha\big{(}(A+BF_{k})^{T}\tilde{P}+\tilde{P}(A+BF_{k})\big{)}\big{)}x_{k}<0.$

For this inequality to hold independent of $x_{k}$, we need

$Q+F_{k}^{T}RF_{k}+\alpha\big{(}(A+BF_{k})^{T}\tilde{P}+\tilde{P}(A+BF_{k})\big{)}\prec
0.$ (20)

Applying the Schur complement to (20), we obtain (19). ∎

Solving the two LMIs (18) and LMI (19) together with our objective function,
we arrive at a semi-definite program

$\displaystyle
F_{k}^{*}=\underset{F_{k}}{\mathrm{argmin.}}~{}~{}\gamma\|F_{k}\|_{1}+\eta\|F_{k}x_{k}\|_{1}$
(P5) $\displaystyle\hskip
27.03003pt\mathrm{subject~{}to:~{}LMI~{}\eqref{eqlowerform}~{}and~{}LMI~{}\eqref{SchurCondition_0},}$

The gain $F_{k}$ that is obtained from (P5) may not satisfy inequality (15)
for all $\xi\in[0,\delta_{k}]$ that may arise from ignoring the continuous
satisfaction of the constraint. To remedy this, we leverage (17) and modify
the value of $\delta_{k}$ for the updated value of $F_{k}$ (see the next
subsection). In the rest of this paper, we suppose that there exists a
procedure $\mathrm{FeedbackGain}(\delta_{k})$ that solves (P5) for a fixed
$\delta_{k}$ and returns $F_{k}^{*}$.

Algorithm 1 Self-Triggered Sparse Optimal Control

$A$, $B$, $Q$, $R$, $\gamma$, $\eta$, and $\alpha$.

$\\{F_{k}\\}_{k=0}^{\infty}$, $\\{\delta_{k}\\}_{k=0}^{\infty}$, and
$\\{t_{k}\\}_{k=0}^{\infty}$

$t_{0}=0$, $N_{0}\leftarrow I$, evaluate $\e^{A\xi}$, $H_{0}(\xi)$,
$H_{1}(\xi)$, $H_{2}(\xi)$, and $Z(\xi)$ offline and save them in a look-up
table

for $k=0$ to $\infty$ do

$F_{k}\leftarrow\tilde{F}$

$\delta_{k}\leftarrow\mathrm{InterExec}(F_{k})$

$F_{k}\leftarrow\mathrm{FeedbackGain}(\delta_{k})$

$\delta_{k}\leftarrow\mathrm{InterExec}(F_{k})$

$M_{k}(\delta_{k})\leftarrow\e^{A\delta_{k}}(I+Z(\delta_{k})BF_{k})$

$N_{k+1}\leftarrow M_{k}(\delta_{k})N_{k}$, $t_{k+1}\leftarrow
t_{k}+\delta_{k}$

end for

###  5.3 Algorithm

Based on the discussions of the previous subsection, our control design
algorithm consists of the following three steps: (i) solve (P4) to find the
first estimate of $\delta_{k}$ for fixed $F_{k}=\tilde{F}$, (ii) for a fixed
value of $\delta_{k}$, find a value for $F_{k}$ using (P5), and (iii) for the
updated value of feedback gain $F_{k}$, correct the value of
$\delta_{k}=\delta_{k}^{*}$ using (P4). These steps are summarized in
Algorithm 1.

######  Theorem 6.

The sequences of feedback gains $\\{F_{k}\\}_{k=0}^{\infty}$ and triggering
times $\\{t_{k}\\}_{k=0}^{\infty}$ resulting from Algorithm 1 gives a
stabilizing control law

$u(t)=F_{k}\,x_{k},$ (21)

for all $t\in[t_{k},t_{k+1})$. Moreover, for all $x_{0}\in\mathbb{R}^{n}$, the
relative performance loss $\nu(x_{0})$ satisfies

$\displaystyle\nu(x_{0})\,\leq\,\alpha-1.$ (22)

###### Proof.

Because LMI (19) has been incorporated, for the derived value of $F_{k}$,
$g^{\prime}(0;F_{k})<0$, thus, the corresponding value of $\delta_{k}^{*}$ is
strictly positive. Since for $F_{k}$ and $\delta_{k}=\delta_{k}^{*}$, the
constraint of the problem (16) is satisfied, (22) holds. ∎

Optimal control problem (P5) can be solved by a convex optimization toolbox
such as CVX [5].

######  Remark 2.

Matrices $\mathrm{e}^{A\xi}$, $Z(\xi)$, $H_{0}(\xi)$, $H_{1}(\xi)$ and
$H_{2}(\xi)$ can be computed offline and over a grid with arbitrary
resolution, as their values only depend on the state-space matrices, $A$ and
$B$, weight matrices, $Q$ and $R$, and time argument, $\xi$. Then, during the
execution, we can use look-up tables to enhance the computational complexity
of our algorithm.

##  6 Numerical Simulations

Figure 1: Randomly generated positions of $N=10$ subsystems.

To assess the effectiveness of the proposed procedure, we consider a class of
spatially distributed systems (see [10] for more details). We consider a
system of $N=10$ subsystems, which are randomly placed in the region
$[0,10]\times[0,10]$; see Fig. 1, in which each shape represents a subsystem.
Then, the dynamics of $i$’th subsystem is characterized by

$\dot{x}^{(i)}(t)=[A]_{ii}x^{(i)}(t)+\sum_{j=1,\atop j\neq
i}^{N}[A]_{ij}x^{(j)}(t)+[B]_{ii}u^{(i)}(t),\vspace{-5mm}$

where

$\displaystyle[A]_{ii}=\begin{bmatrix}1&~{}~{}~{}~{}\hskip 1.42271pt1\\\
1&~{}~{}~{}~{}\hskip 1.42271pt2\end{bmatrix},[B]_{ii}=\begin{bmatrix}0\\\
1\end{bmatrix}~{}\textrm{for square shapes},$
$\displaystyle[A]_{ii}=\begin{bmatrix}-2&1\\\
1&-3\end{bmatrix},[B]_{ii}=\begin{bmatrix}0\\\ 1\end{bmatrix}~{}\textrm{for
circle shapes},$
$\displaystyle[A]_{ij}=\frac{1}{\e^{\beta\mathrm{dis}(i,j)}}\begin{bmatrix}1&~{}~{}~{}~{}0\\\
0&~{}~{}~{}~{}1\end{bmatrix},[B]_{ij}=\begin{bmatrix}0\\\
0\end{bmatrix},~{}\forall j\neq i,$

where $\beta$ determines the rate of decay in the couplings and
$\mathrm{dis}(i,j)$ denotes the Euclidean distance between nodes $i$ and $j$
in $\mathbb{R}^{2}$. The resulting system is of dimension $n=20$. We set
$\beta=1$, $\gamma=\eta=0.001$, $\alpha=1.15$, $Q=I$, and $R=2I$. We apply
Algorithm 1 with $k_{\mbox{{\tiny max}}}=49$ iterations in the for loop and
obtain Figures 2a and 2b, which depict the relative cardinality of feedback
gains given by

$\kappa_{k}:=100\,{\big{\|}F_{k}\big{\|}_{0}}/{\big{\|}\tilde{F}\big{\|}_{0}},$
(23)

and relative cardinality of control input vectors:

$\mu_{k}:=100\,{\big{\|}u_{k}\big{\|}_{0}}/{\big{\|}\tilde{u}_{k}\big{\|}_{0}},$
(24)

respectively. Figures 2a and 2b illustrate that this method improves the
spatial sparsity compared to the periodic feedback control with
$F_{k}=\tilde{F}$ for all time intervals. In average, the cardinalities of
$F_{k}$ and $u_{k}$ have almost decreased by $38\%$ and $50\%$, respectively,
while we lose about $15\%$ in performance. Fig. 3a compares $\|x\|_{2}$ for
two designs: our method and periodic time-triggered LQR design. In Fig. 3b, we
illustrate the state of the system using the time-triggered design, starting
from an initial state. Fig. 4 shows the inter-execution times $\delta_{k}$
versus time $t$. To measure the average relative cardinality of feedback gains
over time, we define

${R}_{F}:={\displaystyle\sum_{k=0}^{k_{\mbox{{\tiny
max}}}}\delta_{k}\kappa_{k}}\bigg{/}{\displaystyle\sum_{k=0}^{k_{\mbox{{\tiny
max}}}}\delta_{k}},$

which represents the average sensor activity throughout time. Likewise, the
average relative cardinality of control inputs over time is defined as

${R}_{u}:={\displaystyle\sum_{k=0}^{k_{\mbox{{\tiny
max}}}}\delta_{k}\mu_{k}}\bigg{/}{\displaystyle\sum_{k=0}^{k_{\mbox{{\tiny
max}}}}\delta_{k}},$

which represents the average actuator activity throughout time. Also, the
average inter-execution time is ${D}:=\big{(}\sum_{k=0}^{k_{\mbox{{\tiny
max}}}}\delta_{k}\big{)}/k_{\max},$ that quantifies the time sparsity of
(state) sampling. The dependency of these quantities on $\alpha$ is captured
in Table 1, which demonstrates that by accepting higher levels of performance
loss, we achieve sparser control designs in terms of both sensing and
actuation activity (i.e., smaller values of $R_{F}$ and $R_{u}$) and sampling
activity in time (i.e., larger values of $D$). In the next simulations, we
preserve the parameters except for $\beta$. Figures 5a, 5b, and 6 demonstrate
the dependency of $R_{F}$, $R_{u}$, and $D$ on $\beta$, respectively. Fig. 5a
and Fig. 5b depict that there is a tradeoff between the spatial decay rate
$\beta$ and the indices of sensing and actuating activity, $R_{F}$ and
$R_{u}$: as $\beta$ increases, the system tends to be more localized and,
consequently, $F_{k}$’s and $u_{k}$’s become sparser. Moreover, according to
Fig. 6, as the system gets more localized, the average inter-execution time
increases; i.e., less number of samples are required on average (in time).

(a)  (b)

Figure 2: (a) Relative cardinality of controllers $\kappa_{k}$ defined in (23)
versus triggering times $t_{k}$ (b) Relative cardinality of control inputs
$\mu_{k}$ defined in (24) versus triggering times $t_{k}$.

(a)  (b)

Figure 3: (a) The Euclidean norm of state trajectories of SSOC and periodic time-triggered LQR design $\|x_{k}\|_{2}$ versus triggering times $t_{k}$ (b) state trajectories $x^{(i)}$’s of our design with a random $x_{0}$ Figure 4: Inter-execution times $\delta_{k}$ versus time $t$. Table 1: Dependency of quantities $R_{F}$, $R_{u}$, and $D$ on parameter $\alpha$. $\alpha$ | $R_{F}$ | $R_{u}$ | $D$
---|---|---|---
$1.05$ | $85.56\%$ | $62.54\%$ | $0.2823$
$1.10$ | $72.60\%$ | $66.89\%$ | $0.2715$
$1.15$ | $61.61\%$ | $49.67\%$ | $0.3322$
$1.20$ | $53.86\%$ | $41.04\%$ | $0.3247$
$1.25$ | $50.63\%$ | $36.81\%$ | $0.3629$
$1.30$ | $49.05\%$ | $32.00\%$ | $0.3529$

(a)  (b)

Figure 5: (a) ${R}_{F}$ versus decay rate $\beta$. (b) ${R}_{u}$ versus decay
rate $\beta$. Figure 6: Average inter-execution time $D$ versus spatial decay
rate $\beta$.

We consider the previous setup and study the effect of penalizing parameters
$\gamma$ and $\eta$. To investigate the effect of $\gamma$ on ${R}_{F}$, we
set $\alpha=1.15$ (i.e., accepting at most $15\%$ performance loss),
$\eta=0.001$, and vary $\gamma$ in $\left[10^{-5},10^{-3}\right]$. The result
of Algorithm $1$ shows that ${R}_{F}$ decreases as $\gamma$ grows (see Fig.
7a), which implies that the average sensing requirements of the subsystems
from each other has decreased. We repeat the previous study for fixed value of
$\gamma=10^{-3}$ and by varying $\eta$ in $\left[10^{-5},10^{-3}\right]$. Fig.
7b illustrates the relationship between $\eta$ and ${R}_{u}$. Similar to
$({R}_{F}$,$\gamma)$ relationship, there exists a tradeoff between ${R}_{u}$
and $\eta$: as $\eta$ increases, on average, it enforces more components of
$u_{k}$ to be zero; i.e., less number of actuators on average are utilized (in
time).

(a)  (b)

Figure 7: (a) ${R}_{F}$ versus penalizing parameter $\gamma$. (b) ${R}_{u}$
versus penalizing parameter $\eta$.

##  7 Discussion and Conclusion

We present a control algorithm based on blending ideas from self-triggered
control and $\ell_{1}$-regularized optimal control. The cornerstones of our
algorithm are: estimation of the inter-execution time while feedback gain is
kept fixed (via a nonlinear optimization), design of the sparse optimal
controller while inter-execution time is kept fixed (via a convex
optimization), and correcting the value of the next triggering times using the
resulting feedback gain. The closed-loop stability is guaranteed by imposing a
performance-ensuring constraint. Our extensive numerical simulations assert
that our proposed algorithm improves space-time sparsity in sensing,
communication, and actuation in an interconnected network of multiple systems.
In the case of spatially decaying systems, our simulations show that there
exists a tradeoff between the spatial decay rate and sparsity of the
communication graph (i.e., sparsity of the feedback gains). Likewise, as the
spatial decay rate increases, the average sampling rate decreases. One of our
future research directions is to modify our algorithm and implement it in a
distributed and asynchronous manner using localized information.

## Acknowledgment

The authors would like to thank Prof. Paulo Tabuada for his fruitful comments
and discussions.

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