# Sequencings in Semidirect Products via the Polynomial Method

S. Costa  DICATAM, Sez. Matematica, Università degli Studi di Brescia,
Via Branze 43, I 25123 Brescia, Italy  S. Della Fiore  DI, Università degli
Studi di Salerno, Via Papa Giovanni Paolo II 132,
I 84084 Fisciano, Italy,  M. A. Ollis  Emerson College, 120 Boylston Street,
Boston, MA 02116, USA

###### Abstract

The partial sums of a sequence ${\mathbf{x}}=x_{1},x_{2},\ldots,x_{k}$ of
distinct non-identity elements of a group $(G,\cdot)$ are $s_{0}=id_{G}$ and
$s_{j}=\prod_{i=1}^{j}x_{i}$ for $0<j\leq k$. If the partial sums are all
different then ${\mathbf{x}}$ is a linear sequencing and if the partial sums
are all different when $|i-j|\leq t$ then ${\mathbf{x}}$ is a $t$-weak
sequencing. We investigate these notions of sequenceability in semidirect
products using the polynomial method. We show that every subset of order $k$
of the non-identity elements of the dihedral group of order $2m$ has a linear
sequencing when $k\leq 12$ and either $m>3$ is prime or every prime factor of
$m$ is larger than $k!$, unless $s_{k}$ is unavoidably the identity; that
every subset of order $k$ of a non-abelian group of order three times a prime
has a linear sequencing when $5<k\leq 10$, unless $s_{k}$ is unavoidably the
identity; and that if the order of a group is $pe$ then all sufficiently large
subsets of the non-identity elements are $t$-weakly sequenceable when $p>3$ is
prime, $e\leq 3$ and $t\leq 6$.

##  1 Introduction

Let $(G,\cdot)$ be a group. We continue the investigation of when the elements
of a subset of $G$ may be ordered in such a way that their partial sums are
distinct. We set up a general framework for investigating semidirect product
groups and look in detail at the dihedral groups of order twice a prime and
non-abelian groups of order three times a prime.

######  Definition 1.1.

Let $S$ be a subset of the non-identity elements of a group $G$ with $|S|=k$.
If $x_{1},x_{2},\ldots,x_{k}$ is an ordering of the elements of $S$, define
the $j$th partial sum of the ordering by $s_{0}=id_{G}$ and
$s_{j}=\prod_{i=1}^{j}x_{i}$ for $0<j\leq k$. The ordering of the subset $S$
is a linear sequencing if the partial sums $s_{0},s_{1},\dots,s_{k}$ are all
different. It is a cyclic sequencing if the partial sums are different with
the exception that $s_{0}=s_{k}=id_{G}$. If $S$ has a linear or cyclic
sequencing then it is sequenceable.

If every subset of the non-identity elements of a group is sequenceable then
the group is strongly sequenceable. Combining several results and conjectures
concering subsets of abelian groups dating back to 1961 [8], we have the
conjecture that all abelian groups are strongly sequenceable. See [2, 5, 18]
for details of the history, motivation and current state of this conjecture.

The question for non-abelian groups is much more widely open. In [6, 16] small
counterexamples are presented to show that the potential conjecture that all
non-abelian groups are strongly sequenceable is false.

Keedwell’s Conjecture [12] is that if $S$ contains all the non-identity
elements of a non-abelian group $G$, then a linear sequencing exists if and
only if $|G|\geq 10$. This known to hold for many groups, including dihedral
groups [11, 13], soluble groups with a single involution [3], at least one
group of every odd order at which a non-abelian group exists [17] and all
groups of order at most 255 [15].

Let $S$ be a subset of size $k$ of $G\setminus\\{id_{G}\\}$ whose elements can
be ordered so that their total sum is not the identity. Then $S$ is known to
have a linear sequencing in the following situations [16]:

  * •

$k\leq 4$ and $S$ does not have one of two specific structures,

  * •

$G$ is dihedral of order $2m$ for prime $m>3$ and $k\leq 9$,

  * •

$G$ is dihedral of order $2m$ for even or prime $m>4$ and $k=2m-2$.

In this paper we make two contributions to finding linear sequencings in non-
abelian groups. First, we improve the second item above, showing that it holds
up to $k=12$ and also for $m$ whose prime factors are each larger than $k!$.
(The result that allows for composite $m$ with large prime factors in fact
applies more widely than dihedral groups; see Section 3.) The second
contribution concerns non-abelian groups of order three times a prime $p$;
there is at most one such group for any $p$ and we denote it $G_{3p}$. In this
case we show that subsets $S$ of size $5<k\leq 10$ are sequenceable and,
further, that we can always find a linear sequencing for these subsets when
the sum of all the elements is not unavoidably the identity.

In [4] a generalization of sequenceability is introduced.

######  Definition 1.2.

Let $S$ be a subset of the non-identity elements of a group $G$ with $|S|=k$
and let $x_{1},x_{2},\ldots,x_{k}$ be an ordering of the elements of $S$ with
partial sums $s_{0},s_{1},\ldots,s_{k}$. If $s_{i}\neq s_{j}$ for any distinct
pair of indices $i,j$ with $|i-j|\leq t$ then the ordering is a $t$-weak
sequencing of $S$ and $S$ is $t$-weakly sequenceable.

Observe that a $k$-weak sequencing is exactly a linear sequencing and that a
$(k-1)$-weak sequencing is either a linear or cyclic sequencing.

In [4] it is shown that a subset of non-identity elements of the cyclic group
$\mathbb{Z}_{p}$ of prime order $p$ is $t$-weak sequenceable when $t\leq 6$
and, with the additional condition that there are no pairs of the form
$\\{x,-x\\}$ in the subset, when $t=7$. We extend the methodology to apply to
groups $G$ that are not cyclic of prime order, showing that if $|G|=pe$ then
all sufficiently large subsets of $G\setminus\\{id_{G}\\}$ are $t$-weakly
sequenceable when $p>3$ is prime, $e\leq 3$ and $t\leq 6$.

In the next section we describe the polynomial method and how it can be
implemented for groups that are semidirect products. In Subsection 2.1 and
Section 3 we apply the method to obtain the results on sequenceability and in
Section 4 we apply it to obtain the results on weak sequenceability.

##  2 Polynomial Method in Semidirect Products

In this section, we apply a method that relies on the Non-Vanishing Corollary
of the Combinatorial Nullstellensatz, see [1, 14]. Given a prime $p$ (in the
following $p$ will be always assumed to be a prime), this corollary allows us
to obtain a nonzero point to suitable polynomials on $\mathbb{Z}_{p}$ derived
starting from the ones defined in [10] and in [16].

######  Theorem 2.1.

(Non-Vanishing Corollary) Let $\mathbb{F}$ be a finite field, and let
$f(x_{1},x_{2},\ldots,x_{k})$ be a polynomial in
$\mathbb{F}[x_{1},x_{2},\ldots,x_{k}]$. Suppose the degree $deg(f)$ of $f$ is
$\sum_{i=1}^{k}\gamma_{i}$, where each $\gamma_{i}$ is a nonnegative integer,
and suppose the coefficient of $\prod_{i=1}^{k}x_{i}^{\gamma_{i}}$ in $f$ is
nonzero. If $C_{1},C_{2},\ldots,C_{k}$ are subsets of $\mathbb{F}$ with
$|C_{i}|>\gamma_{i}$, then there are $c_{1}\in C_{1},\ldots,c_{k}\in C_{k}$
such that $f(c_{1},c_{2},\ldots,c_{k})\neq 0$.

In the notation of the Non-Vanishing Corollary, we call the monomial
$x_{1}^{|C_{1}|-1}$ $\cdots$ $x_{k}^{|C_{k}|-1}$ the bounding monomial. The
corollary can be rephrased as requiring the polynomial to include a monomial
of maximum degree that divides the bounding monomial (where by “include” we
mean that it has a nonzero coefficient).

To use the Non-Vanishing Corollary we require a polynomial for which the
nonzeros correspond to successful solutions to the case of the problem under
consideration. Here, we will consider subsets $S$ of a group $(G,\cdot_{G})$
that is the semidirect product of $(\mathbb{Z}_{p},+)$ and another group
$(H,\cdot_{H})$. For such groups we will use the notation
$G=\mathbb{Z}_{p}\rtimes_{\varphi}H$ for some group homomorphism
$\varphi:H\rightarrow Aut(\mathbb{Z}_{p})$.

######  Remark 2.2.

These kind of semidirect products define a wide class of groups. Our
hypotheses are, indeed, equivalent to consider groups $(G,\cdot)$ that admit
an isomorphic copy of $(\mathbb{Z}_{p},+)$ as a normal subgroup $N$ and whose
cosets are of the form $N\cdot id,N\cdot c_{1},\dots,N\cdot c_{e-1}$ where
$e=|G/N|$ and $\\{c_{0}=id,\dots,c_{e-1}\\}$ is a subgroup of $G$.

Now revisit the polynomial method introduced in [16] for dihedral groups and
in [5] for groups of type $\mathbb{Z}_{p}\times H$. To avoid confusion, from
now on, the symbol $+$ will be used only between elements of $\mathbb{Z}_{p}$.
Let $\pi_{2}:\mathbb{Z}_{p}\rtimes_{\varphi}H\rightarrow H$ be the projection
map that picks out the second coordinate of an element and for a subset
$S\subseteq(\mathbb{Z}_{p}\rtimes_{\varphi}H)\setminus\\{(0,0)\\}$ let
$\pi_{2}(S)$ be the multiset $\\{\pi_{2}(s):s\in S\\}$. Also, the generic
element of $\mathbb{Z}_{p}\rtimes_{\varphi}H$ will be indicated by
$x\cdot_{G}a$ (or $y\cdot b$) implying that $x\in\mathbb{Z}_{p}$ and $a\in H$.
We define the type of $S$ to be the sequence
$\bm{\lambda}=(\lambda_{0},\ldots,\lambda_{e-1})$, where $\lambda_{i}$ is the
number of times that $i$ appears in $\pi_{2}(S)$.

Let $T$ be a multiset of elements from $H\setminus\\{0\\}$ with size $k$. Let
$\bm{a}=(a_{1},\ldots,a_{k})$ be an arrangement of the elements of $T$ with
partial sums $\bm{b}=(b_{0},b_{1},\ldots,b_{k})$. If, for each $i$, the
element $i$ appears at most $r$ times in ${\bf b}$, then ${\bf a}$ is a
quotient sequencing of $T$ with respect to $r$. In our setting, given some
$S\subseteq(\mathbb{Z}_{p}\rtimes_{\varphi}H)\setminus\\{(0,0)\\}$ for which
we wish to find a sequencing, we shall be interested in quotient sequencings
of $\pi_{2}(S)$ with respect to $p$.

Given $S\subseteq(\mathbb{Z}_{p}\rtimes_{\varphi}H)\setminus\\{(0,0)\\}$ we
first construct a quotient sequencing of $\pi_{2}(S)$ with respect to $p$. We
then use the polynomial method to show that there is a sequencing for $S$ that
projects elementwise onto that quotient sequencing. Given such a quotient
sequencing $\bm{a}=(a_{1},\ldots a_{k})$ with partial sums
$\bm{b}=(b_{0},b_{1},\ldots,b_{k})$, let

$\bm{x_{a}}=\left(x_{1}\cdot_{G}a_{1},x_{2}\cdot_{G}a_{2},\ldots,x_{k}\cdot_{G}a_{k}\right)$

be a putative arrangement of the elements of $S$ with partial sums

$\bm{y_{a}}=\left(s_{0},\ldots,s_{k}\right)=\left(y_{0}\cdot_{G}b_{0},y_{1}\cdot_{G}b_{1},\ldots,y_{k}\cdot_{G}b_{k}\right).$

We will need the following lemma:

######  Lemma 2.3.

Let $(G,\cdot_{G})$ be a group of type $\mathbb{Z}_{p}\rtimes_{\varphi}H$.
Then, considering $x_{1}\cdot_{G}a_{1}$ and $x_{2}\cdot_{G}a_{2}$ in $G$,
there exists $\varphi_{a_{1}}\in\mathbb{Z}_{p}$ such that:

$(x_{1}\cdot_{G}a_{1})\cdot_{G}(x_{2}\cdot_{G}a_{2})=(x_{1}+(\varphi_{a_{1}}x_{2}))\cdot_{G}(a_{1}\cdot_{H}a_{2})$

where $\varphi_{a_{1}}x_{2}$ represent the product between the elements of
$\mathbb{Z}_{p}$, $\varphi_{a_{1}}$ and $x_{2}$.

###### Proof.

Due to the definition, we have that

$(x_{1}\cdot_{G}a_{1})\cdot_{G}(x_{2}\cdot_{G}a_{2})=(x_{1}+\varphi(a_{1})(x_{2}))\cdot_{G}(a_{1}\cdot_{H}a_{2}).$

Here the map

$x\rightarrow\varphi(a_{1})(x)$

is an automorphism of $(\mathbb{Z}_{p},+)$. It is well known that the
automorphism of $(\mathbb{Z}_{p},+)$ are the multiplications and hence there
exists $\varphi_{a_{1}}\in\mathbb{Z}_{p}$ such that

$\varphi(a_{1})(x)=\varphi_{a_{1}}x.$

Hence we have that

$(x_{1}+\varphi(a_{1})(x_{2}))\cdot_{G}(a_{1}\cdot_{H}a_{2})=(x_{1}+(\varphi_{a_{1}}x_{2}))\cdot_{G}(a_{1}\cdot_{H}a_{2})$

which implies the thesis. ∎

From now on, when it is clear from the context, the index $G$ or $H$ in the
products will be omitted.

Here we note that, if $j>i$, $s_{i}=s_{j}$ only when

$(x_{i+1}\cdot a_{i+1})\cdot(x_{i+2}\cdot a_{i+2})\cdot\ldots\cdot(x_{j}\cdot
a_{j})=id.$

Applying Lemma 2.3, this equality can be written as

$(x_{i+1}+\varphi_{a_{i+1}}x_{i+2})\cdot(a_{i+1}\cdot
a_{i+2})\cdot\ldots\cdot(x_{j}\cdot a_{j})=id$

and, reiterating it, as

$(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{j-1}}x_{j})\cdot(a_{i+1}\cdot
a_{i+2}\cdot\ldots\cdot a_{j})=id.$

This implies that

$a_{i+1}\cdot a_{i+2}\cdot\ldots\cdot a_{j}=id$

which is $b_{i}=b_{j}$, and

$x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{j-1}}x_{j}=0.$

Hence, reasoning as in [5], we can consider the following polynomial in the
variables $x_{1},x_{2},\ldots,x_{k}$

$p_{\bm{a}}=\prod_{\begin{subarray}{c}1\leq i<j\leq k\\\
a_{i}=a_{j}\end{subarray}}(x_{j}-x_{i})\prod_{\begin{subarray}{c}0\leq i<j\leq
k\\\ b_{i}=b_{j}\\\ j\neq
i+1\end{subarray}}(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{j-1}}x_{j})$ (1)

where the variable $x_{i}$ ranges over the values $\\{c:c\cdot a_{i}\in S\\}$
for each $i$. Note that, if $G$ is non-abelian, there is no reason to exclude
the pair $(0,k)$ in the second product of Equation (1). Indeed we can even ask
that $s_{0}\not=s_{k}$. Now, it is clear that, given $S=\\{x_{1}\cdot
a_{1},x_{2}\cdot a_{2},\ldots,x_{k}\cdot a_{k}\\}\subseteq G\setminus\\{0\\}$
of size $k$, $p_{\bm{a}}(x_{1},\dots,x_{k})\not=0$ only if the sequence
$(x_{1}\cdot a_{1},x_{2}\cdot a_{2},\ldots,x_{k}\cdot a_{k})$ is a sequencing
for the set $S$.

###  2.1 Computational Results

The method introduced in the previous paragraph generalizes that used in [16]
to investigate the problem for dihedral groups. We recall the definition of
these groups.

######  Definition 2.4.

The dihedral group $D_{2n}$ is the semidirect product
$\mathbb{Z}_{n}\rtimes_{\varphi}\mathbb{Z}_{2}$ where

$\varphi(0)=id\mbox{ and }\varphi(1)(x)=-x.$

In the following example, we see how the coefficients of Equation (1) behave
for such groups.

######  Example 2.5.

Example Let $p$ be a prime, $p>3$, and suppose $S\subseteq
D_{2p}\setminus\\{0\\}$ with $|S|=5$ and of type $(3,2)$. The sequence
$\bm{a}=(0,1,0,0,1)$ has partial sums $(0,0,1,1,1,0)$ and so is a quotient
sequencing of $\pi_{2}(S)$ with respect to $p$. We desire a sequencing of $S$
of the form

$\left(x_{1}\cdot 0,x_{2}\cdot 1,x_{3}\cdot 0,x_{4}\cdot 0,x_{5}\cdot
1\right).$

The polynomial is

$p_{\bm{a}}=(x_{3}-x_{1})(x_{4}-x_{1})(x_{5}-x_{2})(x_{4}-x_{3})(x_{1}+x_{2}-x_{3}-x_{4}-x_{5})(x_{2}-x_{3}-x_{4}-x_{5})(x_{3}+x_{4})\,.$

To apply the Non-Vanishing Corollary we need a monomial of this polynomial
which divides the bounding monomial $x_{1}^{2}x_{2}x_{3}^{2}x_{4}^{2}x_{5}$
with a nonzero coefficient. One such is
$x_{1}^{2}x_{2}x_{3}^{2}x_{4}x_{5}^{2}$, which has coefficient $6$. Hence
whenever $S$ has this form it has a sequencing.

Another class of groups that we will consider in this discussion is that of
$G_{3p}$.

######  Definition 2.6.

Given $p\equiv 1\pmod{6}$ and $r$ such that $r^{3}\equiv 1\pmod{p}$ and
$r\not\equiv 1\pmod{p}$, $G_{3p}$ is defined as
$\mathbb{Z}_{p}\rtimes_{\varphi}\mathbb{Z}_{3}$ where

$\varphi(i)(x)=r^{i}x\mbox{ for any }i\in[0,1,2].$

In case $r\equiv 1\pmod{p}$, using the same construction we would obtain the
direct product $\mathbb{Z}_{p}\times\mathbb{Z}_{3}$.

In the following example, we see how the coefficients of Equation (1) behave
for such groups.

######  Example 2.7.

Example Let $p$ be a prime, $p>3$, and suppose $S\subseteq
G_{3p}\setminus\\{0\\}$ with $|S|=5$ and of type $(1,3,1)$. The sequence
$\bm{a}=(1,1,0,2,1)$ has partial sums $(0,1,2,2,1,2)$ and so is a quotient
sequencing of $\pi_{2}(S)$ with respect to $p$. We desire a sequencing of $S$
of the form

$\left(x_{1}\cdot 1,x_{2}\cdot 1,x_{3}\cdot 0,x_{4}\cdot 2,x_{5}\cdot
1\right).$

The polynomial is

$p_{\bm{a}}=(x_{2}-x_{1})(x_{5}-x_{1})(x_{5}-x_{2})(x_{2}+rx_{3}+rx_{4})(x_{3}+x_{4}+r^{2}x_{5})(x_{4}+r^{2}x_{5})\,.$

To apply the Non-Vanishing Corollary we need a monomial of this polynomial
which divides the bounding monomial $x_{1}^{2}x_{2}^{2}x_{5}^{2}$ with a
nonzero coefficient. One such is $x_{1}^{2}x_{2}^{2}x_{5}^{2}$, which has
coefficient $-r^{4}=-r$. Hence whenever $S$ has this form it has a sequencing.

We are now ready to prove the main result of this section.

######  Theorem 2.8.

Let $n=pe$ with $p$ prime and let $G$ be a group of size $n$. Then subsets $S$
of size $k$ of $G$ are sequenceable in the following cases:

  1. 1.

$k\not=4,5$ such that $k\leq 10$ and $e\in\\{1,2,3\\}$;

  2. 2.

$k\leq 12$, $p>3$ and $e\in\\{1,2\\}$.

###### Proof.

The case $e=1$ follows from the results of [10] and [5]. Since
$\mathbb{Z}_{p}\rtimes_{\varphi}\\{0\\}$ is isomorphic to $\mathbb{Z}_{p}$,
from the same papers, it follows the sequenceability of subsets $S$ of
$D_{2p}$ (resp. $G_{3p}$) whose type is $(k,0)$ (resp. $(k,0,0)$).

Note that a group of size $2p$ is either $\mathbb{Z}_{p}\times\mathbb{Z}_{2}$
or the dihedral group $D_{2p}$. Analogously, a group of size $3p$ is either
$\mathbb{Z}_{p}\times\mathbb{Z}_{3}$ or $G_{3p}$ for some $r$. Since the
direct product case has already been considered in [5], we consider here just
the cases of $D_{2p}$ and $G_{3p}$. Also, for the dihedral groups, we can
suppose that $k\geq 10$ since by [16] we know that subsets $S$ of size $k\leq
9$ in $D_{2p}$ (with $p>3$) are sequenceable. We also know from [16, Lemma
3.1] that subsets $S\subseteq D_{2p}$ of size $k$ and of type $(0,k)$ are
sequenceable for every $k$. Then, in all of the other cases stated in the
theorem, we can use the Non-Vanishing Corollary for suitable quotient
sequences and nonzero monomial (in Tables LABEL:tab:12_2 and LABEL:tab:10_3
below we report the extreme cases). These results are obtained using the
Python framework SageMath [19].

Here, when considering $G_{3p}$, for each type we provide a set of
coefficients whose expressions depend on $r$. We have checked that, for any
prime $p\equiv 1\pmod{6}$, at least one of these coefficients is nonzero
proceeding as follows. The coefficients can be written in the form $ar+b$ for
some $a,b$ whose expression do not depend on $p$. If $a\not\equiv 0\pmod{p}$,
the coefficient is zero if and only if $r\equiv-b\cdot a^{-1}\pmod{p}$.
Recalling that $r$ satisfies $r^{2}+r+1\equiv 0\pmod{p}$ we have that
$a^{-2}(b^{2}-ab+a^{2})\equiv 0\pmod{p}$ and hence,

$b^{2}-ab+a^{2}\equiv 0\pmod{p}.$ (2)

On the other hand, if $a\equiv 0\pmod{p}$ the coefficient is zero only when
also $b$ is zero modulo $p$ and hence Equation (2) is satisfied also in this
case. Since Equation (2) is satisfied for finitely many $p$, one can check
that, for any prime $p$, at least one of the coefficients presented in Table
LABEL:tab:10_3 is nonzero. ∎

######  Remark 2.9.

Since in Equation (1) we have imposed also $s_{0}\not=s_{k}$, we have that in
case $G=D_{2p}$ (resp. $G=G_{3p}$), $S$ is not of type $(k,0)$ (resp.
$(k,0,0)$) and $k$ satisfies the hypotheses of the previous theorem, then $S$
admits a linear sequencing.

This implies that, under the hypothesis of Theorem 2.8, $S$ admits a linear
sequencing unless $s_{k}$ is unavoidably the identity.

The following Tables LABEL:tab:12_2 and LABEL:tab:10_3 contain the required
monomials and their coefficients for the proof of Theorem 2.8 in the cases
$(G,k)=(D_{2p},12)$ and $(G,k)=(G_{3p},10)$.

Table 1: Monomials and their coefficients for the proof of Theorem 2.8 in the case $|S|=12$ and $G=D_{2p}$. $\bm{\lambda}$ | $\mathbf{a}$ | deg | monomial/s | coefficient/s
---|---|---|---|---
$(11,1)$ |  | $(0,0,0,0,0,1,0,0,0,0,0,0)$
---
| 71
---
|
$x_{2}^{3}x_{3}^{5}x_{4}^{7}x_{5}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}x_{11}^{10}x_{12}^{10}$
---
$x_{2}^{3}x_{3}^{5}x_{4}^{8}x_{5}^{8}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}x_{11}^{10}x_{12}^{10}$
| $2\cdot 3\cdot 7^{2}\cdot 131$
---
$2\cdot 5\cdot 7\cdot 593$
$(10,2)$ |  | $(0,0,0,0,0,1,0,0,0,0,0,1)$
---
| 64
---
|
$x_{2}^{4}x_{3}^{5}x_{4}^{9}x_{5}^{8}x_{6}x_{7}^{8}x_{8}^{9}x_{9}^{9}x_{10}^{9}x_{11}^{9}x_{12}$
---
$x_{2}^{4}x_{3}^{5}x_{4}^{8}x_{5}^{9}x_{6}x_{7}^{8}x_{8}^{9}x_{9}^{9}x_{10}^{9}x_{11}^{9}x_{12}$
| $-2^{7}\cdot 3\cdot 29\cdot 41$
---
$-2^{5}\cdot 3\cdot 101\cdot 227$
$(9,3)$ |  | $(0,0,0,0,1,1,1,0,0,0,0,0)$
---
| 57
---
|
$x_{2}^{2}x_{3}^{8}x_{4}^{8}x_{5}^{2}x_{6}^{2}x_{7}^{2}x_{8}^{8}x_{9}^{8}x_{10}^{8}x_{11}^{8}x_{12}^{8}$
---
$x_{2}^{3}x_{3}^{7}x_{4}^{8}x_{5}^{2}x_{6}^{2}x_{7}^{2}x_{8}^{8}x_{9}^{8}x_{10}^{8}x_{11}^{8}x_{12}^{8}$
| $-2^{3}\cdot 3^{3}\cdot 5\cdot 31$
---
$-2^{2}\cdot 3^{2}\cdot 61\cdot 73$
$(8,4)$ |  | $(0,0,0,0,1,1,1,0,0,0,0,1)$
---
| 54
---
|
$x_{2}^{6}x_{3}^{7}x_{4}^{7}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{7}x_{9}^{7}x_{10}^{7}x_{11}^{7}x_{12}^{3}$
---
$x_{2}^{7}x_{3}^{7}x_{4}^{7}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{6}x_{9}^{7}x_{10}^{7}x_{11}^{7}x_{12}^{3}$
| $2^{6}\cdot 7\cdot 7669$
---
$-2^{6}\cdot 11\cdot 28591$
$(7,5)$ |  | $(0,0,0,1,1,1,1,1,0,0,0,0)$
---
| 51
---
|
$x_{2}^{6}x_{3}^{6}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{6}x_{10}^{6}x_{11}^{6}x_{12}^{6}$
---
$x_{1}^{6}x_{3}^{6}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{6}x_{10}^{6}x_{11}^{6}x_{12}^{6}$
| $-2^{3}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 401$
---
$-2^{2}\cdot 5^{2}\cdot 179\cdot 389$
$(6,6)$ |  | $(0,0,0,1,1,1,1,1,0,0,0,1)$
---
| 52
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{5}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{5}x_{10}^{5}x_{11}^{5}x_{12}^{5}$
---
$x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{5}x_{10}^{5}x_{11}^{5}x_{12}^{5}$
| $2\cdot 3^{3}\cdot 181\cdot 481373$
---
$-2\cdot 3^{2}\cdot 13\cdot 22399121$
$(4,8)$ |  | $(0,0,1,1,1,1,1,1,1,0,0,1)$
---
| 58
---
|
$x_{2}^{3}x_{3}^{4}x_{4}^{5}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{3}x_{11}^{3}x_{12}^{7}$
---
$x_{2}^{3}x_{3}^{4}x_{4}^{6}x_{5}^{6}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{3}x_{11}^{3}x_{12}^{7}$
| $2^{4}\cdot 3\cdot 79\cdot 179\cdot 18979$
---
$-2^{4}\cdot 3^{2}\cdot 5\cdot 109\cdot 119839$
$(3,9)$ |  | $(0,1,1,1,1,1,1,1,1,1,0,0)$
---
| 63
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{6}x_{4}^{8}x_{5}^{8}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}x_{11}^{2}x_{12}^{2}$
---
$x_{1}^{2}x_{2}^{5}x_{3}^{5}x_{4}^{8}x_{5}^{8}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}x_{11}^{2}x_{12}^{2}$
| $3\cdot 62701\cdot 131111$
---
$-2\cdot 3\cdot 5\cdot 13\cdot 17\cdot 3699679$
$(2,10)$ |  | $(0,1,1,1,1,1,1,1,1,1,0,1)$
---
| 72
---
|
$x_{1}x_{2}^{4}x_{3}^{4}x_{4}^{8}x_{5}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}x_{11}x_{12}^{9}$
---
$x_{1}x_{2}^{4}x_{3}^{5}x_{4}^{7}x_{5}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}x_{11}x_{12}^{9}$
| $2\cdot 5\cdot 71\cdot 1427\cdot 1033463$
---
$-2\cdot 3^{2}\cdot 7\cdot 23\cdot 215947813$
$(1,11)$ |  | $(1,1,1,1,1,1,1,1,1,1,1,0)$
---
| 80
---
|
$x_{2}^{5}x_{3}^{5}x_{4}^{7}x_{5}^{7}x_{6}^{8}x_{7}^{8}x_{8}^{10}x_{9}^{10}x_{10}^{10}x_{11}^{10}$
---
$x_{2}^{4}x_{3}^{6}x_{4}^{7}x_{5}^{7}x_{6}^{8}x_{7}^{8}x_{8}^{10}x_{9}^{10}x_{10}^{10}x_{11}^{10}$
| $2^{3}\cdot 7\cdot 101\cdot 81083329$
---
$-2^{5}\cdot 5\cdot 31\cdot 3557\cdot 21647$
Table 2: Monomials and their coefficients for the proof of Theorem 2.8 in the case $|S|=10$ and $G=G_{3p}$. $\bm{\lambda}$ | $\mathbf{a}$ | deg | monomial/s | coefficient/s
---|---|---|---|---
$(9,1,0)$ |  | $(0,0,0,0,0,1,0,0,0,0)$
---
| 52
---
| $x_{2}^{2}x_{3}^{4}x_{4}^{7}x_{5}^{8}x_{7}^{7}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $-2^{2}$
---
$(8,2,0)$ |  | $(0,0,0,1,0,0,0,1,0,0)$
---
| 36
---
| $x_{2}^{2}x_{3}^{4}x_{5}^{3}x_{6}^{6}x_{7}^{7}x_{8}x_{9}^{6}x_{10}^{7}$
---
| $-2$
---
$(8,1,1)$ |  | $(0,0,0,0,1,0,0,0,0,2)$
---
| 45
---
| $x_{1}x_{2}^{4}x_{3}^{6}x_{4}^{7}x_{6}^{6}x_{7}^{7}x_{8}^{7}x_{9}^{7}$
---
$x_{1}^{2}x_{2}^{3}x_{3}^{6}x_{4}^{7}x_{6}^{6}x_{7}^{7}x_{8}^{7}x_{9}^{7}$
| $-1791r+573$
---
$-235r-746$
$(7,3,0)$ |  | $(0,0,1,0,0,0,1,0,0,1)$
---
| 32
---
|
$x_{1}^{2}x_{2}^{3}x_{4}^{2}x_{5}^{5}x_{6}^{6}x_{7}x_{8}^{5}x_{9}^{6}x_{10}^{2}$
---
$x_{1}^{2}x_{2}^{2}x_{4}^{3}x_{5}^{5}x_{6}^{6}x_{7}x_{8}^{5}x_{9}^{6}x_{10}^{2}$
| $14r+73$
---
$12r+21$
$(7,2,1)$ |  | $(0,0,0,1,0,0,1,0,0,2)$
---
| 30
---
| $x_{1}x_{2}^{2}x_{3}^{4}x_{4}x_{5}^{5}x_{6}^{5}x_{7}x_{8}^{5}x_{9}^{6}$
---
| $1$
---
$(6,4,0)$ |  | $(0,0,1,0,0,1,0,0,1,1)$
---
| 30
---
|
$x_{2}^{2}x_{3}^{2}x_{4}^{4}x_{5}^{5}x_{6}^{2}x_{7}^{4}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
$x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{3}x_{5}^{5}x_{6}^{2}x_{7}^{4}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
| $-1168r+72$
---
$-436r-1078$
$(6,3,1)$ |  | $(0,0,0,1,0,0,1,0,2,1)$
---
| 27
---
| $x_{2}^{2}x_{3}^{5}x_{4}x_{5}^{5}x_{6}^{5}x_{7}^{2}x_{8}^{5}x_{10}^{2}$
---
| $r+1$
---
$(6,2,2)$ |  | $(0,0,1,0,0,1,0,0,2,2)$
---
| 26
---
|
$x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{4}x_{6}x_{7}^{4}x_{8}^{5}x_{9}x_{10}$
---
$x_{1}^{2}x_{2}^{5}x_{4}^{3}x_{5}^{4}x_{6}x_{7}^{4}x_{8}^{5}x_{9}x_{10}$
| $10r-64$
---
$155r+583$
$(5,5,0)$ |  | $(0,0,1,0,0,1,0,1,1,1)$
---
| 30
---
|
$x_{2}^{2}x_{3}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
$x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{2}x_{5}^{3}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
| $1965r+2715$
---
$385r-190$
$(5,4,1)$ |  | $(0,0,1,0,0,1,0,2,1,1)$
---
| 26
---
| $x_{1}x_{2}^{4}x_{3}x_{4}^{3}x_{5}^{4}x_{6}^{3}x_{7}^{4}x_{9}^{3}x_{10}^{3}$
---
$x_{1}^{2}x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{4}x_{6}^{3}x_{7}^{4}x_{9}^{3}x_{10}^{3}$
| $30r+156$
---
$42r+72$
$(5,3,2)$ |  | $(0,0,0,1,0,1,0,2,1,2)$
---
| 24
---
|
$x_{1}^{4}x_{2}^{4}x_{3}^{3}x_{4}^{2}x_{5}^{4}x_{6}^{2}x_{7}^{2}x_{8}x_{9}^{2}$
---
| $2$
---
$(4,6,0)$ |  | $(0,1,0,0,1,0,1,1,1,1)$
---
| 32
---
|
$x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{3}x_{7}^{4}x_{8}^{4}x_{9}x_{10}$
---
$x_{1}^{2}x_{2}^{5}x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{3}x_{7}^{5}x_{8}^{4}x_{9}x_{10}$
| $-11377r+11374$
---
$-21587r-27358$
$(4,5,1)$ |  | $(0,0,1,0,1,0,1,1,1,2)$
---
| 27
---
| $x_{2}^{2}x_{3}^{3}x_{4}^{3}x_{5}^{4}x_{6}^{3}x_{7}^{4}x_{8}^{4}x_{9}^{4}$
---
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{4}x_{9}^{4}$
| $-1286r+68$
---
$-333r-119$
$(4,4,2)$ |  | $(0,0,0,1,0,1,2,1,2,1)$
---
| 24
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}x_{8}^{3}x_{9}x_{10}^{3}$
---
| $-2r$
---
$(4,3,3)$ |  | $(0,0,1,0,1,0,2,1,2,2)$
---
| 23
---
|
$x_{1}^{2}x_{2}^{3}x_{3}^{2}x_{4}^{3}x_{5}^{2}x_{6}^{3}x_{7}^{2}x_{8}^{2}x_{9}^{2}x_{10}^{2}$
---
$x_{1}^{3}x_{2}^{3}x_{3}^{2}x_{4}^{3}x_{5}^{2}x_{6}^{2}x_{7}^{2}x_{8}^{2}x_{9}^{2}x_{10}^{2}$
| $-1452r+1686$
---
$-759r-348$
$(3,7,0)$ |  | $(0,1,0,1,0,1,1,1,1,1)$
---
| 36
---
| $x_{2}x_{3}^{2}x_{4}x_{5}^{2}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{6}$
---
$x_{2}x_{3}^{2}x_{4}^{3}x_{5}^{2}x_{6}^{4}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{6}$
| $-19616r+331264$
---
$802469r-381850$
$(3,6,1)$ |  | $(0,0,1,0,1,1,1,1,2,1)$
---
| 30
---
| $x_{1}^{2}x_{2}x_{3}^{5}x_{4}^{2}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{4}x_{10}$
---
$x_{1}^{2}x_{2}x_{3}^{3}x_{4}^{2}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{4}x_{10}^{3}$
| $3396r-230$
---
$133r+344$
$(3,5,2)$ |  | $(0,0,1,0,1,2,1,2,1,1)$
---
| 26
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{2}x_{5}^{4}x_{6}x_{7}^{4}x_{8}x_{9}^{4}x_{10}^{4}$
---
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{2}x_{5}^{3}x_{6}x_{7}^{4}x_{8}x_{9}^{4}x_{10}^{4}$
| $-3r+24$
---
$21r+21$
$(3,4,3)$ |  | $(0,0,1,0,1,1,2,2,1,2)$
---
$(0,0,1,1,0,1,2,2,1,2)$
| 24
---
24
|
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{2}x_{5}^{3}x_{6}^{3}x_{7}^{2}x_{8}^{2}x_{9}^{3}x_{10}^{2}$
---
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{3}x_{5}^{2}x_{6}^{3}x_{7}^{2}x_{8}^{2}x_{9}^{3}x_{10}^{2}$
| $1358r+1812$
---
$-1742r-1035$
$(2,8,0)$ |  | $(0,1,0,1,1,1,1,1,1,1)$
---
| 42
---
| $x_{3}x_{4}^{2}x_{5}^{4}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{7}$
---
$x_{3}x_{4}^{2}x_{5}^{5}x_{6}^{6}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{7}$
| $65374r+67999$
---
$-160333r-7057$
$(2,7,1)$ |  | $(0,1,0,1,1,1,1,2,1,1)$
---
| 35
---
| $x_{1}x_{3}x_{4}^{3}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{9}^{6}x_{10}^{6}$
---
$x_{1}x_{3}x_{4}^{4}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{9}^{6}x_{10}^{6}$
| $17861r-27263$
---
$12821r+26143$
$(2,6,2)$ |  | $(2,0,1,1,1,1,1,2,1,0)$
---
| 30
---
| $x_{3}^{3}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}x_{9}^{5}x_{10}$
---
$x_{3}^{4}x_{4}^{4}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}x_{9}^{5}x_{10}$
| $-292r+246$
---
$85r+126$
$(2,5,3)$ |  | $(2,1,1,1,0,2,2,1,0,1)$
---
| 27
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{6}^{2}x_{7}^{2}x_{8}^{4}x_{9}x_{10}^{4}$
---
$x_{1}^{2}x_{2}^{3}x_{3}^{4}x_{4}^{4}x_{5}x_{6}^{2}x_{7}^{2}x_{8}^{4}x_{9}x_{10}^{4}$
| $17651r-23546$
---
$-10860r-152465$
$(2,4,4)$ |  | $(0,1,0,1,1,2,2,1,2,2)$
---
$(0,1,1,0,1,2,2,1,2,2)$
| 26
---
26
|
$x_{1}x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{3}x_{9}^{3}x_{10}^{3}$
---
$x_{1}x_{2}^{3}x_{3}^{3}x_{4}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{3}x_{9}^{3}x_{10}^{3}$
| $136392r+247068$
---
$-175716r-167652$
$(1,9,0)$ |  | $(1,0,1,1,1,1,1,1,1,1)$
---
| 50
---
| $x_{3}^{2}x_{4}^{4}x_{5}^{5}x_{6}^{7}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
$x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{7}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
| $992906r+1427608$
---
$-1815167r-733348$
$(1,8,1)$ |  | $(0,1,1,1,1,1,1,1,1,2)$
---
| 42
---
| $x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}$
---
$x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{6}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}$
| $63854r-6907$
---
$38518r+68818$
$(1,7,2)$ |  | $(0,1,1,1,1,1,1,2,2,1)$
---
| 36
---
| $x_{3}^{4}x_{4}^{6}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}x_{9}x_{10}^{6}$
---
$x_{3}^{5}x_{4}^{5}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}x_{9}x_{10}^{6}$
| $-16124r-27940$
---
$3718r+13449$
$(1,6,3)$ |  | $(1,1,0,1,1,1,2,1,2,2)$
---
| 32
---
|
$x_{1}x_{2}^{5}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{2}x_{8}^{5}x_{9}^{2}x_{10}^{2}$
---
$x_{1}^{2}x_{2}^{4}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{2}x_{8}^{5}x_{9}^{2}x_{10}^{2}$
| $31065r+30039$
---
$1764r+9147$
$(1,5,4)$ |  | $(0,1,1,1,1,1,2,2,2,2)$
---
| 30
---
|
$x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{3}x_{8}^{3}x_{9}^{3}x_{10}$
---
$x_{2}^{2}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{3}x_{8}^{3}x_{9}^{3}x_{10}^{3}$
| $211692r+376028$
---
$78700r-731650$
$(0,10,0)$ |  | $(1,1,1,1,1,1,1,1,1,1)$
---
| 60
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{4}x_{4}^{7}x_{5}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
$x_{1}^{2}x_{2}^{2}x_{3}^{5}x_{4}^{6}x_{5}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
| $-250444r-433547$
---
$1040607r+497898$
$(0,9,1)$ |  | $(1,1,1,1,1,2,1,1,1,1)$
---
| 51
---
| $x_{2}^{2}x_{3}^{3}x_{4}^{6}x_{5}^{8}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
$x_{2}^{2}x_{3}^{4}x_{4}^{5}x_{5}^{8}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
| $21957r-14546$
---
$37855r+47642$
$(0,8,2)$ |  | $(1,1,1,1,1,1,1,1,2,2)$
---
| 44
---
| $x_{2}^{3}x_{3}^{4}x_{4}^{7}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}x_{10}$
---
$x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}x_{10}$
| $1698588r+406932$
---
$201528r+1213386$
$(0,7,3)$ |  | $(2,1,1,1,1,1,1,1,2,2)$
---
| 39
---
|
$x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{6}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{2}x_{10}^{2}$
---
$x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{5}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{2}x_{10}^{2}$
| $824103r+258808$
---
$89629r+972050$
$(0,6,4)$ |  | $(2,1,2,1,1,1,1,1,2,2)$
---
| 38
---
|
$x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
$x_{2}^{4}x_{3}^{3}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
| $1390432r+1536910$
---
$-792953r-108625$
$(0,5,5)$ |  | $(1,1,1,1,1,2,2,2,2,2)$
---
| 35
---
|
$x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
| $-710283r+3022047$
---
$2272296r+657732$

##  3 To Infinity and Back

In this section we apply the method of [7] (see also [5]) to generalize the
sequenceability results here obtained to a more general class of semidirect
products.

First, given a subset $S$ of a group $G$, we define the set

$\Upsilon(S)=S\cup\Delta(S).$

Then we can adapt [7, Lemma 2.1] as

######  Lemma 3.1.

Let $G_{1}$ and $G_{2}$ be groups. Given a subset $S$ of
$G_{1}\setminus\\{0_{G_{1}}\\}$ of size $k$, suppose there exists a
homomorphism $\phi:G_{1}\to G_{2}$ such that
$\ker(\phi)\cap\Upsilon(S)=\emptyset$. Then $\phi(S)$ is a subset of
$G_{2}\setminus\\{0_{G_{2}}\\}$ of size $k$ and the subset $S$ is sequenceable
whenever $\phi(S)$ is sequenceable.

In the following, we consider groups of the form $(G\rtimes_{\varphi}H,\cdot)$
where $(G,+)$ is abelian, $H=\\{h_{0}=0_{H},h_{1},\dots,h_{e-1}\\}$ is finite,
$(H,\cdot_{H})$111Also in this section, when it is clear from the context, the
index $H$ will be omitted. is not necessarily abelian and where $\varphi$ is a
group homomorphism $\varphi:H\rightarrow Aut(G)$. We will also assume that the
homomorphism $\varphi$ satisfies the following property:

  * $(*)$

given $h\in H,\varphi(h)$ is either the identity or $\varphi(h)(x)=-x$ for any
$x\in G$.

Note that the latter is an automorphism of $G$ because $G$ is abelian.

######  Remark 3.2.

Two classes of semidirect product that satisfy property $(*)$ are the
generalized dihedral groups and the standard direct product. Indeed:

  * 1)

The generalized dihedral group $Dih(G)$ can be defined as
$G\rtimes_{\varphi}\mathbb{Z}_{2}$ where

$\varphi(0)=id\mbox{ and }\varphi(1)(x)=-x.$

  * 2)

Any direct product $G\times H$ can be defined as $G\rtimes_{\varphi}H$ where

$\varphi(h)=id\mbox{ for any }h\in H.$

Given a multiset $S$ of elements of $G\rtimes_{\varphi}H$, we can define the
type of $S$ also when $G$ is a generic group. It is the sequence
$\bm{\lambda}=(\lambda_{0},\ldots,\lambda_{e-1})$, where $\lambda_{i}$ is the
number of times that $h_{i}$ appears in $\pi_{2}(S)$ and $\pi_{2}$ is the
projection over $H$. Note that two multisets $S_{1}$ of
$G_{1}\rtimes_{\varphi}H$ and $S_{2}$ of $G_{2}\rtimes_{\varphi}H$ can have
the same type even though $G_{1}$ is not equal to $G_{2}$.

Given a semidirect product $G_{1}\rtimes_{\varphi}H$, we can define the
corresponding product $G_{2}\rtimes_{\varphi^{\prime}}H$ where
$\varphi^{\prime}(h)$ is the identity on $G_{2}$ (resp
$\varphi^{\prime}(h)(x)=-x$ for any $x\in G_{2}$) if and only if $\varphi(h)$
is the identity on $G_{1}$ (resp $\varphi(h)(x)=-x$ for any $x\in G_{1}$). For
simplicity, we will denote, with abuse of notation, by
$G_{2}\rtimes_{\varphi}H$ this semidirect product.

######  Proposition 3.3.

Let $H$ be a finite group, let $\varphi$ satisfy property $(*)$, and suppose
that, for infinitely many primes $p$, any subset of a given type
$\bm{\lambda}=(\lambda_{0},\ldots,\lambda_{e-1})$ of
$\mathbb{Z}_{p}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{p}\rtimes_{\varphi}H}\\}$
is sequenceable. Then also any subset of type $\bm{\lambda}$ of
$\mathbb{Z}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}\rtimes_{\varphi}H}\\}$
is sequenceable.

###### Proof.

Consider a subset $S$ of
$\mathbb{Z}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}\rtimes_{\varphi}H}\\}$
of type $\bm{\lambda}$. Let
$p>\max\limits_{z\in\Upsilon(S)}|\pi_{\mathbb{Z}}(z)|$ be a prime such that
any subset of
$\mathbb{Z}_{p}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{p}\rtimes_{\varphi}H}\\}$
of type $\bm{\lambda}$ is sequenceable.

Here we note that, since $\mathbb{Z}\rtimes_{\varphi}H$ and
$\mathbb{Z}_{p}\rtimes_{\varphi}H$ are defined with corresponding maps
$\varphi$, the map $\pi_{p}\times
id:\mathbb{Z}\rtimes_{\varphi}H\rightarrow\mathbb{Z}_{p}\rtimes_{\varphi}H$ is
a group homomorphism. Indeed in this case we have that

$(\pi_{p}\times id)(x_{1},k_{1})\cdot(\pi_{p}\times
id)(x_{2},k_{2})=(x_{1}\pmod{p},k_{1})\cdot(x_{2}\pmod{p},k_{2})=$
$=((x_{1}+\varphi(k_{1})(x_{2}))\pmod{p},k_{1}\cdot_{H}k_{2})=(\pi_{p}\times
id)[(x_{1},k_{1})\cdot(x_{2},k_{2})].$

Then $\Upsilon(S)$ and the kernel of the map $\pi_{p}\times
id:\mathbb{Z}\rtimes_{\varphi}H\to\mathbb{Z}_{p}\rtimes_{\varphi}H$ are
disjoint sets. Therefore, since $\pi_{p}\times id$ is the identity on $H$ and
$S$ is of type $\bm{\lambda}$, we have $(\pi_{p}\times id)(S)$ also is of type
$\bm{\lambda}$. It then follows from Lemma 3.1 that $S$ also is sequenceable.
∎

######  Remark 3.4.

In the previous proposition, we can see why property $(*)$ is needed in this
discussion. Indeed we need that $\varphi(a)$ is a group automorphism of
$\mathbb{Z}$ and hence it maps $1$ in a generator of $\mathbb{Z}$. It follows
that $\varphi(a)(1)\in\\{1,-1\\}$ or, equivalently, $\varphi$ satisfies
property $(*)$.

We now consider the free abelian group $\mathbb{Z}^{n}$ of rank $n$. We obtain
the following proposition whose proof is omitted since it is the same as
Proposition 4.3 of [5].

######  Proposition 3.5.

Let $H$ be a finite group, let $\varphi$ satisfy property $(*)$, and suppose
that any subset of a given type
$\bm{\lambda}=(\lambda_{0},\ldots,\lambda_{e-1})$ of
$\mathbb{Z}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}^{n}\rtimes_{\varphi}H}\\}$
is sequenceable. Then also any subset of type $\bm{\lambda}$ of
$\mathbb{Z}^{n}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}^{n}\rtimes_{\varphi}H}\\}$
(for any $n\geq 2$) is sequenceable.

From the previous proposition, we deduce this result. Also in this case we
omit the proof since it is the same as Theorem 4.4 of [5]

######  Theorem 3.6.

Let $H$ be a finite group, let $\varphi$ satisfy property $(*)$, and suppose
that, for infinitely many primes $p$, any subset of a given type
$\bm{\lambda}=(\lambda_{0},\ldots,\lambda_{e-1})$ of
$\mathbb{Z}_{p}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{p}\rtimes_{\varphi}H}\\}$
is sequenceable. Then also any subset of
$G\rtimes_{\varphi}H\setminus\\{0_{G\rtimes_{\varphi}H}\\}$ of type
$\bm{\lambda}$ is sequenceable provided that $G$ is torsion-free and abelian.

######  Corollary 3.7.

Let $H$ be a finite group, let $\varphi$ satisfy property $(*)$, $k$ be a
positive integer, and suppose that, for infinitely many primes $p$, any subset
of size $k$ of
$\mathbb{Z}_{p}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{p}\rtimes_{\varphi}H}\\}$
is sequenceable. Then also any subset of size $k$ of
$G\rtimes_{\varphi}H\setminus\\{0_{G\rtimes_{\varphi}H}\\}$ is sequenceable
provided that $G$ is torsion-free and abelian.

Given an element $g$ of an abelian group $G$, we denote by $o(g)$ the
cardinality of the cyclic subgroup $\langle g\rangle$ generated by $g$.
Furthermore, we set

$\vartheta(G)=\min_{0_{G}\neq g\in G}o(g).$

As a consequence of Theorem 3.6, and with essentially the same proof of
Theorem 4.6 of [5], we can prove the following result.

######  Theorem 3.8.

Let $H$ be a finite group, let $\varphi$ satisfy property $(*)$, and suppose
that, for infinitely many primes $p$, any subset of
$\mathbb{Z}_{p}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{p}\rtimes_{\varphi}H}\\}$
of a given type $\bm{\lambda}=(\lambda_{0},\ldots,\lambda_{e-1})$ is
sequenceable. Then there exists a positive integer $N(\bm{\lambda})$ such that
any subset of type $\bm{\lambda}$ of
$G\rtimes_{\varphi}H\setminus\\{0_{G\rtimes_{\varphi}H}\\}$ is sequenceable
provided that $G$ is abelian and $\vartheta(G)>N(\bm{\lambda})$.

######  Corollary 3.9.

Let $H$ be a finite group, let $\varphi$ satisfy property $(*)$, $k$ be a
positive integer, and suppose that, for infinitely many primes $p$, any subset
of size $k$ of
$\mathbb{Z}_{p}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{p}\rtimes_{\varphi}H}\\}$
is sequenceable. Then, there exists a positive integer $N(k)$ such that any
subset of size $k$ of
$G\rtimes_{\varphi}H\setminus\\{0_{G\rtimes_{\varphi}H}\\}$ is sequenceable
provided that $G$ is abelian and $\vartheta(G)>N(k)$.

Therefore, as a consequence of Theorem 2.8, and recalling that the only
exceptions to that theorem, in case of dihedral groups and $k\leq 12$, are for
$k=4$ and $p=3$ (see Theorem 3.4 of [16]), we can prove our first asymptotic
result.

######  Theorem 3.10.

Let $G$ be a group such that $\vartheta(G)>N(k)$. Then any subset $S$ of
$Dih(G)$ such that $|S|\leq 12$ is sequenceable.

######  Remark 3.11.

Because of Remark 2.9, with essentially the same proof of Theorem 3.10, we can
prove that if $S$ is a subset of $Dih(G)$ not of type $(k,0)$, $k\leq 12$ and
$\vartheta(G)>N(k)$, then $S$ admits a linear sequencing.

This implies that, under the hypothesis of Theorem 3.10, $S$ admits a linear
sequencing unless $s_{k}$ is unavoidably the identity.

###  3.1 An Explicit Upper Bound

Now, in case $G=\mathbb{Z}_{m}$, and adapting again the procedure of [5], we
provide an explicit upper bound on the numbers $N(\bm{\lambda})$ and $N(k)$.
First, we need to recall some elementary facts about the solutions of a linear
system over fields and commutative rings.

######  Theorem 3.12 (Corollary of Cramer’s Theorem).

Let

$\begin{cases}m_{1,1}x_{1}+\dots+m_{1,k}x_{k}=b_{1}\\\ \dots\\\
m_{l,1}x_{1}+\dots+m_{l,k}x_{k}=b_{l}\end{cases}$ (3)

be a linear system over a commutative ring $R$ whose associated matrix
$M=(m_{i,j})$ is an $l\times k$ matrix over $R$. Let $(x_{1},\dots,x_{k})$ be
a solution of the system and let $M^{\prime}$ be a $k^{\prime}\times
k^{\prime}$ square, nonsingular (i.e. $\det(M^{\prime})$ is invertible in
$R$), submatrix of $M$. We assume, without loss of generality, that
$M^{\prime}$ is obtained by considering the first $k^{\prime}$ rows and the
first $k^{\prime}$ columns of $M$. Then $x_{1},\dots,x_{k}$ satisfy the
following relation

$\begin{bmatrix}x_{1}\\\ x_{2}\\\ \vdots\\\
x_{k^{\prime}}\end{bmatrix}=\begin{bmatrix}m_{1,1}&m_{1,2}&m_{1,3}&\dots&m_{1,k^{\prime}}\\\
m_{2,1}&m_{2,2}&m_{2,3}&\dots&m_{2,k^{\prime}}\\\
\vdots&\vdots&\vdots&&\vdots\\\
m_{k^{\prime},1}&m_{k^{\prime},2}&m_{k^{\prime},3}&\dots&m_{k^{\prime},k^{\prime}}\end{bmatrix}^{-1}\begin{bmatrix}b_{1}-(m_{1,k^{\prime}+1}x_{k^{\prime}+1}+\dots+m_{1,k}x_{k})\\\
b_{2}-(m_{2,k^{\prime}+1}x_{k^{\prime}+1}+\dots++m_{2,k}x_{k})\\\ \vdots\\\
b_{k^{\prime}}-(m_{k^{\prime},k^{\prime}+1}x_{k^{\prime}+1}+\dots+m_{k^{\prime},k}x_{k})\end{bmatrix}.$

Equivalently, set

$M^{\prime}_{j}:=\begin{bmatrix}m_{1,1}&m_{1,2}&\dots&m_{1,j-1}&b_{1}-(m_{1,k^{\prime}+1}x_{k^{\prime}+1}+\dots+m_{1,k}x_{k})&\dots&m_{1,k^{\prime}}\\\
m_{2,1}&m_{2,2}&\dots&m_{2,j-1}&b_{2}-(m_{2,k^{\prime}+1}x_{k^{\prime}+1}+\dots++m_{2,k}x_{k})&\dots&m_{2,k^{\prime}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&&\vdots\\\
m_{k^{\prime},1}&m_{k^{\prime},2}&\dots&m_{k^{\prime},j-1}&b_{k^{\prime}}-(m_{k^{\prime},k^{\prime}+1}x_{k^{\prime}+1}+\dots+m_{k^{\prime},k}x_{k})&\dots&m_{k^{\prime},k^{\prime}}\end{bmatrix}$

we have that, for any $j\in[1,k^{\prime}]$

$x_{j}=\frac{\det(M^{\prime}_{j})}{\det(M^{\prime})}.$ (4)

Now assume that $R$ is a field and that the system (3) admits at least one
solution. Then, given a maximal square nonsingular submatrix $M^{\prime}$ of
$M$, and fixed $x_{k^{\prime}+1},\dots,x_{k}$ in $R$, we have that
$x_{1},x_{2},\dots,x_{k}$ are a solution of the system (3) if and only if
$x_{1},\dots,x_{k^{\prime}}$ satisfy the relation (4).

######  Remark 3.13.

In case the system (3) of Theorem 3.12 is homogeneous (i.e.
$b_{1}=b_{2}=\dots=b_{l}=0$), due to the Laplace expansion of the determinant,
$\det(M^{\prime}_{j})$ can be written as a linear combination of
$x_{k^{\prime}+1},\dots,x_{k}$. More precisely, denoting by $M^{\prime}_{i,j}$
the matrix obtained by deleting the $i$-th row and the $j$-th column of
$M^{\prime}$, we have that

$\det(M^{\prime}_{j})=\sum_{i=1}^{k^{\prime}}(-1)^{j+i+1}\\\
det(M^{\prime}_{i,j})(m_{i,k^{\prime}+1}x_{k^{\prime}+1}+\dots+m_{i,k}x_{k}).$
(5)

Now we are ready to provide, in case of groups of type
$\mathbb{Z}_{m}\rtimes_{\varphi}H$ (where $\varphi$ satisfies property $(*)$),
a quantitative version of Theorem 3.8. In the following proof, developing the
notations of the previous sections, given a set $S$ of size $k$ and an
ordering $\bm{x}=(x_{1},\dots,x_{k})$ of $S$, we denote by $s_{i}(\bm{x})$ the
partial sum $x_{1}\cdot x_{2}\dots x_{i-1}\cdot x_{i}$ where $s_{0}(\bm{x})$
is always zero.

######  Theorem 3.14.

Let $H$ be a finite group, let $\varphi$ satisfy property $(*)$, and suppose
that, for infinitely many primes $p$, any subset of a given type
$\bm{\lambda}=(\lambda_{0},\ldots,\lambda_{e-1})$ of
$\mathbb{Z}_{p}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{p}\rtimes_{\varphi}H}\\}$
is sequenceable. Then also any subset of type $\bm{\lambda}$ of
$\mathbb{Z}_{m}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{m}\rtimes_{\varphi}H}\\}$
is sequenceable provided that the prime factors of $m$ are all greater than
$k!$ where $k=\lambda_{0}+\lambda_{1}+\dots+\lambda_{e-1}$.

###### Proof.

Let us assume, by contradiction, that there exists a subset
$S=\\{x_{1},\dots,x_{k}\\}$ of type $\bm{\lambda}$ in
$\mathbb{Z}_{m}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{m}\rtimes_{\varphi}H}\\}$
that is not sequenceable and all the prime factors of $m$ are larger than
$k!$. Here $S$ would be a set of type $\bm{\lambda}$ whose elements do not
admit an ordering with different partial sums.

This implies that for any possible permutation $\omega\in\mathrm{Sym}(k)$,
considered the ordering

$\bm{x}_{\omega}=(x_{\omega(1)},x_{\omega(2)},\dots,x_{\omega(k)}),$

there exist $h$ and $t$, with $h,t\in[0,k]$, $\\{h,t\\}\not=\\{0,k\\}$ and
$h\not=t$, such that $s_{h}(\bm{x}_{\omega})=s_{t}(\bm{x}_{\omega})$, that is,
assuming $h<t$,

$x_{\omega(h+1)}\cdot x_{\omega(h+2)}\dots x_{\omega(t-1)}\cdot
x_{\omega(t)}=0.$ (6)

Therefore we would have a solution
$(\pi_{\mathbb{Z}_{m}}(x_{1}),\dots,\pi_{\mathbb{Z}_{m}}(x_{k}))^{T}$ to the
system of equations derived from (6) by considering the projection over
$\mathbb{Z}_{m}$. In the following, for simplicity, we will denote
$\pi_{\mathbb{Z}_{m}}(x_{j})$ by $x_{j}^{1}$ and, using this notation, we are
considering the system:

$x^{1}_{\omega(h+1)}+\varphi_{\pi_{H}(x_{\omega(h+1)})}x^{1}_{\omega(h+2)}+\dots+\varphi_{\pi_{H}(x_{\omega(h+1)})\cdots\pi_{H}(x_{\omega(t-1)})}x^{1}_{\omega(t)}=0$
(7)

where, using the notation of the previous sections, and since $\varphi$
satisfies property $(*)$, we set $\varphi(a)(x)=\varphi_{a}x$ for some
constant $\varphi_{a}\in\\{1,-1\\}$. Denoted by $M$ the $(k!)\times k$ matrix
of the coefficients of this system, we note that, since $\varphi$ satisfies
property $(*)$, the coefficients of $M$ belong to $\\{-1,0,1\\}$. It follows
that the determinant of any square submatrix of $M$ (that is big at most
$k\times k$) is a sum of at most $k!$ terms that belong to $\pm 1$. Therefore,
if all the prime factors of $m$ are larger than $k!$, a square submatrix
$M^{\prime}$ of $M$ is nonsingular in $\mathbb{Z}_{m}$ if and only if it is
nonsingular in $\mathbb{Q}$.

Following the proof of Thorem 4.12 of [5] and assuming the existence a subset
$S=\\{x_{1},\dots,x_{k}\\}$ of type $\bm{\lambda}$ in
$\mathbb{Z}_{m}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{m}\rtimes_{\varphi}H}\\}$
that is not sequenceable, we find a subset of the same type in
$\mathbb{Q}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Q}\rtimes_{\varphi}H}\\}$
that is also not sequenceable. However, because of Theorem 3.6 and since
$\mathbb{Q}$ is a torsion-free abelian group, any subset $\widetilde{S}$ of
type $\bm{\lambda}$ of
$G\rtimes_{\varphi}H\setminus\\{0_{G\rtimes_{\varphi}H}\\}$ is sequenceable.
But this is a contradiction and hence the thesis is verified. ∎

######  Corollary 3.15.

Let $H$ be a finite group, let $\varphi$ satisfy property $(*)$, $k$ be a
positive integer, and suppose that, for infinitely many primes $p$, any subset
of size $k$ of
$\mathbb{Z}_{p}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{p}\rtimes_{\varphi}H}\\}$
is sequenceable. Then also any subset of size $k$ of
$\mathbb{Z}_{m}\rtimes_{\varphi}H\setminus\\{0_{\mathbb{Z}_{m}\rtimes_{\varphi}H}\\}$
is sequenceable provided that the prime factors of $m$ are all greater than
$k!$.

If we consider direct products of type $\mathbb{Z}_{m}\times H$, we can
improve the result of Theorem 3.14. Indeed, with essentially the same proof of
Theorem 22 of [5], we can state the following.

######  Theorem 3.16.

Let $H$ be a finite group, $k$ be a positive integer, and suppose that, for
infinitely many primes $p$, any subset of size $k$ of $\mathbb{Z}_{p}\times
H\setminus\\{0_{\mathbb{Z}_{p}\times H}\\}$ is sequenceable. Then also any
subset of size $k$ of $\mathbb{Z}_{m}\times
H\setminus\\{0_{\mathbb{Z}_{m}\times H}\\}$ is sequenceable provided that the
prime factors of $m$ are all greater than $k!/2$.

As regards dihedral groups, as a consequence of Theorem 2.8, we can prove
Theorem 3.17.

######  Theorem 3.17.

Let $m$ be such that all its prime factors are larger than $k!$. Then any
subset $S$ of $D_{2m}$ such that $|S|\leq 12$ is sequenceable.

######  Remark 3.18.

Because of Remark 2.9, with essentially the same proof of Theorem 3.17, we can
prove that if $S$ is a subset of $D_{2m}$ not of type $(k,0)$, $k\leq 12$ and
all the prime factors of $m$ are larger than $k!$, then $S$ admits a linear
sequencing.

This implies that, under the hypothesis of Theorem 3.17, $S$ admits a linear
sequencing unless $s_{k}$ is unavoidably the identity.

##  4 Weak Sequenceability

In this section we consider again subsets $S$ of a group $(G,\cdot)$ of the
form $G=\mathbb{Z}_{p}\rtimes_{\varphi}H$ which is the semidirect product of
$(\mathbb{Z}_{p},+)$ and another group $(H,\cdot)$. As in the previous section
we indicate by $x\cdot a$ (or by $y\cdot b)$, where $x\in\mathbb{Z}_{p}$ and
$a\in H$, a generic element of $G$. We investigate the $t$-weak
sequenceability of subsets of semidirect products. Recall that in addition to
requiring that $x_{i}-x_{j}\not=0$ for $1\leq i<j\leq k$ and $a_{i}=a_{j}$, we
seek an ordering having no two of its partial sums $s_{i},s_{j}$ equal for
$1\leq i<j\leq k$ and $|i-j|\leq t$. Hence, modifying the expression of
Equation (1), we define, for $t<k$, the following polynomial

$q_{\bm{a}}=\prod_{\begin{subarray}{c}1\leq i<j\leq k\\\
a_{i}=a_{j}\end{subarray}}(x_{j}-x_{i})\prod_{\begin{subarray}{c}0\leq i<j\leq
k\\\ b_{i}=b_{j}\\\ j-i\leq t\\\ j\neq
i+1\end{subarray}}(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{j-1}}x_{j}).$ (8)

In this case we have that a set $S=\\{x_{1}\cdot a_{1},x_{2}\cdot
a_{2},\ldots,x_{k}\cdot a_{k}\\}\subseteq G\setminus\\{0\\}$ of size $k$ is
$t$-weak sequenceable if there exists an ordering $\bm{x_{a}}=\left(x_{1}\cdot
a_{1},x_{2}\cdot a_{2},\ldots,x_{k}\cdot a_{k}\right)$ of its elements such
that $q_{\bm{a}}(x_{1},\dots,x_{k})\not=0$.

Now, given a set $S=\\{x_{1}\cdot a_{1},x_{2}\cdot a_{2},\ldots,x_{k}\cdot
a_{k}\\}$ of $k$ elements, the idea is to fix, a priori, the first $h$
elements $(x_{1}\cdot a_{1},x_{2}\cdot a_{2},\ldots,x_{h}\cdot a_{h})$, where
$h$ is not too big, of the ordering in such a way that no of its partial sums
$s_{i},s_{j}$ are equal when $|i-j|\leq t$ and $1\leq i<j\leq h$. This can be
expressed by requiring that $q_{\bm{a}^{\prime}}(x_{1},\cdots,x_{h})\not=0$
where $\bm{a}^{\prime}=(a_{1},\dots,a_{h})$ and we show that this can be done
due the following Proposition whose proof is inspired by that of Proposition
2.2 of [4].

######  Proposition 4.1.

Let $S=\\{x_{1}\cdot a_{1},x_{2}\cdot a_{2},\ldots,x_{k}\cdot
a_{k}\\}\subseteq G\setminus\\{0\\}$ be a set of size $k$ and let $h$ and $t$
be positive integers such that $k-h\geq 2(t-1)$ and $h\geq t-1$. Denoted by
$\bar{a}$ the element that appears the most in $\pi_{2}(S)$, we assume that at
least $2(t-1)$ elements of $S$ have $\bar{a}$ as second coordinate. Then there
exists an ordering of $h$-elements of $S$ that we denote, up to relabeling,
with $(x_{1}\cdot a_{1},x_{2}\cdot a_{2},\ldots,x_{h}\cdot a_{h})$, such that

  * (a)

$q_{\bm{a}^{\prime}}(x_{1},\cdots,x_{h})\not=0$ where
$\bm{a}^{\prime}=(a_{1},\dots,a_{h})$;

  * (b)

$a_{h},a_{h-1},\dots,a_{h-t+2}$ are all equals to $\bar{a}$;

  * (c)

at least $t-1$ elements of $\\{x_{h+1}\cdot a_{h+1},\ldots,x_{k}\cdot
a_{k}\\}$ have $\bar{a}$ as second coordinate.

###### Proof.

Here we choose a set $S^{\prime}$ of $2(t-1)$ elements that have $\bar{a}$ as
the second coordinate, and we set $S^{\prime\prime}:=S\setminus S^{\prime}$.
Now we prove that we can choose, inductively on $\ell$, $(x_{1}\cdot
a_{1},x_{2}\cdot a_{2},\ldots,x_{h-t}\cdot a_{h-t})$ in $S^{\prime\prime}$ and
$(x_{h-t+1}\cdot a_{h-t+1},\ldots,x_{h}\cdot a_{h})$ in $S^{\prime}$ such
that, for any $\ell\leq h$, we have
$q_{(a_{1},\dots,a_{\ell})}(x_{1},\cdots,x_{\ell})\not=0$. Then, properties
$(b)$ and $(c)$ will be guaranteed by our choice for $x_{1}\cdot
a_{1},\ldots,x_{h}\cdot a_{h}$.

BASE CASE: Let $\ell=1$. Since $q_{(a)}(x)=1$ for any $t$ and for any $x\cdot
a$ of both $S^{\prime}$ and $S^{\prime\prime}$, the statement is realized for
$\ell=1$.

INDUCTIVE CASE: Let us assume the statement for $\ell\in\\{1,\dots,m\\}$ and
let us prove it for $\ell=m+1$ where $m+1\leq h$. Since the statement is true
for $\ell=m$, there exists an $m$-tuple $(x_{1}\cdot a_{1},\dots,x_{m}\cdot
a_{m})$ such that

$q_{(a_{1},\dots,a_{m})}(x_{1},\cdots,x_{m})\not=0.$

Given $x\cdot a\in G$, we set $s_{m}:=(x_{1}\cdot a_{1})\cdots(x_{m}\cdot
a_{m})=y_{m}\cdot b_{m}$ and we define $y$ and $s$ such that
$s=s_{m}\cdot(x\cdot a)=y\cdot b$ where $b=b_{m}\cdot a$. Then we have

$\frac{q_{(a_{1},\dots,a_{m},a)}(x_{1},\dots,x_{m},x)}{q_{(a_{1},\dots,a_{m})}(x_{1},\dots,x_{m})}=$
$=\prod_{1\leq i<m+1,\ a=a_{i}}(x-x_{i})\prod_{\begin{subarray}{c}0\leq i<m\\\
m+1-i\leq t,\
b_{i}=b\end{subarray}}(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{m}}x).$

Here, any element $x\cdot a$ of $S\setminus\\{x_{1}\cdot
a_{1},\dots,x_{m}\cdot a_{m}\\}$ satisfies $\prod_{1\leq
i<m+1}(x-x_{i})\not=0$ if $a=a_{i}$. Hence, to have
$\frac{q_{(a_{1},\dots,a_{m})}(x_{1},\dots,x_{m},x)}{q_{(a_{1},\dots,a_{m},a)}(x_{1},\dots,x_{m})}\not=0$,
it suffice to find $x$ such that

$\prod_{\max(0,m+1-t)\leq i<m,\
b_{i}=b}(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{m}}x)\neq 0.$

Note that there is at most one $x\cdot a\in S\setminus\\{x_{1}\cdot
a_{1},\dots,x_{m}\cdot a_{m}\\}$ such that

$(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{m}}x)=0$

and $b_{i}=b$. Indeed this would imply that $s_{i}=s$ in $G$ that is
$(x_{i+1}\cdot a_{i+1})\cdot\ldots\cdot(x_{m}\cdot a_{m})\cdot(x\cdot a)=0$
which is satisfied by exactly one element $x\cdot a$ of $G$. Since those
relations are at most $t-1$, we have at most $t-1$ values $x\cdot a$ in
$S\setminus\\{x_{1}\cdot a_{1},\dots,x_{m}\cdot a_{m}\\}$ such that

$\prod_{\max(0,m+1-t)\leq i<m,\
b_{i}=b}(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{m}}x)=0.$

Here we have that either $h-t+1>m$ or $h-t+1\leq m$. In the first case, we
have that

$|S^{\prime\prime}\setminus\\{x_{1}\cdot a_{1},\dots,x_{m}\cdot
a_{m}\\}|=k-2(t-1)-m\geq t+1$

where the inequality holds since $k-2(t-1)-m=(k-m)-2(t-1)$ and $k-m\geq
k-h\geq 2(t-1)$. It follows that there exists $x\cdot a\in
S^{\prime\prime}\setminus\\{x_{1}\cdot a_{1},\dots,x_{m}\cdot a_{m}\\}$ such
that

$\prod_{\max(0,m+1-t)\leq i<m,\
b_{i}=b}(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{m}}x)\not=0.$

In the second case, we have that

$|S^{\prime}\setminus\\{x_{1}\cdot a_{1},\dots,x_{m}\cdot
a_{m}\\}|=2(t-1)-(m-(h-t+1))\geq t$

where the last inequality holds since $m\leq h-1$ and hence
$2(t-1)-(m-(h-t+1))\geq 2(t-1)-(h-1-(h-t+1))=t$. It follows that also in this
case, there exists $x\cdot a\in S^{\prime}\setminus\\{x_{1}\cdot
a_{1},\dots,x_{m}\cdot a_{m}\\}$ such that

$\prod_{\max(0,m+1-t)\leq i<m,\
b_{i}=b}(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{m}}x)\not=0.$

In both cases, we can find $x_{m+1}\cdot a_{m+1}$ that satisfies property
$(b)$ and $(c)$ and which it is such that

$\frac{q_{(a_{1},\dots,a_{m},a_{m+1})}(x_{1},\dots,x_{m},x_{m+1})}{q_{(a_{1},\dots,a_{m})}(x_{1},\dots,x_{m})}=\prod_{1\leq
i<m+1,\ a_{m+1}=a_{i}}(x_{m+1}-x_{i})\cdot$ $\cdot\prod_{\max(0,m+1-t)\leq
i<m,\ b_{i}=b_{m+1}}(x_{i+1}+\varphi_{a_{i+1}}x_{i+2}+\varphi_{a_{i+1}\cdot
a_{i+2}}x_{i+3}+\dots+\varphi_{a_{i+1}\dots a_{m}}x_{m+1})\not=0.$

Since $\mathbb{Z}_{p}$ is a field and due to the inductive hypothesis
$q_{(a_{1},\dots,a_{m},a)}(x_{1},\dots,x_{m})\neq 0$, we also have that

$\frac{q_{(a_{1},\dots,a_{m},a_{m+1})}(x_{1},\dots,x_{m},x_{m+1})}{q_{(a_{1},\dots,a_{m})}(x_{1},\dots,x_{m})}\cdot\\\
q_{(a_{1},\dots,a_{m})}(x_{1},\dots,x_{m})=q_{(a_{1},\dots,a_{m},a_{m+1})}(x_{1},\dots,x_{m+1})\not=0$
(9)

which completes the proof. ∎

######  Remark 4.2.

Note that the hypothesis that at least $2(t-1)$ elements of $S$ has the same
second coordinate is always satisfied when

$k>(2t-3)\frac{|G|}{p}=(2t-3)|H|.$

This means that the approach of Proposition 4.1 can be always tried assuming
that $k$ is big enough. For smaller values of $k$ one could fix the first
elements differently, as done for example in Proposition 2.2 of [4], but,
since then one should consider a huge number of cases (see also the following
Remark 4.3), we consider explicitly only this asymptotic case.

In the following we assume that we have fixed, according to Proposition 4.1,

$\\{x_{1}\cdot a_{1},x_{2}\cdot a_{2},\ldots,x_{h}\cdot a_{h}\\}\subseteq S$

that satisfies properties $(a),(b)$ and $(c)$ of such proposition. We note
that every $x\cdot a\in S\setminus\\{x_{1}\cdot a_{1},x_{2}\cdot
a_{2},\ldots,x_{h}\cdot a_{h}\\}$ is such that $x-x_{i}\not=0$ for any
$i\in\\{1,\dots,h\\}$ such that $a=a_{i}$. Therefore, if $\bm{a}$ is a
quotient sequencing that extends $\bm{a}^{\prime\prime}$, it is left to find a
nonzero point for the polynomial

$\frac{q_{\bm{a}}(x_{1},\dots,x_{h},x_{h+1},\dots,x_{k})}{q_{\bm{a}^{\prime}}(x_{1},\cdots,x_{h})\prod_{1\leq
i\leq h<j\leq k,\ a_{j}=a_{i}}(x_{j}-x_{i})}.$

Since the free variables are now $x_{h+1},\dots,x_{k}$, we set $\ell:=k-h$ and
$z_{i}:=x_{i+h}$; here the constrain $k-h\geq 2(t-1)$ of Proposition 4.1
becomes $\ell\geq 2(t-1)$. Then we denote by $h_{\bm{a},\ell}$ the polynomial

$h_{\bm{a},\ell}(z_{1},\dots,z_{\ell}):=\frac{q_{\bm{a}}(x_{1},\dots,x_{h},z_{1},\dots,z_{\ell})}{q_{\bm{a}^{\prime}}(x_{1},\cdots,x_{h})\prod_{1\leq
i\leq 1<j\leq\ell,\ a_{j}=a_{i}}(z_{j}-x_{i})}.$ (10)

Since we are also assuming that $k-\ell\geq t-1$, that is,
$k-(t-1)\geq\ell\geq 2(t-1)$, denoting by $\bm{a}^{\prime\prime}$ the quotient
sequencing given by the last $\ell$ elements of $\bm{a}$, we obtain the
following expression

$h_{\bm{a},\ell}(z_{1},\cdots,z_{\ell})=q_{\bm{a}^{\prime\prime}}(z_{1},\cdots,z_{\ell})\cdot\\\
\prod_{\begin{subarray}{c}0\leq i\leq t-1;\\\ 1\leq j\leq t-i-1;\\\
b_{k-\ell-i-1}=b_{(k-\ell)+j}\end{subarray}}(x_{k-\ell-i}+\varphi_{\bar{a}}x_{k-\ell-i+1}+\cdots+\varphi_{\bar{a}^{i}}x_{k-\ell}+\varphi_{\bar{a}^{i+1}}z_{1}+\cdots+\varphi_{\bar{a}^{i+1}\cdot
a_{k-\ell+1}\cdot\dots\cdot a_{k-\ell+j-1}}z_{j}).$ (11)

Now we aim to apply the Non-Vanishing Corollary (of the Combinatorial
Nullstellensatz) to the polynomial $h_{\bm{a},\ell}$. For this purpose, it is
enough to consider the terms of $h_{\bm{a},\ell}$ of maximal degree in the
variables $z_{1},\dots,z_{\ell}$ that are the ones where no $x_{i}$ appears.
We denote by $r_{\bm{a},\ell}$ the polynomial given by those terms, that is

$r_{\bm{a},\ell}:=q_{\bm{a}^{\prime\prime}}(z_{1},\cdots,z_{\ell})\cdot\\\
\prod_{\begin{subarray}{c}0\leq i\leq t-1;\\\ 1\leq j\leq t-i-1;\\\
b_{k-\ell-i-1}=b_{(k-\ell)+j}\end{subarray}}(\varphi_{\bar{a}^{i+1}}z_{1}+\varphi_{\bar{a}^{i+1}\cdot
a_{k-\ell+1}}z_{2}+\cdots+\varphi_{\bar{a}^{i+1}\cdot
a_{k-\ell+1}\cdot\dots\cdot a_{k-\ell+j-1}}z_{j}).$ (12)

Since $b_{k-\ell-i-1}=b_{(k-\ell)+j}$ if and only if

$a_{k-\ell-i}\cdot a_{k-\ell-1}\cdot\ldots\cdot a_{k-\ell}\cdot
a_{k-\ell+1}\cdot\ldots\cdot a_{k-\ell+j}=id$

and $a_{k-\ell-i}=a_{k-\ell-1}=\ldots=a_{k-\ell}=\bar{a}$, we can state the
following, simple but very powerful, remark.

######  Remark 4.3.

Since we are assuming that $a_{h},a_{h-1},\dots,a_{h-t+1}$ are all equals to
$\bar{a}$, the expression of $r_{\bm{a},\ell}$ does not depend on all the
quotient sequence $\bm{a}$ but only by $\bm{a}^{\prime\prime}$ and $\bar{a}$.
Here property $b$ of Proposition 4.1 is crucial since, otherwise, given
$\bm{a}^{\prime\prime}$, we would have $|H|^{t-1}$ possibilities for the
polynomial $r_{\bm{a},\ell}(z_{1},\cdots,z_{\ell})$ according to the sequence
$a_{h},a_{h-1},$ $\ldots,a_{h-t+1}$.

In the following, we just denote this polynomial by
$r_{\bm{a}^{\prime\prime},\bar{a},\ell}$.

Indeed Remark 4.3 means that, after these manipulations, we are left to
consider a polynomial that does not depend on $k=|S|$, and hence we have
chances to get a result that is very general on $k$.

To apply the Non-Vanishing Corollary, we also need to find, for any quotient
sequencing $\bm{a}^{\prime\prime}$ of length $\ell$, a nonzero coefficient in
$r_{\bm{a}^{\prime\prime},\bar{a},\ell}$ that divides the bounding monomial.
Moreover, due to property $c$ of Proposition 4.1, we may assume that $\bar{a}$
appears at least $t-1$ times in $\bm{a}^{\prime\prime}$.

###  4.1 Computational Results

As done for the sequenceability problem in the more general setting, we
consider here groups of type $D_{2p}$ and $G_{3p}$. Also, since the case
$G=\mathbb{Z}_{p}\times\mathbb{Z}_{e}$ has not been considered in [4], we
consider these groups here.

######  Example 4.4.

Example Let $t=4$, $\ell=2(t-1)=6$ and $\bar{a}=0$. Let $p$ be a prime and
suppose $S\subseteq D_{2p}\setminus\\{0\\}$ with $|S|=k\geq 11$ and suppose
that $S$ has at least $6$ elements with coset $\bar{a}$. Let
$S^{\prime}=S\setminus\\{x_{1}\cdot a_{1},\ldots,x_{k-\ell}\cdot
a_{k-\ell}\\}$ be of type $(3,3)$. The sequence
$\bm{a^{\prime\prime}}=(1,0,1,0,1,0)$ has partial sums $(0,1,1,0,0,1,1)$.

By Proposition 4.1, we desire a sequencing of $S^{\prime}$ of the form

$\left(z_{1}\cdot 1,z_{2}\cdot 0,z_{3}\cdot 1,z_{4}\cdot 0,z_{5}\cdot
1,z_{6}\cdot 0\right).$

The polynomial we need to study is the one defined in (12). Hence we get

$\displaystyle
r_{\bm{a}^{\prime\prime},\bar{a},\ell}=(z_{1}-z_{2}-z_{3})(z_{3}-z_{1})(z_{5}-z_{1})(z_{4}-z_{2})(z_{6}-z_{2})(z_{5}-z_{3})(z_{6}-z_{4})$
$\displaystyle(z_{1}-z_{2}-z_{3})(z_{1}-z_{2}-z_{3}+z_{4})(z_{2}+z_{3}-z_{4}-z_{5})(z_{3}-z_{4}-z_{5})(z_{3}-z_{4}-z_{5}+z_{6})$

To apply the Non-Vanishing Corollary we need a monomial of this polynomial
which divides the bounding monomial
$z_{1}^{2}z_{2}^{2}z_{3}^{2}z_{4}^{2}z_{5}^{2}z_{6}^{2}$ with a nonzero
coefficient. One such is
$z_{1}^{2}z_{2}^{2}z_{3}^{2}z_{4}^{2}z_{5}^{2}z_{6}^{2}$, which has
coefficient $-12$. Hence whenever $S$ has this form it is $t$-weakly
sequenceable.

We are now ready to prove the main result of this section.

######  Theorem 4.5.

Let $n=pe$ with $p>3$ prime and let $G$ be a group of size $n$. Then subsets
$S$ of size $k$ of $G$ are $t$-weakly sequenceable whenever $e\in\\{1,2,3\\}$,
$k$ is large enough and $t\leq 6$.

###### Proof.

The case $e=1$ follows from the results of [4].

As already noted in the proof of Theorem 2.8, a group of size $2p$ is either
$\mathbb{Z}_{p}\times\mathbb{Z}_{2}$ or the dihedral group $D_{2p}$ and a
group of size $3p$ is either $\mathbb{Z}_{p}\times\mathbb{Z}_{3}$ or $G_{3p}$
for some value of $r$. Here, due to Remark 4.2, if $k\geq(2t-3)e+1$, $S$
admits a coset whose size is at least $2(t-1)$ and hence, due to Proposition
4.1, we can apply Non-Vanishing Corollary to the polynomial of Equation (12)
for suitable quotient sequences and nonzero monomial (see Tables
LABEL:tab:6_Prod, LABEL:tab:6_Prod1, LABEL:tab:6_D and LABEL:tab:6_G below).
These results are obtained using SageMath [19].

Here we note that the coefficients presented in Table LABEL:tab:6_G can be
zero only for finitely many primes $p$ (see also the discussion in the proof
of Theorem 2.8). On the other hand, since $3p>k$, if $k$ is large enough all
these coefficients must be nonzero. ∎

The following Tables LABEL:tab:6_Prod, LABEL:tab:6_Prod1, LABEL:tab:6_D and
LABEL:tab:6_G contain the required monomials and their coefficients for the
proof of Theorem 4.5 in the cases
$G\in\\{\mathbb{Z}_{p}\times\mathbb{Z}_{2},\mathbb{Z}_{p}\times\mathbb{Z}_{3},D_{2p},G_{3p}\\}$.

Table 3: Monomials and their coefficients sufficient for the proof of Theorem 4.5 in the case $t=6$ and $G=\mathbb{Z}_{p}\times\mathbb{Z}_{2}$. $\bar{a}$ | $\bm{\lambda}$ | $\mathbf{a^{\prime\prime}}$ | deg | monomial/s | coefficient/s
---|---|---|---|---|---
$0$ | $(5,5)$ |  | $(0,1,0,0,1,1,1,0,0,1)$
---
| 40
---
|
$x_{1}^{4}x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
| $-5\cdot 467$
---
$0$ | $(6,4)$ |  | $(1,0,0,0,1,0,1,0,1,0)$
---
| 38
---
|
$x_{2}^{4}x_{3}^{5}x_{4}^{5}x_{5}^{3}x_{6}^{5}x_{7}^{3}x_{8}^{5}x_{9}^{3}x_{10}^{5}$
---
$x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{5}x_{5}^{3}x_{6}^{5}x_{7}^{3}x_{8}^{5}x_{9}^{3}x_{10}^{5}$
| $-1063$
---
$2^{4}\cdot 503$
$0$ | $(7,3)$ |  | $(1,0,0,0,1,0,0,0,1,0)$
---
| 36
---
| $x_{3}^{2}x_{4}^{6}x_{5}^{2}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{2}x_{10}^{6}$
---
| $2$
---
$0$ | $(8,2)$ |  | $(1,0,0,0,1,0,0,0,0,0)$
---
| 45
---
| $x_{3}^{2}x_{4}^{7}x_{5}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{7}$
---
| $-2^{2}$
---
$0$ | $(9,1)$ |  | $(0,0,0,0,1,0,0,0,0,0)$
---
| 66
---
|
$x_{1}^{5}x_{2}^{7}x_{3}^{7}x_{4}^{8}x_{6}^{7}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $-2^{5}\cdot 3$
---
$1$ | $(4,6)$ |  | $(0,1,1,1,0,0,1,0,1,1)$
---
| 42
---
|
$x_{1}^{3}x_{2}^{5}x_{3}^{5}x_{4}^{5}x_{5}^{3}x_{6}^{3}x_{7}^{5}x_{8}^{3}x_{9}^{5}x_{10}^{5}$
---
| $-1187$
---
$1$ | $(3,7)$ |  | $(1,0,1,1,1,1,1,0,1,0)$
---
| 47
---
|
$x_{1}^{5}x_{2}^{2}x_{3}^{6}x_{4}^{6}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{2}x_{9}^{6}x_{10}^{2}$
---
$x_{1}^{6}x_{2}x_{3}^{6}x_{4}^{6}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{2}x_{9}^{6}x_{10}^{2}$
| $2\cdot 3^{2}\cdot 1511$
---
$23\cdot 499$
$1$ | $(2,8)$ |  | $(1,0,1,1,1,1,1,1,1,0)$
---
| 54
---
|
$x_{1}^{4}x_{2}x_{3}^{6}x_{4}^{7}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
$x_{1}^{5}x_{2}x_{3}^{5}x_{4}^{7}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
| $-2^{5}\cdot 5^{2}$
---
$-4337$
$1$ | $(1,9)$ |  | $(1,0,1,1,1,1,1,1,1,1)$
---
| 62
---
|
$x_{1}^{5}x_{3}^{3}x_{4}^{6}x_{5}^{8}x_{6}^{8}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $-2$
---
$1$ | $(0,10)$ |  | $(1,1,1,1,1,1,1,1,1,1)$
---
| 75
---
|
$x_{1}^{3}x_{2}^{4}x_{3}^{5}x_{4}^{9}x_{5}^{9}x_{6}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
$x_{1}^{3}x_{2}^{5}x_{3}^{5}x_{4}^{8}x_{5}^{9}x_{6}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
| $-2^{4}\cdot 5^{2}$
---
$-7\cdot 853$
Table 4: Monomials and their coefficients sufficient for the proof of Theorem 4.5 in the case $t=6$ and $G=\mathbb{Z}_{p}\times\mathbb{Z}_{3}$. $\bar{a}$ | $\bm{\lambda}$ | $\mathbf{a^{\prime\prime}}$ | deg | monomial/s | coefficient/s
---|---|---|---|---|---
$0$ | $(5,3,2)$ |  | $(1,0,0,1,0,0,0,2,1,2)$
---
| 25
---
|
$x_{1}^{2}x_{2}x_{3}^{4}x_{4}^{2}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}x_{9}^{2}x_{10}$
---
| $2\cdot 3^{2}$
---
$0$ | $(5,4,1)$ |  | $(1,0,0,1,0,0,0,1,1,2)$
---
| 22
---
| $x_{3}^{4}x_{4}x_{5}^{3}x_{6}^{4}x_{7}^{4}x_{8}^{3}x_{9}^{3}$
---
| $1$
---
$0$ | $(5,5,0)$ |  | $(0,1,0,1,0,0,0,1,1,1)$
---
| 33
---
|
$x_{1}^{2}x_{2}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{2}x_{9}^{4}x_{10}^{4}$
---
| $3\cdot 5^{2}$
---
$0$ | $(6,2,2)$ |  | $(1,0,0,1,0,0,0,0,2,2)$
---
| 25
---
| $x_{3}^{3}x_{4}x_{5}^{4}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}x_{10}$
---
| $-1$
---
$0$ | $(6,3,1)$ |  | $(1,0,0,1,0,0,0,0,1,2)$
---
| 30
---
| $x_{2}x_{3}^{5}x_{4}^{2}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{2}$
---
| $-2^{4}$
---
$0$ | $(6,4,0)$ |  | $(1,0,0,1,0,0,0,0,1,1)$
---
| 28
---
| $x_{3}^{3}x_{4}x_{5}^{4}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{2}x_{10}^{3}$
---
| $-1$
---
$0$ | $(7,2,1)$ |  | $(1,0,0,1,0,0,0,0,0,2)$
---
| 33
---
| $x_{3}^{3}x_{4}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}$
---
| $-2^{2}$
---
$0$ | $(7,3,0)$ |  | $(1,0,0,1,0,0,0,0,0,1)$
---
| 35
---
| $x_{3}^{3}x_{4}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $-2^{2}$
---
$0$ | $(8,1,1)$ |  | $(1,0,0,0,2,0,0,0,0,0)$
---
| 44
---
| $x_{3}^{2}x_{4}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{7}$
---
| $-2^{2}$
---
$0$ | $(8,2,0)$ |  | $(1,0,0,0,1,0,0,0,0,0)$
---
| 42
---
| $x_{3}^{2}x_{4}^{5}x_{5}x_{6}^{6}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{7}$
---
| $-2^{2}$
---
$0$ | $(9,1,0)$ |  | $(0,0,0,1,0,0,0,0,0,0)$
---
| 66
---
|
$x_{1}^{5}x_{2}^{6}x_{3}^{8}x_{5}^{7}x_{6}^{8}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $2^{2}\cdot 7$
---
$1$ | $(3,5,2)$ |  | $(0,1,1,0,1,1,1,2,0,2)$
---
| 27
---
|
$x_{1}x_{2}^{4}x_{3}^{4}x_{4}^{2}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}x_{9}^{2}x_{10}$
---
$x_{1}^{2}x_{2}^{3}x_{3}^{4}x_{4}^{2}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}x_{9}^{2}x_{10}$
| $-73$
---
$-233$
$1$ | $(2,5,3)$ |  | $(0,1,0,1,2,1,1,1,2,2)$
---
| 28
---
|
$x_{1}x_{2}^{4}x_{3}x_{4}^{4}x_{5}^{2}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{2}x_{10}^{2}$
---
| $7$
---
$1$ | $(4,5,1)$ |  | $(0,1,1,0,1,1,1,0,0,2)$
---
| 31
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{4}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{3}x_{9}^{3}$
---
$x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{3}x_{9}^{3}$
| $-5\cdot 43$
---
$-3\cdot 13\cdot 17$
$1$ | $(1,5,4)$ |  | $(1,2,2,0,2,2,1,1,1,1)$
---
| 32
---
|
$x_{1}^{4}x_{2}^{3}x_{3}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
| $-7$
---
$1$ | $(0,5,5)$ |  | $(1,2,1,2,1,1,1,2,2,2)$
---
| 38
---
|
$x_{1}^{3}x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{3}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
| $-2\cdot 3\cdot 5$
---
$1$ | $(2,6,2)$ |  | $(0,1,0,1,1,2,2,1,1,1)$
---
| 30
---
|
$x_{1}x_{2}^{2}x_{3}x_{4}^{4}x_{5}^{5}x_{6}x_{7}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $-5$
---
$1$ | $(3,6,1)$ |  | $(0,1,0,1,1,0,2,1,1,1)$
---
| 32
---
|
$x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{5}x_{5}^{5}x_{6}^{2}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $2^{2}$
---
$1$ | $(1,6,3)$ |  | $(0,1,0,1,1,0,2,1,1,1)$
---
| 32
---
|
$x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{5}x_{5}^{5}x_{6}^{2}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $2^{2}$
---
$1$ | $(4,6,0)$ |  | $(0,1,1,0,1,1,1,1,0,0)$
---
| 34
---
|
$x_{1}x_{2}x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
| $-2\cdot 3$
---
$1$ | $(0,6,4)$ |  | $(2,1,1,2,1,1,1,1,2,2)$
---
| 37
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{4}x_{4}^{3}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
| $2^{2}$
---
$1$ | $(2,7,1)$ |  | $(0,1,1,0,1,1,1,1,1,2)$
---
| 35
---
| $x_{1}x_{2}x_{3}^{3}x_{4}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}$
---
| $-2$
---
$1$ | $(1,7,2)$ |  | $(1,0,1,2,1,1,1,1,1,2)$
---
| 40
---
| $x_{1}^{3}x_{3}^{5}x_{4}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}$
---
| $3^{2}$
---
$1$ | $(3,7,0)$ |  | $(0,1,1,0,1,1,1,1,1,0)$
---
| 36
---
|
$x_{1}x_{2}x_{3}^{2}x_{4}^{2}x_{5}^{4}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $1$
---
$1$ | $(0,7,3)$ |  | $(2,1,1,2,1,1,1,1,1,2)$
---
| 40
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{2}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $-2^{2}$
---
$1$ | $(1,8,1)$ |  | $(1,0,1,1,1,1,1,1,1,2)$
---
| 43
---
| $x_{1}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{6}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}$
---
| $1$
---
$1$ | $(2,8,0)$ |  | $(1,0,1,1,1,1,1,1,1,0)$
---
| 43
---
|
$x_{1}^{2}x_{2}x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
| $-1$
---
$1$ | $(0,8,2)$ |  | $(1,2,1,1,1,1,1,1,1,2)$
---
| 47
---
|
$x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{3}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
| $2^{2}$
---
$1$ | $(1,9,0)$ |  | $(1,0,1,1,1,1,1,1,1,1)$
---
| 51
---
|
$x_{1}^{3}x_{3}^{2}x_{4}^{4}x_{5}^{5}x_{6}^{5}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $2$
---
$1$ | $(0,9,1)$ |  | $(1,2,1,1,1,1,1,1,1,1)$
---
| 54
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
| $2$
---
$1$ | $(0,10,0)$ |  | $(1,1,1,1,1,1,1,1,1,1)$
---
| 65
---
|
$x_{1}^{3}x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{6}x_{6}^{8}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
| $2\cdot 5$
---
$2$ | $(3,2,5)$ |  | $(0,2,2,0,2,2,2,1,0,1)$
---
| 27
---
|
$x_{1}x_{2}^{4}x_{3}^{4}x_{4}^{2}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}x_{9}^{2}x_{10}$
---
$x_{1}^{2}x_{2}^{3}x_{3}^{4}x_{4}^{2}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}x_{9}^{2}x_{10}$
| $-73$
---
$-233$
$2$ | $(2,3,5)$ |  | $(0,2,0,2,1,2,2,2,1,1)$
---
| 28
---
|
$x_{1}x_{2}^{4}x_{3}x_{4}^{4}x_{5}^{2}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{2}x_{10}^{2}$
---
| $7$
---
$2$ | $(4,1,5)$ |  | $(0,2,2,0,2,2,2,0,0,1)$
---
| 31
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{4}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{3}x_{9}^{3}$
---
$x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{3}x_{9}^{3}$
| $-5\cdot 43$
---
$-3\cdot 13\cdot 17$
$2$ | $(1,4,5)$ |  | $(2,1,1,0,1,1,2,2,2,2)$
---
| 32
---
|
$x_{1}^{4}x_{2}^{3}x_{3}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
| $-7$
---
$2$ | $(5,0,5)$ |  | $(0,2,0,2,0,0,0,2,2,2)$
---
| 33
---
|
$x_{1}^{2}x_{2}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{2}x_{9}^{4}x_{10}^{4}$
---
| $3\cdot 5^{2}$
---
$2$ | $(2,2,6)$ |  | $(0,2,0,2,2,1,1,2,2,2)$
---
| 30
---
|
$x_{1}x_{2}^{2}x_{3}x_{4}^{4}x_{5}^{5}x_{6}x_{7}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $-5$
---
$2$ | $(3,1,6)$ |  | $(0,2,0,2,2,0,1,2,2,2)$
---
| 32
---
|
$x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{5}x_{5}^{5}x_{6}^{2}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $2^{2}$
---
$2$ | $(1,3,6)$ |  | $(0,2,0,2,2,0,1,2,2,2)$
---
| 32
---
|
$x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{5}x_{5}^{5}x_{6}^{2}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $2^{2}$
---
$2$ | $(4,0,6)$ |  | $(0,2,2,0,2,2,2,2,0,0)$
---
| 34
---
|
$x_{1}x_{2}x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
| $-2\cdot 3$
---
$2$ | $(0,4,6)$ |  | $(1,2,2,1,2,2,2,2,1,1)$
---
| 37
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{4}x_{4}^{3}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
| $2^{2}$
---
$2$ | $(2,1,7)$ |  | $(0,2,2,0,2,2,2,2,2,1)$
---
| 35
---
| $x_{1}x_{2}x_{3}^{3}x_{4}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}$
---
| $-2$
---
$2$ | $(1,2,7)$ |  | $(2,0,2,1,2,2,2,2,2,1)$
---
| 40
---
| $x_{1}^{3}x_{3}^{5}x_{4}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}$
---
| $3^{2}$
---
$2$ | $(3,0,7)$ |  | $(0,2,2,0,2,2,2,2,2,0)$
---
| 36
---
|
$x_{1}x_{2}x_{3}^{2}x_{4}^{2}x_{5}^{4}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $1$
---
$2$ | $(0,3,7)$ |  | $(1,2,2,1,2,2,2,2,2,1)$
---
| 40
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{2}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $-2^{2}$
---
$2$ | $(1,1,8)$ |  | $(2,0,2,2,2,2,2,2,2,1)$
---
| 43
---
| $x_{1}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{6}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}$
---
| $1$
---
$2$ | $(2,0,8)$ |  | $(2,0,2,2,2,2,2,2,2,0)$
---
| 43
---
|
$x_{1}^{2}x_{2}x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
| $-1$
---
$2$ | $(0,2,8)$ |  | $(2,1,2,2,2,2,2,2,2,1)$
---
| 47
---
|
$x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{3}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
| $2^{2}$
---
$2$ | $(1,0,9)$ |  | $(2,0,2,2,2,2,2,2,2,2)$
---
| 51
---
|
$x_{1}^{3}x_{3}^{2}x_{4}^{4}x_{5}^{5}x_{6}^{5}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $2$
---
$2$ | $(0,1,9)$ |  | $(2,1,2,2,2,2,2,2,2,2)$
---
| 54
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{3}x_{4}^{3}x_{5}^{5}x_{6}^{6}x_{7}^{6}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
| $2$
---
$2$ | $(0,0,10)$ |  | $(2,2,2,2,2,2,2,2,2,2)$
---
| 65
---
|
$x_{1}^{3}x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{6}x_{6}^{8}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
| $2\cdot 5$
---
Table 5: Monomials and their coefficients sufficient for the proof of Theorem 4.5 in the case $t=6$ and $G=D_{2p}$. $\bar{a}$ | $\bm{\lambda}$ | $\mathbf{a^{\prime\prime}}$ | deg | monomial/s | coefficient/s
---|---|---|---|---|---
$0$ | $(5,5)$ |  | $(1,0,0,0,1,1,1,0,1,0)$
---
| $34$
---
|
$x_{2}^{2}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
$x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
| $2^{6}\cdot 5\cdot 13$
---
$2^{4}\cdot 17\cdot 23$
$0$ | $(6,4)$ |  | $(1,0,0,0,1,0,1,0,1,0)$
---
| $38$
---
|
$x_{2}^{4}x_{3}^{5}x_{4}^{5}x_{5}^{3}x_{6}^{5}x_{7}^{3}x_{8}^{5}x_{9}^{3}x_{10}^{5}$
---
$x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{5}x_{5}^{3}x_{6}^{5}x_{7}^{3}x_{8}^{5}x_{9}^{3}x_{10}^{5}$
| $-3\cdot 41543$
---
$2^{2}\cdot 3\cdot 31\cdot 617$
$0$ | $(7,3)$ |  | $(1,0,0,0,1,0,0,0,1,0)$
---
| $36$
---
| $x_{3}^{2}x_{4}^{6}x_{5}^{2}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{2}x_{10}^{6}$
---
| $-2^{3}\cdot 3$
---
$0$ | $(8,2)$ |  | $(1,0,0,0,1,0,0,0,0,0)$
---
| $45$
---
| $x_{3}^{2}x_{4}^{7}x_{5}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{7}$
---
| $-2^{2}$
---
$0$ | $(9,1)$ |  | $(0,0,0,0,1,0,0,0,0,0)$
---
| $66$
---
|
$x_{1}^{5}x_{2}^{7}x_{3}^{7}x_{4}^{8}x_{6}^{7}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $-2^{5}\cdot 3$
---
$1$ | $(5,5)$ |  | $(1,0,1,1,1,0,0,0,1,0)$
---
| $39$
---
|
$x_{1}^{3}x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
$x_{1}^{4}x_{2}^{3}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
| $43\cdot 97$
---
$-2\cdot 3^{3}\cdot 257$
$1$ | $(4,6)$ |  | $(0,1,1,1,1,1,0,0,1,0)$
---
| $39$
---
|
$x_{1}^{2}x_{2}^{3}x_{3}^{5}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{3}x_{8}^{3}x_{9}^{5}x_{10}^{3}$
---
$x_{1}^{3}x_{2}^{2}x_{3}^{5}x_{4}^{5}x_{5}^{5}x_{6}^{5}x_{7}^{3}x_{8}^{3}x_{9}^{5}x_{10}^{3}$
| $-11\cdot 23\cdot 61$
---
$-7^{2}\cdot 953$
$1$ | $(3,7)$ |  | $(1,0,1,1,1,1,1,0,1,0)$
---
| $43$
---
|
$x_{1}^{3}x_{2}^{2}x_{3}^{4}x_{4}^{6}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{2}x_{9}^{6}x_{10}^{2}$
---
$x_{1}^{4}x_{2}^{2}x_{3}^{3}x_{4}^{6}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{2}x_{9}^{6}x_{10}^{2}$
| $-5\cdot 23\cdot 61$
---
$2\cdot 32117$
$1$ | $(2,8)$ |  | $(1,0,1,1,1,1,1,1,1,0)$
---
| $48$
---
|
$x_{1}^{4}x_{2}x_{3}^{3}x_{4}^{4}x_{5}^{7}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
$x_{1}^{4}x_{2}x_{3}^{3}x_{4}^{5}x_{5}^{6}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
| $-2^{2}\cdot 5\cdot 61$
---
$2^{4}\cdot 3^{2}\cdot 29$
$1$ | $(1,9)$ |  | $(1,0,1,1,1,1,1,1,1,1)$
---
| $55$
---
|
$x_{1}^{5}x_{3}^{3}x_{4}^{4}x_{5}^{4}x_{6}^{7}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
$x_{1}^{5}x_{3}^{3}x_{4}^{4}x_{5}^{5}x_{6}^{6}x_{7}^{8}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
| $2\cdot 3\cdot 61$
---
$-2^{4}\cdot 83$
$1$ | $(0,10)$ |  | $(1,1,1,1,1,1,1,1,1,1)$
---
| $66$
---
|
$x_{1}^{4}x_{2}^{4}x_{3}^{5}x_{4}^{8}x_{6}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
$x_{1}^{4}x_{2}^{4}x_{3}^{5}x_{4}^{9}x_{6}^{8}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
| $-2\cdot 457$
---
$-2^{8}\cdot 3^{2}$
Table 6: Monomials and their coefficients sufficient for the proof of Theorem 4.5 in the case $t=6$ and $G=G_{3p}$. $\bar{a}$ | $\bm{\lambda}$ | $\mathbf{a^{\prime\prime}}$ | deg | monomial/s | coefficient/s
---|---|---|---|---|---
$0$ | $(5,3,2)$ |  | $(1,0,0,1,0,0,0,2,1,2)$
---
| 25
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{4}x_{4}^{2}x_{5}^{2}x_{6}^{4}x_{7}^{4}x_{9}^{2}x_{10}$
---
| $-r-1$
---
$0$ | $(5,4,1)$ |  | $(1,0,0,1,0,0,0,1,1,2)$
---
| 22
---
| $x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{3}x_{5}x_{6}^{2}x_{7}^{4}x_{8}x_{9}$
---
| $1$
---
$0$ | $(5,5,0)$ |  | $(0,1,0,1,0,0,0,1,1,1)$
---
| 33
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
| $124r+294$
---
$0$ | $(6,2,2)$ |  | $(1,0,0,1,0,0,0,0,2,2)$
---
| 25
---
| $x_{1}x_{2}^{5}x_{3}^{2}x_{4}x_{5}^{3}x_{6}^{2}x_{7}^{5}x_{8}^{5}x_{10}$
---
| $1$
---
$0$ | $(6,3,1)$ |  | $(1,0,0,1,0,0,0,0,1,2)$
---
| 30
---
|
$x_{1}^{2}x_{2}^{3}x_{3}^{4}x_{4}x_{5}^{5}x_{6}^{5}x_{7}^{4}x_{8}^{4}x_{9}^{2}$
---
| $1$
---
$0$ | $(6,4,0)$ |  | $(1,0,0,1,0,0,0,0,1,1)$
---
| 28
---
|
$x_{2}^{2}x_{3}^{3}x_{4}x_{5}^{4}x_{6}^{4}x_{7}^{5}x_{8}^{4}x_{9}^{2}x_{10}^{3}$
---
| $2$
---
$0$ | $(7,2,1)$ |  | $(1,0,0,1,0,0,0,0,0,2)$
---
| 33
---
| $x_{3}^{3}x_{4}x_{5}^{6}x_{6}^{5}x_{7}^{6}x_{8}^{6}x_{9}^{6}$
---
| $2$
---
$0$ | $(7,3,0)$ |  | $(1,0,0,1,0,0,0,0,0,1)$
---
| 35
---
| $x_{3}^{3}x_{4}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{5}x_{9}^{6}x_{10}^{2}$
---
| $2$
---
$0$ | $(8,1,1)$ |  | $(1,0,0,0,2,0,0,0,0,0)$
---
| 44
---
| $x_{2}^{3}x_{3}^{4}x_{4}^{2}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}^{7}$
---
| $12r+12$
---
$0$ | $(8,2,0)$ |  | $(1,0,0,0,1,0,0,0,0,0)$
---
| 42
---
|
$x_{2}^{3}x_{3}^{4}x_{4}^{4}x_{5}x_{6}^{5}x_{7}^{5}x_{8}^{7}x_{9}^{7}x_{10}^{6}$
---
| $5$
---
$0$ | $(9,1,0)$ |  | $(0,0,0,1,0,0,0,0,0,0)$
---
| 66
---
|
$x_{1}^{5}x_{2}^{7}x_{3}^{7}x_{5}^{8}x_{6}^{8}x_{7}^{7}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $2^{2}\cdot 5^{2}$
---
$1$ | $(3,5,2)$ |  | $(0,1,1,0,1,1,1,2,0,2)$
---
| 27
---
|
$x_{1}x_{2}^{4}x_{3}^{4}x_{4}^{2}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}x_{9}^{2}x_{10}$
---
| $-3055r-1872$
---
$1$ | $(2,5,3)$ |  | $(0,1,1,0,1,1,1,2,2,2)$
---
| 28
---
|
$x_{1}x_{2}^{4}x_{3}^{4}x_{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{2}x_{9}^{2}x_{10}^{2}$
---
| $3993r+2619$
---
$1$ | $(4,5,1)$ |  | $(0,1,1,0,1,1,1,0,0,2)$
---
| 31
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{4}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{3}x_{9}^{3}$
---
| $3130r+21$
---
$1$ | $(1,5,4)$ |  | $(1,2,2,1,2,2,1,1,1,0)$
---
| 32
---
|
$x_{1}^{4}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{3}x_{6}^{3}x_{7}^{4}x_{8}^{4}x_{9}^{4}$
---
| $-4043r-1625$
---
$1$ | $(0,5,5)$ |  | $(1,2,1,2,1,1,1,2,2,2)$
---
| 38
---
|
$x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
| $7128r-12852$
---
$1$ | $(2,6,2)$ |  | $(0,1,0,1,1,2,2,1,1,1)$
---
| 30
---
|
$x_{1}x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{5}x_{6}x_{7}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $-424r-595$
---
$1$ | $(3,6,1)$ |  | $(0,1,0,1,1,0,2,1,1,1)$
---
| 36
---
|
$x_{1}^{2}x_{2}^{5}x_{3}^{2}x_{4}^{5}x_{5}^{5}x_{6}^{2}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $-566r+1653$
---
$1$ | $(1,6,3)$ |  | $(1,0,1,1,2,2,2,1,1,1)$
---
| 34
---
|
$x_{1}^{4}x_{3}^{4}x_{4}^{5}x_{5}^{2}x_{6}^{2}x_{7}^{2}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $61364r+25951$
---
$1$ | $(4,6,0)$ |  | $(0,1,1,0,1,1,1,1,0,0)$
---
| 34
---
|
$x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
| $-1272r+653$
---
$1$ | $(0,6,4)$ |  | $(2,1,1,2,1,1,1,1,2,2)$
---
| 37
---
|
$x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{4}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
| $-5321r-5949$
---
$1$ | $(2,7,1)$ |  | $(0,1,1,0,1,1,1,1,1,2)$
---
| 35
---
| $x_{1}x_{2}^{3}x_{3}^{3}x_{4}x_{5}^{3}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}$
---
| $-291r+1$
---
$1$ | $(1,7,2)$ |  | $(1,0,1,2,1,1,1,1,1,2)$
---
| 40
---
| $x_{1}^{4}x_{3}^{4}x_{4}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}$
---
| $11711r+1287$
---
$1$ | $(3,7,0)$ |  | $(0,1,1,0,1,1,1,1,1,0)$
---
| 36
---
|
$x_{1}^{2}x_{2}^{3}x_{3}^{3}x_{4}^{2}x_{5}^{3}x_{6}^{3}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $5839r+1451$
---
$1$ | $(0,7,3)$ |  | $(2,1,1,2,1,1,1,1,1,2)$
---
| 40
---
|
$x_{1}^{2}x_{2}^{3}x_{3}^{3}x_{4}^{2}x_{5}^{4}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $-1365r-1577$
---
$1$ | $(1,8,1)$ |  | $(1,0,1,1,1,1,1,1,1,2)$
---
| 43
---
| $x_{1}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{6}x_{7}^{7}x_{8}^{7}x_{9}^{7}$
---
| $239r-1811$
---
$1$ | $(2,8,0)$ |  | $(1,0,1,1,1,1,1,1,1,0)$
---
| 43
---
|
$x_{1}^{4}x_{2}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
| $-10919r-9004$
---
$1$ | $(0,8,2)$ |  | $(1,2,1,1,1,1,1,1,1,2)$
---
| 47
---
|
$x_{1}^{4}x_{2}x_{3}^{4}x_{4}^{4}x_{5}^{5}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
| $5278r+39848$
---
$1$ | $(1,9,0)$ |  | $(1,0,1,1,1,1,1,1,1,1)$
---
| 51
---
|
$x_{1}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $-12212r-8422$
---
$1$ | $(0,9,1)$ |  | $(1,2,1,1,1,1,1,1,1,1)$
---
| 54
---
|
$x_{1}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $-82561r-26061$
---
$1$ | $(0,10,0)$ |  | $(1,1,1,1,1,1,1,1,1,1)$
---
| 65
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{6}x_{6}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
| $7028r+8686$
---
$2$ | $(3,2,5)$ |  | $(0,2,2,0,2,2,2,1,0,1)$
---
| 27
---
|
$x_{1}x_{2}^{4}x_{3}^{4}x_{4}^{2}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}x_{9}^{2}x_{10}$
---
| $3055r+1183$
---
$2$ | $(2,3,5)$ |  | $(0,2,2,0,2,2,2,1,1,1)$
---
| 28
---
|
$x_{1}x_{2}^{4}x_{3}^{4}x_{4}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{2}x_{9}^{2}x_{10}^{2}$
---
| $-3993r-1374$
---
$2$ | $(4,1,5)$ |  | $(0,2,2,0,2,2,2,0,0,1)$
---
| 31
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{4}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{3}x_{9}^{3}$
---
| $-3130r-3109$
---
$2$ | $(1,4,5)$ |  | $(2,1,1,2,1,1,2,2,2,0)$
---
| 32
---
|
$x_{1}^{4}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{3}x_{6}^{3}x_{7}^{4}x_{8}^{4}x_{9}^{4}$
---
| $4043r+2418$
---
$2$ | $(5,0,5)$ |  | $(0,2,0,2,0,0,0,2,2,2)$
---
| 33
---
|
$x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{3}x_{5}^{4}x_{6}^{4}x_{7}^{4}x_{8}^{4}x_{9}^{4}x_{10}^{4}$
---
| $-124r+170$
---
$2$ | $(2,2,6)$ |  | $(0,2,0,2,2,1,1,2,2,2)$
---
| 30
---
|
$x_{1}x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{5}x_{6}x_{7}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $424r-171$
---
$2$ | $(3,1,6)$ |  | $(0,2,0,2,2,0,1,2,2,2)$
---
| 36
---
|
$x_{1}^{2}x_{2}^{5}x_{3}^{2}x_{4}^{5}x_{5}^{5}x_{6}^{2}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $566r+2219$
---
$2$ | $(1,3,6)$ |  | $(2,0,2,2,1,1,1,2,2,2)$
---
| 34
---
|
$x_{1}^{4}x_{3}^{4}x_{4}^{5}x_{5}^{2}x_{6}^{2}x_{7}^{2}x_{8}^{5}x_{9}^{5}x_{10}^{5}$
---
| $-61364r-35413$
---
$2$ | $(4,0,6)$ |  | $(0,2,2,0,2,2,2,2,0,0)$
---
| 34
---
|
$x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
| $1272r+1925$
---
$2$ | $(0,4,6)$ |  | $(1,2,2,1,2,2,2,2,1,1)$
---
| 37
---
|
$x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{4}x_{6}^{5}x_{7}^{5}x_{8}^{5}x_{9}^{3}x_{10}^{3}$
---
| $5321r-628$
---
$2$ | $(2,1,7)$ |  | $(0,2,2,0,2,2,2,2,2,1)$
---
| 35
---
| $x_{1}x_{2}^{3}x_{3}^{3}x_{4}x_{5}^{3}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}$
---
| $291r+29$
---
$2$ | $(1,2,7)$ |  | $(2,0,2,1,2,2,2,2,2,1)$
---
| 40
---
| $x_{1}^{4}x_{3}^{4}x_{4}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}$
---
| $-11711r-10424$
---
$2$ | $(3,0,7)$ |  | $(0,2,2,0,2,2,2,2,2,0)$
---
| 36
---
|
$x_{1}^{2}x_{2}^{3}x_{3}^{3}x_{4}^{2}x_{5}^{3}x_{6}^{3}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $-5839r-4388$
---
$2$ | $(0,3,7)$ |  | $(1,2,2,1,2,2,2,2,2,1)$
---
| 40
---
|
$x_{1}^{2}x_{2}^{3}x_{3}^{3}x_{4}^{2}x_{5}^{4}x_{6}^{6}x_{7}^{6}x_{8}^{6}x_{9}^{6}x_{10}^{2}$
---
| $1365r-212$
---
$2$ | $(1,1,8)$ |  | $(2,0,2,2,2,2,2,2,2,1)$
---
| 43
---
| $x_{1}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{6}x_{7}^{7}x_{8}^{7}x_{9}^{7}$
---
| $-239r-2050$
---
$2$ | $(2,0,8)$ |  | $(2,0,2,2,2,2,2,2,2,0)$
---
| 43
---
|
$x_{1}^{4}x_{2}x_{3}^{4}x_{4}^{4}x_{5}^{4}x_{6}^{4}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
| $10919r+1915$
---
$2$ | $(0,2,8)$ |  | $(2,1,2,2,2,2,2,2,2,1)$
---
| 47
---
|
$x_{1}^{4}x_{2}x_{3}^{4}x_{4}^{4}x_{5}^{5}x_{6}^{7}x_{7}^{7}x_{8}^{7}x_{9}^{7}x_{10}$
---
| $-5278r+34570$
---
$2$ | $(1,0,9)$ |  | $(2,0,2,2,2,2,2,2,2,2)$
---
| 51
---
|
$x_{1}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{5}x_{6}^{5}x_{7}^{5}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $12212r+3790$
---
$2$ | $(0,1,9)$ |  | $(2,1,2,2,2,2,2,2,2,2)$
---
| 54
---
|
$x_{1}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{6}x_{6}^{6}x_{7}^{6}x_{8}^{8}x_{9}^{8}x_{10}^{8}$
---
| $82561r+56500$
---
$2$ | $(0,0,10)$ |  | $(2,2,2,2,2,2,2,2,2,2)$
---
| 65
---
|
$x_{1}^{2}x_{2}^{4}x_{3}^{4}x_{4}^{4}x_{5}^{6}x_{6}^{9}x_{7}^{9}x_{8}^{9}x_{9}^{9}x_{10}^{9}$
---
| $-7028r+1658$
---

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