# Counting Gray codes for an improved upper bound of the
Grünbaum–Hadwiger–Ramos problem

Jonathan Kliem  Institut für Mathematik, FU Berlin, Arnimallee 2, 14195
Berlin, Germany<EMAIL_ADDRESS>
###### Abstract.

We give an improved upper bound for the Grünbaum–Hadwiger–Ramos problem: Let
$d,n,k\in\mathbb{N}$ such that $d\geq 2^{n}(1+2^{k-1})$. Given $2^{n+1}$
masses on $\mathbb{R}^{d}$, there exist $k$ hyperplanes in $\mathbb{R}^{d}$
that partition it into $2^{k}$ sets of equal size with respect to all
measures. This is an improvement to the previous bound $d\geq 2^{n+k}$ by
Mani-Levitska, Vrećica & Živaljević in 2006.

This is achieved by classifying the number of certain Gray code patterns
modulo 2. The reduction was developed by Blagojević, Frick, Haase & Ziegler in
2016. It utilizes the group action of the symmetric group
$(\mathbb{Z}/2)^{k}\rtimes\mathfrak{S}_{k}$ of $k$ oriented hyperplanes. If we
restrict to the subgroup $(\mathbb{Z}/2)^{k}$ as Mani-Levitska et al. we
retrieve their bound.

J.K. receives funding by the Deutsche Forschungsgemeinschaft DFG under
Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+
(EXC-2046/1, project ID: 390685689).

###### Contents

  1. 1 Introduction
  2. 2 Counting Gray codes
    1. 2.1 Gray codes
    2. 2.2 Equiparting matrices
      1. 2.2.1 Proof of Theorem 2.15
  3. 3 Reduction to Equiparting matrices

##  1\. Introduction

Consider the $k$-dimensional hypercube $[0,1]^{k}$. Its graph has several
Hamiltonian paths that is, paths of vertices and edges that visit every vertex
exactly once. A collection of $0,1$-vectors representing such a Hamiltonian
path is known as $k$-bit Gray code. The simplest and for our purposes most
important such path is the standard $k$-bit Gray code, which is defined
inductively as traversing the front facet by the standard $(k-1)$-bit Gray
code and the back facet by the standard $(k-1)$-bit Gray code in reverse
order.

The numbers of $k$-bit Gray codes with marked starting node for $k=1,\dots,5$
are $1,2,18,5712$ and $5859364320$111https://oeis.org/A003043. For $k\geq 6$
this number is not known. Invariants of a Gray code are the transition counts
$c_{1},\dots,c_{k}$, where $c_{i}$ counts the number of edges parallel to the
$i$-th standard base vector for each $i=1,\dots,k$. We can classify for which
transition counts the number of Gray codes with fixed starting node is odd:

######  Theorem 1.1.

The standard $k$-bit Gray code has transition counts
$(2^{k-1},2^{k-2},\dots,4,2,1)$ and it is the only Gray code (up to choosing a
starting node) with those transition counts. Up to permutation those are the
only transition counts that are represented by an odd number of Gray codes.

Given a tuple of $j>1$ $k$-bit Gray codes. One can add their transition
counts. We try to minimize $\max(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$
such that the number of $j$-tuples of $k$-bit Gray codes with fixed starting
node and sum of transition counts equal to
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ is odd. If $j=2^{n}+r$ with $0\leq
r<2^{n}$ we find that this value is exactly $2^{k-1}2^{n}+r$; see Theorem 2.15
(1).

Now the symmetric group of the $k$-dimensional hypercube acts on Hamiltonian
paths not just by changing the starting node, but also by permuting
coordinates. It acts on tuples of Gray codes by diagonal action. Hence, the
number of $2$-tuples of $4$-bit Gray codes with transition counts
$(17,17,8,4)$ is even, as those transition counts are invariant with respect
to transposing the first two coordinates. We will see however, that this
number is not divisible by four; see Proposition 2.26.

We can relax our condition to minimize
$\max(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ such that the number of orbits
of those $j$-tuples of Gray codes is odd. Theorem 2.15 establishes that if
$j=2^{n}+r$ with $0<r\leq 2^{n}$, this value is exactly $2^{k-1}2^{n}+r$; see
Theorem 2.15 (2). This differs from the above when $j=2^{n}+2^{n}$ and this
value is $(1+2^{k-1})2^{n}$ instead of $2^{k}2^{n}$. Using methods developed
by Blagojević, Frick, Haase, & Ziegler in [3] our combinatorial observations
lead to a new bound for the Grünbaum–Hadwiger–Ramos problem:

A mass on $\mathbb{R}^{d}$ is a finite Borel measure that vanishes on every
affine hyperplane. Hugo Steinhaus conjectured that any $d$ masses on
$\mathbb{R}^{d}$ can be simultaneously bisected by an appropriate (affine)
hyperplane [2]. This is now known as the Ham–Sandwich theorem. Several authors
have studied the following generalization:

An arrangment of $k$ hyperplanes in $\mathbb{R}^{d}$ partitions its complement
into $2^{k}$ (possibly empty) orthants. A mass $\mu$ is equiparted by this
arrangment, if each orthant has measure $\frac{1}{2^{k}}\mu(\mathbb{R}^{d})$.
The Grünbaum–Hadwiger–Ramos problem is to find the minimal $d=\Delta(j,k)$
such that for any $j$ masses on $\mathbb{R}^{d}$ there exists an arrangment of
$k$ hyperplanes simultaneously equiparting all masses. For a brief history of
this problem, we refer the reader to [3].

The lower bound developed by Avis [1] and Ramos [8] is

$\Delta(j,k)\geq\frac{2^{k}-1}{k}j.$

It is conjectured to be tight and for $k=1$ this is the case by the
Ham–Sandwich theorem. However, even for $k=2$ exact bounds are elusive: In
this case it is shown to be tight for $j=2^{n}-1,2^{n},2^{n}+1$ by Blagojević,
Frick, Haase, & Ziegler [3, Thm. 1.5]. Partial results were known before, see
[3].

For $k>2$ the only tight bounds are $\Delta(1,3)=3$ by Hadwiger [4] and
$\Delta(2,3)=5$, $\Delta(4,3)=10$ by [3]. The reduction of Hadwiger and Ramos
$\Delta(j,k)\leq\Delta(2j,k-1)$ has been used to give (non-optimal) bounds
$\Delta(1,4)\leq 5$ and $\Delta(2,4)\leq 10$ and $\Delta(1,5)\leq 10$; see
[3].

We complete the list of all previous known bounds with the general upper bound

$\Delta(j,k)\leq j+(2^{k-1}-1)2^{\lfloor\log_{2}j\rfloor}$

or equivalently

$\Delta(2^{n}+r,k)\leq 2^{n+k-1}+r\quad\text{for }0\leq r<2^{n}$

by Mani-Levitska, Vrećica & Živaljević [6]. We will use the methods develeoped
in [3] and our combinatorial results briefly described above to generalize
this bound:

######  Theorem 1.2.

Let $j,k\geq 1$ be integers. It holds that

$\Delta(j,k)\leq\Big{\lceil}j+(2^{k-1}-1)2^{\lfloor\log_{2}(j-\frac{1}{2})\rfloor}\Big{\rceil}.$

Equivalently, for integers $j=2^{n}+r$, $n\geq 0$, and $1\leq r\leq 2^{n}$ it
holds that

$\Delta(2^{n}+r,k)\leq
2^{n+k-1}+r\quad\text{and}\quad\Delta(1,k)\leq\begin{cases}2^{k-2}+1,&\text{if
}k\geq 2\\\ 1,&\text{if }k=1.\end{cases}$

Note that [3, Thm. 1.5 (iii)] provides a better and tight bound for
$\Delta(2^{n}+1,2)$, which cannot be generalized for $k>2$. Other than this,
the above theorem collects all currently known upper bounds. To illustrate the
improvements, we give a few examples:

######  Corollary 1.3.

  1. (1)

$\Delta(2^{n+1},2)\leq 2^{n+1}+2^{n}=3\cdot 2^{n}$, shown before in [3].

  2. (2)

$\Delta(2,3)\leq 5$, shown before in [3].

  3. (3)

$\Delta(4,3)\leq 10$, shown before in [3].

  4. (4)

$\Delta(8,3)\leq 20$, where the previous best bound is $32$; see [6].

  5. (5)

$\Delta(16,3)\leq 40$, where the previous best bound is $64$; see [6].

  6. (6)

$\Delta(1,4)\leq 5$, shown before in [3].

  7. (7)

$\Delta(2,4)\leq 9$, where the previous best bound is $10$; see [3].

  8. (8)

$\Delta(4,4)\leq 18$, where the previous best bound is $32$; see [6].

  9. (9)

$\Delta(8,4)\leq 36$, where the previous best bound is $64$; see [6].

  10. (10)

$\Delta(1,5)\leq 9$, where the previous best bound is $10$; see [3].

  11. (11)

$\Delta(2,5)\leq 17$, where the previous best bound is $32$; see [6].

  12. (12)

$\Delta(4,5)\leq 34$, where the previous best bound is $64$; see [6].

  13. (13)

$\Delta(2^{n+1},k)\leq(2^{k}+1)\cdot 2^{n}$, where the previous best bound is
$2^{k+1}\cdot 2^{n}$; see [6].

If $j$ is not a power of two, our bound coincides with the bound in [6] and we
provide an alternative proof of their result. Their bound was derived from a
Fadell–Husseini index calculation using the product scheme, parametrizing
arrangements of $k$ hyperplanes by $(S^{d})^{k}$. Our proof relies on
equivariant obstruction theory with the join scheme, which parametrizes
arrangements of $k$ hyperplanes by $(S^{d})^{*k}$. (See [3, Section 1.1] for
more detail.) Using the join scheme instead of the product scheme, allows to
use the full symmetric group $(\mathbb{Z}/2)^{k}\rtimes\mathfrak{S}_{k}$ of
$k$ oriented hyperplanes. If we restrict to the subgroup $(\mathbb{Z}/2)^{k}$
we retrieve exactly the result by [6].

To conclude the introduction, we remark that for $j=1$ and $k>1$ our new bound

$\Delta(1,k)\leq 2^{k-2}+1$

is already implied by the reduction of Hadwiger and Ramos
$\Delta(1,k)\leq\Delta(2,k-1)$ and our new bound

$\Delta(2,k-1)\leq 2^{k-2}+1.$

##  2\. Counting Gray codes

Blagojević, Frick, Haase and Ziegler [3] have approached the
Grünbaum–Hadwiger–Ramos problem with obstruction theory and developed a
reduction of this problem to a combinatorial problem. The topological problem
can be solved by counting equivalence classes of certain $0,1$-matrices modulo
$2$ with number of rows equal to $k$. They have classified this parity for the
matrices corresponding to $k=2$. For $j\leq 9$, $k=3$ and $j\leq 2$, $k=4$
they have counted some of them with the help of a computer to obtain new
bounds for $\Delta(2,3)$ and $\Delta(4,3)$, see [3, Table 1]. By the reduction
of Hadwiger and Ramos they could derive bounds for $\Delta(1,4)$ and
$\Delta(2,4)$.

Instead of listing all those matrices, one can use an iterator to count more
matrix classes. This helped to understand the underlying structure: For $j>2$
we provide a full classification, when the number of equivalence classes is
odd.

###  2.1. Gray codes

We will use notation from Knuth [5, pp. 292–294] but start indexing with $1$
instead of $0$.

######  Definition 2.1 ([3, Sec. 1.3]).

A $k$-bit Gray code is a $k\times 2^{k}$ binary matrix containing all column
vectors in $\\{0,1\\}^{k}$ such that two consecutive vectors differ in only
one entry.

A Gray code can be seen as a Hamiltonian path on the edge graph of a
hypercube. This is a path visiting each vertex. Knuth restricts to Hamiltonian
paths that can be completed to a cycle. We will not restrict to this case.

######  Definition 2.2.

The _standard $k$-bit Gray code_ is a path obtained inductively. The standard
$1$-bit Gray code is
$\left(\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix}\right)$.
For $k>1$ the standard $k$-bit Gray code is given by

  * •

each vector of the standard $(k-1)$-bit Gray code with $0$ appended,

  * •

in reverse order each vector of the standard $(k-1)$-bit Gray code with $1$
appended.

Thus the standard $k$-bit Gray code traverses the front facet by the standard
$(k-1)$-bit Gray code and then in reverse order the back facet.

######  Example 2.3.

The standard $3$-bit Gray code is

$G=\begin{pmatrix}0&1&1&0&0&1&1&0\\\ 0&0&1&1&1&1&0&0\\\
0&0&0&0&1&1&1&1\end{pmatrix}.$

We will see later that for $j>2$ we only need to consider standard $k$-bit
Gray codes.

######  Definition 2.4.

Let $G=(g_{1},\dots,g_{2^{k}})$ be a $k$-bit Gray code. Then the _delta
sequence of_ $G$

$\delta(G):=(\delta_{1}(G),\dots,\delta_{2^{k}-1}(G))$

is defined by $\delta_{i}(G)$ being the row of the bit change from $g_{i}$ to
$g_{i+1}$ for each $i=1,\dots,2^{k}-1$.

A _$k$ -bit delta sequence_ is a vector, which is the delta sequence of some
$k$-bit Gray code.

######  Example 2.5.

The standard $3$-bit Gray code has delta sequence $(1,2,1,3,1,2,1)$. The
standard $4$-bit Gray code has delta sequence
$(1,2,1,3,1,2,1,4,1,2,1,3,1,2,1)$. In general, $\delta_{i}$ of the standard
$k$-bit Gray code is equal to one plus the dyadic order of $i$.

Let $D$ be a $k$-bit delta seqence. For each choice of a first column
$g_{1}\in\\{0,1\\}^{k}$ there exists exactly one Gray code $G$ with first
column $g_{1}$ and $\delta(G)=D$.

The symmetric group of the $k$-dimensional hypercube,
$(\mathbb{Z}/2)^{k}\rtimes\mathfrak{S}_{k}=:\mathfrak{S}_{k}^{\pm}$, acts on
the set of all Gray codes by permuting rows and inverting all bits in one row.
The group $\mathfrak{S}_{k}^{\pm}$ also acts on $[k]^{2^{k}-1}$ by

$((\beta_{1},\dots,\beta_{k})\rtimes\tau)\cdot(d_{1},\dots,d_{2^{k}-1})=\big{(}\tau(d_{1}),\dots,\tau(d_{2^{k}-1})\big{)}.$

With this action, the map $\delta$ from all $k$-bit Gray codes to
$[k]^{2^{k}-1}$ is $\mathfrak{S}_{k}^{\pm}$-equivariant.

Note that $\mathfrak{S}_{k}^{\pm}$ acts freely on $k$-bit Gray codes, that is
$\sigma\in\mathfrak{S}_{k}^{\pm}$ and $G$ a gray code with $\sigma\cdot G=G$
implies that $\sigma$ is the unit element. On the other hand,
$\mathfrak{S}_{k}$ acts freely on $k$-bit delta sequences (but not on
$[k]^{2^{k}-1}$) and $(\mathbb{Z}/2)^{k}$ acts trivially on $k$-bit delta
sequences. So $(\mathbb{Z}/2)^{k}$ acts on Gray codes by changing the first
column while stabilizing the delta sequence.

######  Definition 2.6.

Let $D$ be a $k$-bit delta sequence. The _transition counts_

$c(D):=(c_{1}(D),\dots,c_{k}(D))$

are defined by letting $c_{i}(D)$ be the number of $i$’s in $D$, for every
$i=1,\dots,k$.

The transition counts of a Gray code are the transition counts of its delta
sequence.

######  Example 2.7.

The transition counts of the standard $k$-bit Gray code are
$(2^{k-1},2^{k-2},\dots,4,2,1)$.

$\mathfrak{S}_{k}^{\pm}$ acts on transition counts by trivial action of
$(\mathbb{Z}/2)^{k}$ and by $\mathfrak{S}_{k}$ permuting coordinates. With
this action, taking the transition counts is
$\mathfrak{S}_{k}^{\pm}$-equivariant.

######  Lemma 2.8.

Let $D=(d_{1},\dots,d_{2^{k}-1})$ be a $k$-bit delta sequence. Suppose for
some $0\leq i<k$ it holds that

$c_{k}(D)=1,\quad c_{k-1}(D)=2,\quad\dots,\quad c_{k-i+1}(D)=2^{i-1}.$

Then, $c_{k-i}(D)\geq 2^{i}$.

###### Proof.

We induct on $i$ for all $k$. For $i=0$ the statement holds as $c_{k}(D)\geq
1$. Suppose now that $i>0$ and

$c_{k}(D)=1,\quad c_{k-1}(D)=2,\quad\dots,\quad c_{k-i+1}(D)=2^{i-1}.$

As $c_{k}(D)=1$ it follows that the $k$-th bit must change at the central
position. Then, $\overline{D}_{1}:=(d_{1},\dots,d_{2^{k-1}-1})$ and
$\overline{D}_{2}:=(d_{2^{k-1}+1},\dots,d_{2^{k}-1})$ are both $(k-1)$-bit
delta sequences such that

$c_{k-1}(\overline{D}_{1})+c_{k-1}(\overline{D}_{2})=2,\quad
c_{k-2}(\overline{D}_{1})+c_{k-2}(\overline{D}_{2})=4,\quad\dots,\quad
c_{k-i+1}(\overline{D}_{1})+c_{k-i+1}(\overline{D}_{2})=2^{i-1},$

which implies by induction that

$c_{k-1}(\overline{D}_{1})=c_{k-1}(\overline{D}_{2})=1,\quad
c_{k-2}(\overline{D}_{1})=c_{k-2}(\overline{D}_{2})=2,\quad\dots,\quad
c_{k-i+1}(\overline{D}_{1})=c_{k-i+1}(\overline{D}_{2})=2^{i-2},$

and therefore $c_{k-i}(\overline{D}_{1})\geq 2^{i-1}$ and
$c_{k-i}(\overline{D}_{2})\geq 2^{i-1}$. ∎

######  Definition 2.9.

Let $D$ be a $k$-bit delta sequence. The _shuffle count_ of $D$ is the number
of delta sequences $\widetilde{D}$ with $c(D)=c(\widetilde{D})$.

We will now proof 1.1 by the following two lemmas:

######  Lemma 2.10.

Let $D$ be the delta sequence of the standard $k$-bit Gray code. The shuffle
count of $D$ is $1$.

###### Proof.

For $k=1$ the statement is clear. Let
$\widetilde{D}=(\tilde{d}_{1},\dots,\tilde{d}_{2^{k}-1})$ be a $k$-bit delta
sequence with $c(\widetilde{D})=c(D)$. The equality
$c_{k}(\widetilde{D})=c_{k}(D)=1$ implies that the only $k$-bit change must be
at central position and $\tilde{d}_{2^{k-1}}=k$ as desired. By Lemma 2.8 it
follows that $\overline{D}_{1}:=(\tilde{d}_{1},\dots,\tilde{d}_{2^{k-1}-1})$
and $\overline{D}_{2}:=(\tilde{d}_{2^{k-1}+1},\dots,\tilde{d}_{2^{k}-1})$ are
both $(k-1)$-bit delta sequences such that
$c(\overline{D}_{1})=c(\overline{D}_{2})=(2^{k-2},2^{k-3},\dots,1)$. By
induction this implies that $\overline{D}_{1}$ and $\overline{D}_{2}$ are the
delta sequence of the standard $(k-1)$-bit Gray code. ∎

######  Lemma 2.11.

Let $D=(d_{1},\dots,d_{2^{k}-1})$ be a $k$-bit delta sequence. If the shuffle
count of $D$ is odd, then $D$ lies in the $\mathfrak{S}_{k}^{\pm}$-orbit of
the delta sequence of the standard $k$-bit Gray code.

###### Proof.

Denote by $D^{R}$ the sequence of $D$ in reversed order. If $D$ is a delta
sequence, then so is $D^{R}$. Hence the parity of the shuffle count is equal
to the parity of the number of symmetric delta sequences $\widetilde{D}$ with
$c(\widetilde{D})=c(D)$. If the shuffle count of $D$ is odd, then w.l.o.g. $D$
itself is symmetric. Let $G$ be a Gray code with $\delta(G)=D$. As the
statement is invariant under the group action of $\mathfrak{S}_{k}^{\pm}$, we
may assume that the central entry of $D$ is $k$.

As $D$ is symmetric, it follows that for each $i=1,\dots,2^{k-1}$ the columns
$G_{i}$ and $G_{2^{k}-i+1}$ differ exactly in the row $k$. This implies that
$c_{k}(D)=1$ and $\overline{D}_{1}:=(d_{1},\dots,d_{2^{k-1}-1})$ and
$\overline{D}_{2}:=(d_{2^{k-1}+1},\dots,d_{2^{k}-1})=\overline{D}_{1}^{R}$ are
$(k-1)$-bit delta sequences. As the shuffle count of $D$ is odd, we may assume
that $\overline{D}_{1}$ itself has odd shuffle count. By induction this
implies that $\overline{D}_{1}$ lies in the $\mathfrak{S}_{k-1}^{\pm}$-orbit
of the standard $(k-1)$-bit Gray code and then $D$ lies in the
$\mathfrak{S}_{k}^{\pm}$-orbit of the standard $k$-bit Gray code. ∎

###  2.2. Equiparting matrices

We generalize [3, Def. 1.1] as follows:

######  Definition 2.12.

Let $\mathfrak{c}_{1}+\dots+\mathfrak{c}_{k}=j(2^{k}-1)$. A binary matrix
$\mathcal{G}$ of size $k\times j2^{k}$ is a
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrix if

  1. (1)

$\mathcal{G}=(G_{1},\dots,G_{j})$ for Gray codes $G_{1},\dots,G_{j}$ with the
property that the last column of $G_{i}$ is equal to the first column of
$G_{i+1}$ for $1\leq i<j$; and

  2. (2)

the transition counts of $\mathcal{G}$ are given by

$c(\mathcal{G})=c(G_{1})+\dots+c(G_{j})=(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}).$

If
$\mathfrak{c}_{1}>\mathfrak{c}_{2}=\mathfrak{c}_{3}=\dots=\mathfrak{c}_{k}$,
we have a $(\mathfrak{c}_{1}-\mathfrak{c}_{2})$-equiparting matrix from [3,
Def. 1.1].

Let $\mathfrak{S}_{k}$ act on $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ by
permutation and let
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$ denote the
stabilizer subgroup. Then
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}^{\pm}$ acts on
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices by acting on
all Gray codes simultaneously. Note that
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}^{\pm}$ acts
freely on $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices.

######  Definition 2.13.

  1. (1)

We say that two $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting
matrices $\mathcal{G}$ and $\mathcal{G}^{\prime}$ are _isomorphic_ , if they
are in one $(\mathbb{Z}/2)^{k}$-orbit.

  2. (2)

We say that two $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting
matrices $\mathcal{G}$ and $\mathcal{G}^{\prime}$ are _equivalent_ , if they
are in one
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}^{\pm}$-orbit.

######  Definition 2.14.

Let $j\geq 1$, $k\geq 2$ be integers.

  1. (1)

Consider the set $\mathcal{I}(j,k)$ of all tuples of integers
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ with
$\mathfrak{c}_{1}+\dots+\mathfrak{c}_{k}=j(2^{k}-1)$ such that the number of
non-isomorphic $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting
matrices is odd.

Denote by $I(j,k)$ the minimum of
$\max\left(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}\right)$ for all
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})\in\mathcal{I}(j,k)$.

  2. (2)

Consider the set $\mathcal{E}(j,k)$ of all tuples of integers
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ with
$\mathfrak{c}_{1}+\dots+\mathfrak{c}_{k}=j(2^{k}-1)$ such that the number of
non-equivalent $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting
matrices is odd().

Denote by $E(j,k)$ the minimum of
$\max\left(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}\right)$ for all
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})\in\mathcal{E}(j,k)$.

Clearly, $E(j,k)\leq I(j,k)$. We obtain the following classification of the
parity of $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices:

######  Theorem 2.15.

Let $j\geq 1$, $k\geq 2$ be integers.

  1. (1)

Then

$I(2^{n}+r,k)=2^{n+k-1}+r$

for all $0\leq r<2^{n}$ and $n\geq 0$.

  2. (2)

Then

$E(1,k)\leq 2^{k-2}+1$

and

$E(2^{n}+r,k)=2^{n+k-1}+r$

for all $1\leq r\leq 2^{n}$ and $n\geq 0$.

Proposition 3.7 establishes that $\Delta(j,k)\leq I(j,k)$ and $\Delta(j,k)\leq
E(j,k)$ and concludes our proof of Theorem 1.2. This reduction has basically
been shown in [3] and we generalize it in Section 3. Note that these bounds
only differ in the cases

$E(1,k)\leq 2^{k-2}+1,\quad I(1,k)=2^{k-1}$

and

$E(2^{n}+2^{n},k)=2^{n+k-1}+2^{n},\quad I(2^{n+1},k)=2^{n+k}.$

The value of $I(j,k)$ that is obtained by only considering the
$(\mathbb{Z}/2)^{k}$-action is exactly the bound by Mani-Levitska, Vrećica &
Živaljević [6]. This seems natural as they also have restricted their
attention to the $(\mathbb{Z}/2)^{k}$-action [6, Proof of Thm. 38]. The value
of $E(j,k)$ is the improved bound by considering the full
$\mathfrak{S}_{k}^{\pm}$-action with the methods developed in [3].

The exact value of $E(1,k)$ remains unknown as we do not understand the
precise structure of non-equivalent Gray codes (equiparting matrices for
$j=1$). It has been verified to be tight by computation for $k\leq 5$.

####  2.2.1. Proof of Theorem 2.15

The $k$-bit Gray codes $G$ with transition count $c_{k}(G)=1$ are exactly
given by composition of two $(k-1)$-bit Gray codes. Let
$\mathfrak{c}_{1}+\dots+\mathfrak{c}_{k-1}=2(2^{k-1}-1)$ be such that the
number of non-equivalent
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k-1})$-equiparting matrices is odd.
Then it holds that $\mathfrak{c}_{1}+\dots+\mathfrak{c}_{k-1}+1=(2^{k}-1)$ and
the number of non-equivalent
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k-1},1)$-equiparting matrices is odd,
which implies $E(1,k)\leq E(2,k-1)$. $I(1,k)=2^{k-1}$ is a consequence of
Theorem 1.1.

Thus it suffices to verify the values of $E(j,k)$ and $I(j,k)$ for $j>1$. In
order to do so, we use delta sequences and make a few observations:

######  Definition 2.16.

A $j$-tuple of elements in $[k]^{2^{k}-1}$

$\mathcal{D}=(D_{1},\dots,D_{j})$

is a $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequence, if it is a
delta sequence of a $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting
matrix

$\mathcal{G}=(G_{1},\dots,G_{j}),$

i.e. for each $i=1,\dots,j$ it holds that $\delta(G_{i})=D_{i}$.

The number of non-isomorphic
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices is equal to
the number of $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequences.

As before $(\mathbb{Z}/2)^{k}$ acts trivially on
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequences and as above
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$ acts freely
by diagonal action. We say that two
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequences are equivalent, if
they are in one
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$-orbit. Thus
the number of non-equivalent
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices is equal to
the number of non-equivalent $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta
sequences.

We denote by $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ the number of non-
equivalent $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequences modulo
$2$. Understanding this for all $\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}$
determines $E(j,k)$. As $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ is
invariant under the action of $\mathfrak{S}_{k}$, we may ass well assume that
$\mathfrak{c}_{1}\geq\mathfrak{c}_{2}\geq\dots\geq\mathfrak{c}_{k}$.

Understanding $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ for all
$\mathfrak{c}_{1}>\dots>\mathfrak{c}_{k}$ will determine $I(j,k)$:

######  Lemma 2.17.

If the number of non-isomorphic
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}$)-equiparting is odd, then the
$\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}$ are pairwise distinct.

If the $\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}$ are pairwise distinct, then
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices are
isomorphic if and only if the are equivalent.

###### Proof.

As $(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}^{\pm}$ acts
freely on the set of $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting
matrices, the number of
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices is divisible
by
$|(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}^{\pm}|=|(\mathbb{Z}/2)^{k}|\cdot|(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}|$.
Hence the number of non-isomorphic
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices is even
unless $|(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}|$ is
odd, which is equivalent to the $\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}$
being pairwise distinct.

The second part is clear. In this case,
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$ is trivial. ∎

To determine this parity it suffices to consider only the standard $k$-bit
Gray code:

######  Lemma 2.18.

Let $j,k\geq 2$ and let $\mathfrak{c}_{1}+\dots+\mathfrak{c}_{k}=j(2^{k}-1)$.

  1. (1)

Let $(D_{1},\dots,D_{j})$ be a
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequence. Let $1\leq i\leq
j$ and let $\widetilde{D}$ be a delta sequence with
$c(\widetilde{D})=c(D_{i})$. Then

$(D_{1},\dots,D_{i-1},\widetilde{D},D_{i+1},\dots,D_{j})$

is equivalent to $(D_{1},\dots,D_{j})$ if and only if $D_{i}=\widetilde{D}$.

  2. (2)

The parity $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ is determined by
considering only permutations of delta sequences of the standard $k$-bit Gray
code.

###### Proof.

  1. (1)

Let $\tau\in(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$
with

$\tau\cdot(D_{1},\dots,D_{j})=(D_{1},\dots,D_{i-1},\widetilde{D},D_{i+1},\dots,D_{j}).$

The group $\mathfrak{S}_{k}$ acts freely on delta sequences and as $j>1$ it
follows that $\tau=\operatorname{id}$ and $\widetilde{D}=D_{i}$.

  2. (2)

Follows from (1) and Theorem 1.1.

∎

We observe another group action: $\mathfrak{S}_{j}$ acts on
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequences by

$\sigma\cdot(D_{1},\dots,D_{j}):=(D_{\sigma^{-1}(1)},\dots,D_{\sigma^{-1}(j)}).$

This action commutes with the action of
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$ and it is the
key in determining $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$: If the
$\mathfrak{S}_{j}$-orbit of $(D_{1},\dots,D_{j})$ contains an even number of
non-equivalent elements, then $(D_{1},\dots,D_{j})$ need not be considered for
the parity of $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$.

Suppose that $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ is odd. Let
$j=2^{n}+r$. In the next Proposition we will see that we only need to consider
$(D_{1},\dots,D_{j})$, where $2^{n}$ of them are identical. We have already
seen that we only need to consider permutations of the standard $k$-bit Gray
code, which has transition counts $(2^{k-1},\dots,2,1)$. It follows then that
$\mathfrak{c}_{1}\geq 2^{n}2^{k-1}+r\cdot 1$. Before we proceed, recall some
number theory and introduce a term:

The dyadic valuation of an integer $n$ is the largest integer $m$ such that
$2^{m}$ divides $n$. To determine whether a fraction is odd, one can compare
the dyadic valuation of numerator and denominator.

######  Lemma 2.19.

The dyadic valuation of $\binom{j}{a}$ is

$\nu_{2}\left(\binom{j}{a}\right)=s_{2}(a)+s_{2}(j-a)-s_{2}(j),$

where $s_{2}(d)$ is the sum of the dyadic digits of $j$.

###### Proof.

By a formula of Legendre, see [7, Theorem 2.6.4], the dyadic valuation of $j!$
is given by

$\nu_{2}(j!)=j-s_{2}(j).$

∎

######  Lemma 2.20.

The sum of the dyadic digits of positive numbers is subadditive and
submultiplicative: Let $n,m\geq 1$ be integers.

  1. (1)

$s_{2}(n)+s_{2}(m)\geq s_{2}(n+m)$

and the inequality is tight, exactly if the dyadic digits of $n$ and $m$
decompose the dyadic digits of $n+m$.

  2. (2)

$s_{2}(n)s_{2}(m)\geq s_{2}(nm).$

###### Proof.

We assume the dyadic digits of $n$ and $m$ to be given as

$n=\sum_{i=0}^{r}2^{i}n_{i},\quad m=\sum_{i=0}^{r}2^{i}m_{i}.$

  1. (1)

That the inequality is tight, when the dyadic digits are decomposed is clear.

Now by induction on the number of digits of $m$, it suffices to show that for
an integer $j\geq 1$ with $n_{j}=1$ it holds that

$s_{2}(n)+1>s_{2}(n+2^{j}).$

We induce for fixed $n$ on $j$ from above (note that $j$ is bounded by $n$).
It holds that $s_{2}(n)=s_{2}(n-2^{j})+1$. Then

$s_{2}(n+2^{j})=s_{2}(n-2^{j}+2^{j+1})\leq
s_{2}(n-2^{j})+s_{2}(2^{j+1})=s_{2}(n)+1+1.$

If $n_{j+1}=0$, the last inequality holds tight. If $n_{j+1}=1$, the last
inequality holds by induction hypothesis. The later is never the case for the
base case.

  2. (2)

With

$nm=\sum_{k=0}^{2r}\sum_{i+j=k}2^{i}2^{j}m_{i}n_{i}$

the statment follows from (1).

∎

######  Lemma 2.21.

Let $T\geq 1$ be an integer. It holds that

$T-\nu_{2}(T)\geq s_{2}(T)$

and this inequality is tight exactly for $T\in\\{1,2\\}$.

###### Proof.

It is clear that the inequality is tight for $T\in\\{1,2\\}$. If $T$ is odd
and at least $3$ then $T-\nu_{2}(T)=T>s_{2}(T)$. The remaining cases follow by
induction. If $T-\nu_{2}(T)\geq s_{2}(T)$ for some $T\geq 2$, then

$\displaystyle 2T-\nu_{2}(2T)$ $\displaystyle=2T-\nu_{2}(T)-1$
$\displaystyle\geq s_{2}(T)+T-1$ $\displaystyle=s_{2}(2T)+T-1>s_{2}(2T).$

∎

######  Definition 2.22.

The _multiplicity_ of $(D_{1},\dots,D_{j})$ is the multiset formed by:

$|\\{i\in\\{1,\dots,j\\}\colon D_{i}=D\\}|$

for all $D\in\\{D_{1},\dots,D_{j}\\}$.

Note that the multiplicity of a
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequence is
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$-invariant and
$\mathfrak{S}_{j}$-invariant.

######  Example 2.23.

Let $D,D^{\prime},D^{\prime\prime}$ be pairwise distinct.

  * •

$(D,D,D^{\prime},D^{\prime})$ has multiplicity $(2,2)$.

  * •

$(D,D,D,D^{\prime},D^{\prime})$ has multiplicity $(2,3)$.

  * •

$(D,D,D^{\prime},D^{\prime\prime},D^{\prime\prime})$ has multiplicity
$(1,2,2)$.

The multiplicites of those delta sequences we need to consider are limited by
$j$:

######  Proposition 2.24.

The parity $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ is determined by
considering only $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequences
of multiplicity

  1. (1)

$(a_{1},\dots,a_{\ell})$, where the dyadic digits of the $a_{i}$ decompose the
dyadic digits of $j$.

  2. (2)

$(\frac{j}{2},\frac{j}{2})$, where $j$ is a power of two and
$|(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}|\geq 2$.

###### Proof.

Let $\mathcal{D}:=(D_{1},\dots,D_{j})$ be a
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequence of multiplicity
$(a_{1},\dots,a_{\ell})$. As $\mathfrak{S}_{j}$ and
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$ commute, the
number of non-equivalent elements of the $\mathfrak{S}_{j}$-orbit of
$\mathcal{D}$ is equal to the size of the $\mathfrak{S}_{j}$-orbit divided by
the cardinality $T$ of

$\mathcal{T}:=\left\\{\tau\in(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}\mid\exists\sigma\in\mathfrak{S}_{j}\colon\tau\mathcal{D}=\sigma\mathcal{D}\right\\}.$

If $\tau\in\mathcal{T}$ and $\tau(D_{i_{1}})=D_{i_{2}}$, both $D_{i_{1}}$ and
$D_{i_{2}}$ must have the same cardinality in $(D_{1},\dots,D_{j})$. As
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$ acts freely
on delta sequences, $\tau(D_{i_{1}})=D_{i_{2}}$ completely determines $\tau$
and the $(a_{1},\dots,a_{\ell})$ can be sorted as follows:

$(a_{1},\dots,a_{\frac{\ell}{T}},\quad
a_{1},\dots,a_{\frac{\ell}{T}},\quad\dots,\quad
a_{1},\dots,a_{\frac{\ell}{T}}).$

Note that $T(a_{1}+\dots+a_{\ell})=j$ and $j$ must be divisible by $T$. There
are

$S:=\binom{j}{a_{1}}\binom{j-a_{1}}{a_{2}}\dots\binom{j-a_{1}-a_{2}-\dots-
a_{\ell-1}}{a_{\ell}}$

elements in the $\mathfrak{S}_{j}$-orbit of $\mathcal{D}$. Hence

$\displaystyle\nu_{2}(S)$
$\displaystyle=\sum_{i=1}^{\ell}\Big{(}s_{2}(a_{i})+s_{2}(j-a_{1}-\dots-
a_{i})-s_{2}(j-a_{1}-\dots-a_{i-1})\Big{)}$
$\displaystyle=s_{2}(0)-s_{2}(j)+\sum_{i=1}^{\ell}s_{2}(a_{i}).$

If the number of non-equivalent elements of the $\mathfrak{S}_{j}$-orbit of
$\mathcal{D}$ is even, $\mathcal{D}$ may be disregarded for the parity of
$N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$. So suppose that it is odd, it
follows that $\nu_{2}(T)=\nu_{2}(S)$. However,

$\displaystyle\nu_{2}(S)$
$\displaystyle=-s_{2}(j)+T\sum_{i=1}^{\frac{\ell}{T}}s_{2}\left(a_{i}\right)$
$\displaystyle\geq-
s_{2}(T)s_{2}\left(\frac{j}{T}\right)+T\sum_{i=1}^{\frac{\ell}{T}}s_{2}\left(a_{i}\right)$
$\displaystyle\stackrel{{\scriptstyle(\ref{Eq:1})}}{{\geq}}-\left(T-\nu_{2}(T)\right)s_{2}\left(\frac{j}{T}\right)+T\sum_{i=1}^{\frac{\ell}{T}}s_{2}\left(a_{i}\right)$
(1)
$\displaystyle=\nu_{2}(T)s_{2}\left(\frac{j}{T}\right)+T\left(-s_{2}\left(\frac{j}{T}\right)+\sum_{i=1}^{\frac{\ell}{T}}s_{2}\left(a_{i}\right)\right)$
$\displaystyle\stackrel{{\scriptstyle(\ref{Eq:2})}}{{\geq}}\nu_{2}(T)s_{2}\left(\frac{j}{T}\right)$
$\displaystyle\stackrel{{\scriptstyle(\ref{Eq:3})}}{{\geq}}\nu_{2}(T).$ (3)

Note, that (1) follows from Lemma 2.20 and that (2.2.1) follows from Lemma
2.21. So the above inequalities must be tight. This implies:

  * •

$T\in\\{1,2\\}$ by (1) and Lemma 2.20.

  * •

The dyadic digits of $a_{1},\dots,a_{{\frac{\ell}{T}}}$ decompose the dyadic
digits of $\frac{j}{T}$ by (2.2.1) and Lemma 2.21.

  * •

$T$ is odd or $\frac{j}{T}$ is a power of two by (3).

∎

To determine $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ it now suffices to
consider those $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequences,
such that

  1. (1)

each $D_{i}$ is the delta sequence of a permutation of a standard $k$-bit Gray
code by Lemma 2.18 and

  2. (2)

the multiplicity $(a_{1},\dots,a_{\ell})$ of $(D_{1},\dots,D_{j})$ satisfies
the conditions of Propostion 2.24.

Suppose that $(D_{1},\dots,D_{j})$ is a sequence that satisfies both. Recall
that the standard $k$-bit Gray code has transition counts
$(2^{k-1},2^{k-2},\dots,1)$. Hence,

$\max(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})=a_{1}\cdot
2^{k-1}+(j-a_{1})\cdot 1,$

as $a_{1}$ of the $D_{1},\dots,D_{j}$ have transition count $2^{k-1}$ for some
row.

If $j=2^{n}$ and the $\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}$ are not
pairwise distinct, we have that $(a_{1},\dots,a_{\ell})$ is one of $(2^{n})$
and $(2^{n-1},2^{n-1})$. In this case

$\max(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})\geq 2^{n}2^{k-1}+2^{n}.$

Otherwise, if $j=2^{n}+r$ for $0\leq r<2^{n}$, we have that $a_{1}$ is at
least $2^{n}$ and we conclude

$\max(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})\geq 2^{n}2^{k-1}+j.$

The first case implies that

$E(2^{n-1}+2^{n-1},k)\geq 2^{n-1}2^{k-1}+2^{n-1}$

and the second case implies that

$E(2^{n}+r,k)\geq 2^{n}2^{k-1}+r\leq I(2^{n}+r,k).$

This bounds the values of $E(j,k)$ and $I(j,k)$ from below according to
Theorem 2.15. To bound the value from above and conclude our proof, we need to
find some values for which $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$ is
odd.

The case $k=2$ needs special attention, but was also known before:

######  Proposition 2.25 (See [3, Section 4.3.1]).

Let $k=2$ and let $2\leq j=2^{n}+r$ with $0\leq r\leq 2^{n}$. Then

$N(2^{n}+2r,2^{n}2+r)=1.$

Note that $2^{n}+2r\neq 2^{n}2+r$ unless $r=2^{n}$.

###### Proof.

There exist exactly two $2$-bit delta sequences with transition counts $(1,2)$
resp. $(2,1)$. Hence the number of $(2^{n}+2r,2^{n}2+r)$-delta sequences is
$\binom{j}{r}=\binom{2^{n}+r}{r}$. By Lemma 2.19:

$\nu_{2}\left(\binom{2^{n}+r}{r}\right)=s_{2}(r)+s_{2}(2^{n})-s_{2}(2^{n}+r)=s_{2}(r)+1-s_{2}(2^{n}+r)=\begin{cases}0,&\text{if
}0\leq r<2^{n},\\\ 1,&\text{if }r=2^{n}.\end{cases}$

We see that if $r\neq 2^{n}$, then the number of
$(2^{n}+2^{r},2^{n}2+r)$-delta sequences is odd and $N(2^{n}+2r,2^{n}2+r)$
must be odd as well. If $r=2^{n}$, then the number of
$(2^{n}+2^{r},2^{n}2+r)$-delta sequences is even but not divisible by $4$.
However, $|\mathfrak{S}_{(2^{n}+2r,2^{n}2+r)}|=2$, so the delta sequences come
in equivalent pairs and $N(2^{n}+2r+2^{n}2+r)$ is also odd in this case. ∎

The next step is also intuitive. Combining $2^{n}$ delta sequences with
transition counts $(2,4,\dots,2^{k-1},1)$ and $2^{n}$ delta sequences with
transition counts $(2,4,\dots,2^{k-2},1,2^{k-1})$ will give the only case of

$2^{n}\cdot(2+2,4+4,\dots,2^{k-2}+2^{k-2},2^{k-1}+1,2^{k-1}+1)$

-equiparting matrices we need to consider and will show that $E(2^{n+1},k)=2^{n+k-1}+2^{n}$:

######  Proposition 2.26.

Let $k\geq 2$ and $j=2^{n+1}$ Then

$N(2^{n+2},2^{n+3},\dots,2^{n+k-1},2^{n+k-1}+2^{n},2^{n+k-1}+2^{n})$

is odd.

###### Proof.

Note that

$|\mathfrak{S}_{(2^{n+2},2^{n+3},\dots,2^{n+k-1},2^{n+k-1}+2^{n},2^{n+k-1}+2^{n})}|=2.$

We only need to consider those delta sequences of permutations of standard
$k$-bit Gray codes of multiplicity $(2^{n+1})$ or $(2^{n},2^{n})$. Such
sequences do not exist for multiplicity $(2^{n+1})$ so we consider the other
case.

Suppose $\mathcal{D}=(D_{1},\dots,D_{j})$ is such a sequence. As noted above,
the transition count $c_{i}(\mathcal{D})\geq 2^{k-1}2^{n}+2^{n}$ for two of
the $i$. It immediatly follows that $2^{n}$ of the $(D_{1},\dots,D_{j})$ have
transition counts $(2,4,\dots,2^{k-2},1,2^{k-1})$ and the others have
transition counts $(2,4,\dots,2^{k-2},2^{k-1},1)$.

The $\mathfrak{S}_{j}$-orbit has size $\binom{2^{n+1}}{2^{n}}$, which is
divisible by two, but not by four. Those come in pairs of equivalent
permutations, hence the number of non-equivalent elements is odd. ∎

The other remaining cases are a bit more difficult to see. With the help of a
computer we see some examples:

######  Examples 2.27.

Let $k=4$. The following are all $\mathfrak{c}_{1},\dots,\mathfrak{c}_{4}$ for
different $j$ with $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{4})$ odd and

$\mathfrak{c}_{1}<\mathfrak{c}_{2}<\mathfrak{c}_{3}<\mathfrak{c}_{4}$

such that $\mathfrak{c}_{4}$ is minimal with this property:

  1. $j=1$:

$(1,2,4,8)$,

  2. $j=2$:

$(2,4,8,16)$,

  3. $j=3$:

$(4,8,16,17)$,

  4. $j=4$:

$(4,8,16,32)$,

  5. $j=5$:

$(6,12,24,33)$, $(6,16,20,33)$, $(8,10,24,33)$, $(10,12,20,33)$,
$(8,16,18,33)$,

  6. $j=6$:

$(8,16,32,34)$,

  7. $j=7$:

$(12,24,34,35)$, $(16,20,34,35)$,

  8. $j=8$:

$(8,16,32,64)$,

  9. $j=9$:

$(10,20,40,65)$, $(10,24,36,65)$, $(12,18,40,65)$, $(12,24,34,65)$,
$(16,18,36,65)$, $(16,20,34,65)$,

  10. $j=10$:

$(12,24,48,66)$, $(12,32,40,66)$, $(16,20,48,66)$, $(16,32,36,66)$,
$(20,24,40,66)$,

  11. $j=11$:

$(14,28,56,67)$, $(14,32,52,67)$, $(14,36,48,67)$, $(14,40,44,67)$,
$(16,26,56,67)$, $(16,32,50,67)$,
$(16,34,48,67)$, $(16,40,42,67)$, $(18,24,56,67)$, $(18,28,52,67)$,
$(18,36,44,67)$, $(20,22,56,67)$,
$(20,26,52,67)$, $(20,36,42,67)$, $(20,38,40,67)$, $(22,24,52,67)$,
$(22,28,48,67)$, $(22,32,44,67)$,
$(24,26,48,67)$, $(24,32,42,67)$, $(24,34,40,67)$, $(24,36,38,67)$,
$(26,28,44,67)$,

  12. $j=12$:

$(16,32,64,68)$,

  13. $j=13$:

$(18,40,68,69)$, $(20,40,66,69)$, $(24,34,68,69)$, $(24,36,66,69)$,

  14. $j=14$:

$(24,48,68,70)$, $(32,40,68,70)$

  15. $j=15$:

$(28,56,70,71)$, $(32,52,70,71)$, $(36,48,70,71)$, $(40,44,70,71)$

Apparently there are many values, to verify that the lower bound is achieved.
One observation is that there is always a tuple with
$2\mathfrak{c}_{1}=\mathfrak{c}_{2}$. This generalizes:

######  Examples 2.28.

Let $k=5$. The following are all $\mathfrak{c}_{1},\dots,\mathfrak{c}_{5}$ for
different $j$ with $N(\mathfrak{c}_{1},\dots,\mathfrak{c}_{5})$ odd and

$4\mathfrak{c}_{1}=2\mathfrak{c}_{2}=\mathfrak{c}_{3}<\mathfrak{c}_{4}<\mathfrak{c}_{5}$

such that $\mathfrak{c}_{5}$ is minimal with this property:

  1. $j=1$:

$(1,2,4,8,16)$,

  2. $j=2$:

$(2,4,8,16,32)$,

  3. $j=3$:

$(4,8,16,32,33)$,

  4. $j=4$:

$(4,8,16,32,64)$,

  5. $j=5$:

$(6,12,24,48,65)$, $(8,16,32,34,65)$,

  6. $j=6$:

$(8,16,32,64,66)$

  7. $j=7$:

$(12,24,48,66,67)$

  8. $j=8$:

$(8,16,32,64,128)$

  9. $j=9$:

$(10,20,40,80,129)$, $(12,24,48,66,129)$

  10. $j=10$:

$(12,24,48,96,130)$, $(16,32,64,68,130)$

  11. $j=11$:

$(14,28,56,112,131)$, $(16,32,64,98,131)$

  12. $j=12$:

$(16,32,64,128,132)$

  13. $j=13$:

$(20,40,80,130,133)$

  14. $j=14$:

$(24,48,96,132,134)$

Apparently we may start with $2^{n}$ times transition counts $(1,2,4,8,16)$,
fill up with $(2,4,8,16,1)$ until we reach $j=2^{n}+2^{n-1}$ and the remaining
onces are $(4,8,16,2,1)$:

######  Proposition 2.29.

Let $k>2$ and $2\leq j=2^{n}+r$ with $0\leq r<2^{n}$. Define integers
$w=\min(r,2^{n-1})$ and $s=r-w$ such that $j=2^{n}+w+s$. Then

$N\left(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}\right)=1,$

where

$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})=2^{n}\cdot(1,2,4,\dots,2^{k-1})+w\cdot(2,4,\dots,2^{k-1},1)+s\cdot(4,8,\dots,2^{k-1},2,1)$

###### Proof.

It holds that

$\mathfrak{c}_{k-1}-\mathfrak{c}_{k-2}=2^{k-3}2^{n}+2^{k-2}w-(2^{k-1}-2)s>\begin{cases}0,&\text{if
}s=0,\\\ 2^{k-3}2w+2^{k-2}w-2^{k-1}w=0,&\text{otherwise},\end{cases}$

and

$\mathfrak{c}_{k}-\mathfrak{c}_{k-1}=2^{k-2}2^{n}+w+s-2^{k-1}w-2s.$

Hence

$\mathfrak{c}_{1}<\mathfrak{c}_{2}<\dots<\mathfrak{c}_{k-2}<\mathfrak{c}_{k-1}<\mathfrak{c}_{k}.$

as $r<2^{n}$. This implies that

$|(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}|=1.$

By Proposition 2.24 it suffices to consider
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-delta sequences of multiplicity
$(a_{1},a_{2},\dots,a_{\ell})$, where we may assume that $a_{1}\geq 2^{n}$. If
$w=2^{n-1}$, then we may assume that $a_{2}\geq 2^{n-1}$ or that $a_{1}\geq
2^{n}+2^{n-1}$. By Lemma 2.18 it suffices to consider those sequences that
consist of permutations of the standard $k$-bit Gray code.

Suppose $\mathcal{D}=(D_{1},\dots,D_{j})$ is such sequence. W.l.o.g. we
reorder $\mathcal{D}$ such that $D_{1}=\dots=D_{2^{n}}$ and such that $a$ is
maximal with $D_{2^{n}+1}=\dots=D_{2^{n}+a}$. We will show that
$(a_{1},\dots,a_{\ell})=(2^{n},w,s)$ and that $a=w$ and that

$c(D_{m})=\begin{cases}\left(1,2,4,\dots,2^{k-1}\right),&\text{if }m\leq
2^{n},\\\ \left(2,4,\dots,2^{k-1},1\right),&\text{if }2^{n}<m\leq 2^{n}+w,\\\
\left(4,8,\dots,2^{k-1},2,1\right),&\text{otherwise}.\end{cases}$

As above $2^{n}$ of the $(D_{1},\dots,D_{j})$ are equal. Hence at least one of
the transition counts is $2^{k-1}2^{n}+r=2^{k-1}2^{n}+w+s$. As
$\mathfrak{c}_{k}$ has this value and is strictly the largest, we obtain

$c_{k}(D_{m})=\begin{cases}2^{k-1},&\text{if }m\leq 2^{n},\\\
1,&\text{otherwise}.\end{cases}$

Suppose $s>0$. This implies $w=2^{n-1}$ and

$\mathfrak{c}_{k-1}\equiv 2s\mod 2^{n}$

with $2s<2^{n}$. However,

$\nu_{2}\left(c_{k-1}(D_{1})\right)=\dots=\nu_{2}\left(c_{k-1}(D_{2^{n}})\right)\geq
0$

and

$\nu_{2}\left(c_{k-1}(D_{2^{n}+1})\right)=\dots=\nu_{2}\left(c_{k-1}(D_{2^{n}+w})\right)\geq
1.$

Therefore,

$c_{k-1}(D_{1})+\dots+c_{k-1}(D_{j})\equiv
c_{k-1}(D_{2^{n}+w+1})+\dots+c_{k-1}(D_{j})\mod 2^{n}.$

As they are all at least $2$, we conclude that

$c_{k-1}(D_{2^{n}+w+1})=\dots=c_{k-1}(D_{j})=2.$

Next we use induction on $i$ to determine the $c_{i}(D_{m})$ for $i\leq k-2$.
By induction we have that

$c_{i}(D_{m})\geq\begin{cases}2^{i-1},&\text{if }m\leq 2^{n},\\\
2^{i},&\text{if }2^{n}<m\leq 2^{n}+w,\\\ 2^{i+1},&\text{if }2^{n}+w<m\leq
2^{n}+w+s.\end{cases}$

As $\mathfrak{c}_{i}=2^{i-1}2^{n}+2^{i}w+2^{i+1}s$, it follows that the
inequalities are tight. We conclude the claimed values for
$c_{1}(D_{m}),\dots,c_{k-2}(D_{m})$ and $c_{k}(D_{m})$ for all $m$. The values
for $c_{k-1}(D_{m})$ follow as they are the only values left.

The $\mathfrak{S}_{j}$-orbit of $\mathcal{D}$ has size
$\binom{2^{n}+w+s}{s}\binom{2^{n}+w}{w}$, which is not divisible by two. ∎

##  3\. Reduction to Equiparting matrices

We will briefly summarize the approach in [3] along with a slight
generalization. Via the inclusion

$\iota\colon\mathbb{R}^{d}\to\mathbb{R}^{d+1},\quad(y_{1},\dots,y_{d})\mapsto(1,y_{1},\dots,y_{d})$

we parametrize oriented affine hyperplanes in $\mathbb{R}^{d}$ by $S^{d}$ with
two non-proper hyperplanes that cannot correspond to solutions. A vector $v\in
S^{d}$ corresponds to the oriented hyperplane

$\left\\{r\in\mathbb{R}^{d}\colon\langle\iota(r),v\rangle=0\right\\}$

with induced positive and negative side.

A collection $\mathcal{M}=(\mu_{1},\dots,\mu_{j})$ of finite Borel measures on
$\mathbb{R}^{d}$ induces a map

$\displaystyle\psi_{\mathcal{M}}\colon\quad X_{d,k}=(S^{d})^{*k}\quad$
$\displaystyle\to\quad W_{k}\oplus(U_{k})^{\oplus
j}\cong\mathbb{R}^{k-1}\oplus\big{(}\mathbb{R}^{(\mathbb{Z}/2)^{k}}\big{)}^{\oplus
j}$ $\displaystyle\lambda_{1}v_{1}+\dots+\lambda_{k}v_{k}\quad$
$\displaystyle\mapsto\quad(\lambda_{1}-\frac{1}{k},\dots,\lambda_{k}-\frac{1}{k})\oplus(\lambda_{1}\cdots\lambda_{k})\cdot\phi_{\mathcal{M}}(v_{1},\dots,v_{k}),$

where $\phi_{\mathcal{M}}$ assigns $k$ oriented hyperplanes with normals
$(v_{1},\dots,v_{k})$ to each measure evaluated on each of the (possibly
empty) $2^{k}$ regions minus $\frac{1}{2^{k}}\mu_{i}(\mathbb{R}^{d})$. This
map is $\mathfrak{S}_{k}^{\pm}$-equivariant as explained in [3, Section 2.3].
If $0$ is in the image for all collections of measures, then $\Delta(j,k)\leq
d$.

If $0$ lies not in the image, then $\psi_{\mathcal{M}}$ can be composed with
the radial projection $\nu\colon(W_{k}\oplus U_{k}^{\oplus
j})\setminus\\{0\\}\to S(W_{k}\oplus U_{k}^{\oplus j})$.

Denote by $X_{d,k}^{>1}$ the subset of those points in $X_{d,k}$ of non-
trivial stabilizer. The image of $X_{d,k}^{>1}$ does not contain $0$ and in
fact any two maps $\psi_{\mathcal{M}}$ and $\psi_{\mathcal{M}^{\prime}}$
restricted to $X_{d,k}^{>1}$ are $\mathfrak{S}_{k}^{\pm}$-homotopic [3, Prop
2.2]. This yields:

######  Proposition 3.1 ([3, Thm. 2.3 (ii)]).

Let $d,k,j\geq 1$ be integers and let $\mathcal{M}=(\mu_{1},\dots,\mu_{j})$ be
finite Borel measures on $\mathbb{R}^{d}$.

If there is no $\mathfrak{S}_{k}^{\pm}$-equivariant map

$X_{d,k}\to S(W_{k}\oplus U_{k}^{\oplus j})$

whose restriction to $X_{d,k}^{>1}$ is $\mathfrak{S}_{k}^{\pm}$-homotopic to
$\nu\circ\psi_{\mathcal{M}}\mid_{X_{d,k}^{>1}}$, then $\Delta(j,k)\leq d$.

######  Corollary 3.2.

If there is no $(\mathbb{Z}/2)^{k}$-equivariant map

$X_{d,k}\to S(W_{k}\oplus U_{k}^{\oplus j})$

whose restriction to $X_{d,k}^{>1}$ is $(\mathbb{Z}/2)^{k}$-homotopic to
$\nu\circ\psi_{\mathcal{M}}\mid_{X_{d,k}^{>1}}$, then $\Delta(j,k)\leq d$.

There are different approaches, how to show the non-existent of such
equivariant maps. We will use obstruction theory as developed in [3]. As a
first step, one can equip $X_{d,k}$ with the CW-structure developed in [3,
Section 3]. We assume the reader to be familiar with this CW-structure. We
remark that $X_{d,k}^{>1}$ is a subcomplex, which allows us to use relative
equivariant obstruction theory. The definition of a (relatively open) cell is
as follows:

######  Definition 3.3 (See [3, Sections 2.2 and 2.3]).

Let $(\sigma_{1},\dots,\sigma_{k})$ be a permutation of $1,\dots,k$. Let
$(s_{1},\dots,s_{k})\in\\{+1,-1\\}^{k}$ and let
$(i_{1},\dots,i_{k})\in\\{1,\dots,d+2\\}^{k}$. Then

$\displaystyle
C_{i_{1},\dots,i_{k}}^{s_{1},\dots,s_{k}}(\sigma_{1},\dots,\sigma_{k}):=\left\\{(x_{1},\dots,x_{k})\in\mathbb{R}^{(d+1)\times
k}\colon
0<_{i_{1}}s_{1}x_{\sigma_{1}}<_{i_{2}}\dots<_{i_{k}}s_{k}x_{\sigma_{k}}\right\\},$

where $y<_{i}y^{\prime}$ means that $y$ and $y^{\prime}$ agree in the first
$i-1$ coordinates and $y_{i}<y_{i}^{\prime}$.

This induces a (relatively open) cell

$D_{i_{1},\dots,i_{k}}^{s_{1},\dots,s_{k}}(\sigma_{1},\dots,\sigma_{k}):=C_{i_{1},\dots,i_{k}}^{s_{1},\dots,s_{k}}(\sigma_{1},\dots,\sigma_{k})\cap
S(\mathbb{R}^{(d+1)\times k}).$

We proceed as described in [3, Section 2.6]. Let $N_{2}=(2^{k}-1)j+k-2$ be the
dimension of the sphere $S(W_{k}\oplus U_{k}^{\oplus j})$ and let $\theta$ be
some $(N_{2}+1)$-cell of $X_{d,k}$ and let $Z$ be the union of $X_{d,k}^{>1}$
with the $\mathfrak{S}_{k}^{\pm}$-orbit of the relative closure of $\theta$.
It suffices to show that the map
$\nu\circ\psi_{\mathcal{M}}\mid_{X_{d,k}^{>1}}$ cannot be
$\mathfrak{S}_{k}^{\pm}$-equivariantly extended to $Z$. We may also use
Corollary 3.2 and show that this map cannot be
$(\mathbb{Z}/2)^{k}$-equivariantly extended to $Z$.

Let $\mathcal{S}$ be a subgroup of $\mathfrak{S}_{k}^{\pm}$. An extension to
the $N_{2}$-skeleton $Z^{(N_{2})}$ can be $\mathcal{S}$-equivariantly extended
to $Z$ if and only if the obstruction cocycle

$\mathfrak{o}(g)\in\mathcal{C}_{\mathcal{S}}^{N_{2}+1}\big{(}Z,X_{d,k}^{>1};\pi_{N_{2}}(S(W_{k}\oplus
U_{k}^{\oplus j}))\big{)}$

is zero. If this cochain is not a coboundary, i.e.

$0\neq[\mathfrak{o}(g)]\in\mathcal{H}_{\mathcal{S}}^{N_{2}+1}\big{(}Z,X_{d,k}^{>1};\pi_{N_{2}}(S(W_{k}\oplus
U_{k}^{\oplus j}))\big{)},$

then this map cannot be extended to $Z$ independent of the choice on the
$N_{2}$-skeleton. In general it can be difficult to determine the obstruction
cochain and its cohomology class. However, the parity of the cochain can be
determined by counting equiparting matrices for a suitable choice of the cell
$\theta$ and the measures $\mathcal{M}$. Even better, cochains of odd parity
do not vanish in this case:

######  Proposition 3.4.

Let $\mathcal{S}$ be a subgroup of $\mathfrak{S}_{k}^{\pm}$. Let $\theta$ be
an $(N_{2}+1)$-cell of $X_{d,k}$ such that the $N_{2}$-faces
$\eta_{1},\dots,\eta_{n},\zeta_{1},\dots,\zeta_{n}$ in the boundary of
$\theta$ can be grouped in pairs such that $\eta_{i}$ and $\zeta_{i}$ are in
one $\mathcal{S}$-orbit for each $i=1,\dots,n$.

Suppose $\mathcal{M}$ is a collection of measures such that $0$ is not in the
image of $\psi_{\mathcal{M}}$ restricted to the boundary of $\theta$ and that
the number of $0$s in the image of the interior of $\theta$ is finite and odd,
then $\Delta(j,k)\leq d$.

###### Proof.

Denote by $e_{\theta}$ the element in the cellular chain group
$\mathcal{C}_{N_{2}+1}(Z,X_{d,k}^{>1})$ corresponding to $\theta$. The image
of the boundary of $\theta$ under $\psi_{\mathcal{M}}$ does not contain zero.
Then by [3, Section 2.6] $\mathfrak{o}(g)(e_{\theta})$ is the same as the
number of $x\in\operatorname{relint}\theta$ with $\psi_{\mathcal{M}}(x)=0$
modulo $2$.

We proceed as in [3, Proof of Theorem 1.4]. Suppose there exists a cochain

$\mathfrak{h}\in\mathcal{C}_{\mathcal{S}}^{N_{2}}\big{(}Z,X_{d,k}^{>1};\pi_{N_{2}}(S(W_{k}\oplus
U_{k}^{\oplus j}))\big{)}$

such that $\delta\mathfrak{h}=\mathfrak{o}(g)$, then in particular

$\mathfrak{o}(g)(e_{\theta})=\left(\delta\mathfrak{h}\right)(e_{\theta})=\mathfrak{h}(\eta_{1}+\dots+\eta_{n}+\zeta_{1}+\dots+\zeta_{n}).$

However, it holds that $\mathfrak{h}(\eta_{i})=\pm\mathfrak{o}(\zeta_{i})$ for
all $i=1,\dots,n$. This is a contradiction and hence the obstruction cocycle
is not a coboundary and the map cannot be $\mathcal{S}$-equivariantly extended
to $Z$. By Proposition 3.1 this implies that $\Delta(j,k)\leq d$. ∎

Now we will describe a suitable cell $\theta$. Let $d\geq 1,\,j\geq 1$ and
$k\geq 2$ be integers such that $dk\geq(2^{k}-1)j$. Let further
$\ell_{1}\geq\dots\geq\ell_{k}\geq 0$ be integers with
$\ell_{1}+\dots+\ell_{k}=dk-(2^{k}-1)j$.

Consider the cell

$\theta=D_{\ell_{1}+1,\ell_{2}+1,\dots,\ell_{k}+1}^{+,+,\dots,+}(1,2,\dots,k).$

This cell has dimension

$(d+2)k-k-(\ell_{1}+\dots+\ell_{k})-1=k+(2^{k}-1)j-1=N_{2}+1$

as desired.

By definition, $\theta$ parametrizes all arrangements
$\mathcal{H}=(H_{1},\dots,H_{k})$ of $k$ linear oriented hyperplanes in
$\mathbb{R}^{d+1}$ with normal vectors $x_{1},\dots,x_{k}$ such that $x_{i}$
has the first $\ell_{i}$ coordinates zero and coordinate $\ell_{i}+1$ strictly
greater than $x_{i-1}$, where $x_{0}:=0$.

######  Lemma 3.5.

The boundary of $\theta$ can be grouped in pairs of
$\mathfrak{S}_{k}^{\pm}$-orbit elements. If farther $\ell_{1},\dots,\ell_{k}$
are pairwise distinct, the boundary can be grouped in paris of
$(\mathbb{Z}/2)^{k}$-orbit elements.

###### Proof.

The cells in the boundary are obtained by introducing one of the following
equalities:

$0=x_{\ell_{1}+1,1},\quad x_{\ell_{2}+1,1}=x_{\ell_{2}+1,2},\quad\dots,\quad
x_{\ell_{k}+1,k-1}=x_{\ell_{k}+1,k}.$

  1. (A)

The equality $0=x_{\ell_{1}+1,1}$ induces cells:

$\eta_{1}:=D_{\ell_{1}+2,\ell_{2}+1,\dots,\ell_{k}+1}^{+,+,\dots,+}(1,2,\dots,k),\quad\zeta_{1}:=D_{\ell_{1}+2,\ell_{2}+1,\dots,\ell_{k}+1}^{-,+,\dots,+}(1,2,\dots,k)$

that are related, as sets via $\zeta_{1}=\varepsilon_{1}\cdot\eta_{1}$
($\varepsilon_{1}$ is the group element flipping the normal of the first
hyperplane).

  2. (B)

Let $b\in\\{2,\dots,k\\}$ and suppose $\ell_{b-1}\geq\ell_{b}+1$. Let $a$ be
minimal such that $\ell_{a}\leq\ell_{b}+1$ (possibly $a=b$). For the relative
interior of $\theta$ it holds that

$0<_{\ell_{b}+2}x_{a}<_{\ell_{b}+2}x_{a+1}<_{\ell_{b}+2}\dots<_{\ell_{b}+2}x_{b-1}<_{\ell_{b}+1}<x_{b}$

and $a$ is minimal with this property. When introducing the equality
$x_{\ell_{b}+1,b}=x_{\ell_{b}+1,b-1}$ there are $b-a+1$ possible positions for
the $b$-th hyperplane, each position with both signs. This equality induces
the cells

$\displaystyle\eta_{b,b}$
$\displaystyle:=D_{\ell_{1}+1,\dots,\ell_{b-1}+1,\ell_{b}+2,\ell_{b+1}+1,\dots,\ell_{k}+1}^{+,+,\dots,+}(1,\dots,k)$
$\displaystyle\zeta_{b,b}$ $\displaystyle:=\varepsilon_{b}\cdot\eta_{b,b},$
$\displaystyle\eta_{b-1,b}$ $\displaystyle:=\tau_{b-1,b}\cdot\eta_{b,b},$
$\displaystyle\zeta_{b-1,b}$
$\displaystyle:=\varepsilon_{b-1}\cdot\tau_{b-1,b}\cdot\eta_{b,b}=\varepsilon_{b-1}\cdot\eta_{b-1,b},$
$\displaystyle\eta_{b-2,b}$
$\displaystyle:=\tau_{b-2,b-1}\cdot\tau_{b-1,b}\cdot\eta_{b,b},$
$\displaystyle\zeta_{b-2,b}$
$\displaystyle:=\varepsilon_{b-2}\cdot\tau_{b-2,b-1}\cdot\tau_{b-1,b}\cdot\eta_{b,b}=\varepsilon_{b-2}\cdot\eta_{b-2,b},$
$\displaystyle\dots$ $\displaystyle\eta_{a,b}$
$\displaystyle:=\tau_{a,a+1}\cdot\tau_{a+1,a+2}\cdot\dots\cdot\tau_{b-1,b}\cdot\eta_{b,b},$
$\displaystyle\zeta_{a,b}$
$\displaystyle:=\varepsilon_{a}\cdot\tau_{a,a+1}\cdot\tau_{a+1,a+2}\cdot\dots\cdot\tau_{b-1,b}\cdot\eta_{b,b}=\varepsilon_{a}\cdot\eta_{a,b}.$

$\tau_{c,d}$ swaps the hyperplanes $c$ and $d$.

  3. (C)

Let $b\in{2,\dots,k}$ and suppose $\ell_{b-1}=\ell_{b}$, in particular the
$(\ell_{1},\dots,\ell_{k})$ are not pairwise distinct. The equality
$x_{\ell_{b}+1,b}=x_{\ell_{b}+1,b-1}$ induces two cells

$\displaystyle\eta_{b,b}$
$\displaystyle:=D_{\ell_{1}+1,\dots,\ell_{b-1}+1,\ell_{b}+2,\ell_{b+1}+1,\dots,\ell_{k+1}}^{+,+,\dots,+}(1,\dots,k)$
$\displaystyle\zeta_{b-1,b}$ $\displaystyle:=\tau_{b-1,b}\cdot\eta_{b,b}.$

∎

Consider the binomial moment curve

$\gamma\colon\mathbb{R}\to\mathbb{R}^{d},\quad
t\mapsto\left(t,\binom{t}{2},\binom{t}{3},\dots,\binom{t}{d}\right)$

and points

$q_{1}:=\gamma(0),\quad q_{2}:=\gamma(1),\quad\dots,\quad
q_{\ell_{1}+1}=\gamma(\ell_{1}).$

We obtain the following parametrization:

######  Lemma 3.6.

The relative closure of the cell $\theta$ parametrizes those
$(v_{1},\dots,v_{k})$ in $(S^{d})^{k}$ corresponding to affine oriented
hyperplanes $\mathcal{H}=(H_{1},\dots,H_{k})$ in $\mathbb{R}^{d}$ such that

  * •

$\\{q_{1},\dots,q_{\ell_{i}}\\}\subset H_{i}$ for $i=1,\dots,k$,

  * •

if $\ell_{i}=\ell_{i+1}$ for any $i=1,\dots,k-1$, then $v_{\ell_{i}+1,i}\leq
v_{\ell_{i+1}+1,i+1}$, where $v_{\ell_{i}+1,i}$ is the $(\ell_{i}+1)$th-
coordinate of the unit normal vector of the hyperplane $H_{i}$,

  * •

$v_{\ell_{i}+1,i}\geq 0$ for any $i=1,\dots,k$,

and for relative open cell $\theta$ additionally

  * •

$q_{\ell_{i}+1}\not\in H_{i}$ for $i=1,\dots,k$,

  * •

if $\ell_{i}=\ell_{i+1}$ for any $i=1,\dots,k-1$, then
$v_{\ell_{i}+1,i}<v_{\ell_{i+1}+1,i+1}$,

  * •

$v_{\ell_{i}+1,i}>0$ for any $i=1,\dots,k$.

By parametrize we mean the restriction of $X_{d,k}$ to the embedding of
$(S^{d})^{k}$:

$\left\\{(\lambda_{1}v_{1},\dots\lambda_{k}v_{k})\in
X_{d,k}\colon\lambda_{1}=\dots=\lambda_{k}=\frac{1}{k}\right\\}.$

If $\psi_{\mathcal{M}}(x)=0$, then $x$ lies in the image of the embedding.

###### Proof.

This proof is analogous to [3, Lem. 3.13]:

By Definition 3.3 the relatively open cell $\theta$ corresponds to hyperplanes
with normals $v_{1},\dots,v_{k}$ such that

$0<_{\ell_{1}+1}v_{1}<_{\ell_{2}+1}v_{2}<_{\ell_{3}+1}\dots<_{\ell_{k}+1}v_{k}.$

As $\ell_{1}\geq\ell_{2}\geq\dots\geq\ell_{k}$, by induction on $i$ we see
that this is equivalent to

$0=v_{i,1},\quad 0=v_{i,2},\quad\dots,\quad 0=v_{i,\ell_{i}},\quad 0\leq
v_{i-1,\ell_{i}+1}<v_{i,\ell_{i}+1}.$

for each $i=1,\dots,k$, where $v_{0}:=0$. The relative closure is obtained by
allowing the strict inequality to be non-strict.

Observe that a hyperplane with normal $v$, that is zero in the first $\ell$
entries contains $q_{\ell+1}$ if and only if the $\ell+1$th entry is zero as
well. The characterization of the relatively open cell $\theta$ and its
closure follow. ∎

We are now ready to use Theorem 2.15:

######  Proposition 3.7.

  1. (1)

$\Delta(j,k)\leq I(j,k)$.

  2. (2)

$\Delta(j,k)\leq E(j,k)$.

Along with Theorem 2.15 this proves Theorem 1.2. The part 1 is strictly
weaker, but uses only the $(\mathbb{Z}/2)^{k}$-action. Its purpose is to
explain why $I(j,k)$ coincides with the bound provided in [6].

###### Proof.

  1. (2)

Suppose that $d=E(j,k)$. In this case there are
$d=\mathfrak{c}_{1}\geq\mathfrak{c}_{2}\geq\dots\geq\mathfrak{c}_{k}\geq 0$
with

$\mathfrak{c}_{1}+\dots+\mathfrak{c}_{k}=j(2^{k}-1).$

such that the number of non-equivalent
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices is odd.

  2. (1)

If $d=I(j,k)$ there are such $\mathfrak{c}_{1},\dots,\mathfrak{c}_{k}$ as
above which are additionally pairwise distinct. Note that in this case
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}$ is trivial
and
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}^{\pm}=(\mathbb{Z}/2)^{k}$
and non-equivalent $(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting
matrices are the same as non-isomorphic ones.

Either way, define

$\ell_{1}:=d-\mathfrak{c}_{1},\quad\dots,\quad\ell_{k}:=d-\mathfrak{c}_{k}.$

Consider the cell

$\displaystyle\theta$
$\displaystyle=D_{\ell_{1}+1,\ell_{2}+1,\dots,\ell_{k}+1}^{+,+,\dots,+}(1,2,\dots,k)$
$\displaystyle=D_{d-\mathfrak{c}_{1}+1,d-\mathfrak{c}_{2}+1,\dots,d-\mathfrak{c}_{k}+1}^{+,+,\dots,+}(1,2,\dots,k).$

In order to use Proposition 3.4 it remains to find measures $\mathcal{M}$ such
that the image of $\psi_{\mathcal{M}}$

  * •

does not contain $0$ restricted to the boundary of $\theta$,

  * •

contains an odd number of $0$s restricted to the relative interior of
$\theta$.

We proceed as in [3, Lemma 4.1, Lemma 4.2]. However, in [3] it was omitted to
show that the $0$s are in the relative open cell $\theta$ (and not just in the
closure as was done in [3, Thm. 1.3]). Here we provide a proof for
completeness.

A measure on $\mathbb{R}$ induces a measure on $\mathbb{R}^{d}$ via the map
$\gamma\colon\mathbb{R}\to\mathbb{R}^{d}$. For points
$p_{0}<p_{1}<\dots<p_{2^{k}}$ we construct a Borel measure
$\mu_{p_{0},\dots,p_{2^{k}}}$ on $\mathbb{R}^{d}$ induced by $\mu$ on
$\mathbb{R}$ as follows

  * •

$\mu([p_{i},p_{i+1}])=1$ for $i=0,\dots,2^{k}-1$ and $1$ is uniformly
distributed on this interval,

  * •

$\mu((-\infty,p_{0}])=0$,

  * •

$\mu([p_{2^{k}},\infty))=0$.

The measure $\mu_{p_{0},\dots,p_{2^{k}}}$ can only be equiparted into $2^{k}$
pieces by $k$ hyperplanes, if at least $2^{k}-1$ points in
$\gamma((p_{0},p_{2^{k}}))$ are contained in one of the hyperplanes
$H_{1},\dots,H_{k}$. More than $2^{k}-1$ points are needed, unless the
hyperplanes contain exactly the points
$\gamma(p_{1}),\dots,\gamma(p_{2^{k}-1})$. If we encode the intervals
$[p_{0},p_{1}],\dots,[p_{2^{k}-1},p_{2^{k}}]$ with vectors in $\\{0,1\\}^{k}$
according to the region, an equipartition of $\mu_{p_{0},\dots,p_{2^{k}}}$ by
$2^{k}-1$ points corresponds exactly to a $k$-bit Gray code.

Overall we choose points

$\ell_{1}\quad=\quad p_{1,0}<\dots<p_{1,2^{k}}\quad=\quad
p_{2,0}<\dots<p_{2,2^{k}}\quad=\quad\dots\quad=\quad
p_{j,0}<\dots<p_{j,2^{k}}$

and obtain measures

$\mathcal{M}=\left(\mu_{p_{1,0},\dots,p_{1,2^{k}}},\,\dots,\,\mu_{p_{j,0},\dots,p_{j,2^{k}}}\right).$

An equipartition of all measures simultanously can only be obtained with at
least $j(2^{k}-1)$ intersection points of the $H_{1},\dots,H_{k}$ with
$\gamma((p_{1,0},p_{j,2^{k}}))$. The equipartitions with exactly $j(2^{k}-1)$
intersection points are encoded by equiparting matrices.

According to Lemma 3.6, for an arrangement $(H_{1},\dots,H_{k})$ in the
relative closure of $\theta$, the hyperplane $H_{i}$ contains the points
$q_{1},\dots,q_{\ell}$ on $\gamma(\mathbb{R})$ for each $i=1,\dots,k$. Each
hyperplane can have at most $d$ intersection points with $\gamma(\mathbb{R})$
and

$dk=\ell_{1}+\dots+\ell_{k}+j(2^{k}-1).$

So, hyperplane arrangements in the relative closure of $\theta$ have at most
$j(2^{k}-1)$ intersection points with $\gamma((p_{1,0},p_{j,2^{k}}))$. We
conclude that arrangments in the
$(\mathfrak{S}_{k})_{(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})}^{\pm}$-orbit
of the relative closure of $\theta$ that equipart $\mathcal{M}$ into $2^{k}$
pieces are in one-to-one correspondence with
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices.

The number of non-equivalent
$(\mathfrak{c}_{1},\dots,\mathfrak{c}_{k})$-equiparting matrices is odd by
assumption. If each of those arrangements has a representative in the
relatively open cell $\theta$ and not just in the relative closure, we are
done, as $\mathfrak{S}_{k}^{\pm}$ acts free on the $\theta$-orbit.

Clearly, for $i=1,\dots,k$ no hyperplane $H_{i}$ in such an arrangement may
contain the point $q_{\ell_{i}+1}$ and we conclude $v_{\ell_{i}+1,i}\neq 0$.
According to Lemma 3.8, we can choose the points $p_{1,0}<\dots<p_{j,2^{k}}$
such that $\ell_{i}=\ell_{i+1}$ for any $i\in\\{1,\dots,k-1\\}$ implies
$v_{\ell_{i}+1,i}\neq v_{\ell_{i+1}+1,i+1}$ and then indeed the arrangement
lies in the relatively open cell $\theta$ by Lemma 3.6. ∎

######  Lemma 3.8.

Let $d>\ell_{1}>0$ and $n\geq 2(d-\ell_{1})$ be integers. There exist
$\ell_{1}<\pi_{1}<\dots<\pi_{n}$ in $\mathbb{R}$ such that for any two
oriented hyperplanes $H_{v},H_{w}$ parametrized by $v,w\in S^{d}$ and
$\ell\leq\ell_{1}$ with the following properties

  1. (1)

$v_{i}=w_{i}=0$ for $i=1,\dots,\ell$, i.e. $H_{v},H_{w}$ contain
$q_{1},\dots,q_{\ell}$,

  2. (2)

$v_{\ell+1}\neq 0\neq w_{\ell+1}$, i.e. $H_{v},H_{w}$ do not contain
$q_{\ell+1}$,

  3. (3)

the sets

$H_{v}\cap\left\\{\gamma(\pi_{1}),\dots\gamma(\pi_{n})\right\\},\quad
H_{w}\cap\left\\{\gamma(\pi_{1}),\dots\gamma(\pi_{n})\right\\}$

are disjoint and both of cardinality $d-\ell$,

it holds that $v_{\ell+1}\neq w_{\ell+1}$.

###### Proof.

Let $\pi_{1},\dots,\pi_{n}$ be algebraically independent with
$\ell_{1}<\pi_{1}<\dots<\pi_{n}$. Suppose

$H_{v}\cap\left\\{\gamma(\pi_{1}),\dots\gamma(\pi_{n})\right\\}=\left\\{\gamma(\pi_{I_{1}}),\dots\gamma(\pi_{I_{d-\ell}})\right\\},\quad
H_{w}\cap\left\\{\gamma(\pi_{1}),\dots\gamma(\pi_{n})\right\\}=\left\\{\gamma(\pi_{J_{1}}),\dots\gamma(\pi_{J_{d-\ell}})\right\\}.$

This means that $\pi_{I_{1}},\dots,\pi_{I_{d-\ell}}$ are roots of the
polynomial

$1v_{1}+tv_{2}+\binom{t}{2}v_{3}+\dots+\binom{t}{d}v_{d+1}$

in $\mathbb{Q}(v_{1},\dots,v_{d+1})[t]$. As $v_{1}=\dots=v_{\ell}=0$ and
$v_{1}^{2}+\dots+v_{d+1}^{2}=1$, they are algebraic over
$\mathbb{Q}(v_{\ell+1},\dots,v_{d})$.

Likewise, $\pi_{J_{1}},\dots,\pi_{J_{d-\ell}}$ are algebraic over
$\mathbb{Q}(w_{\ell+1},\dots,w_{d})$. By assumptions the transcendence degree
of
$\mathbb{Q}(\pi_{I_{1}},\dots,\pi_{I_{d-\ell}},\pi_{J_{1}},\dots\pi_{J_{d-\ell}})$
is $2(d-\ell)$. This implies that $v_{\ell+1}$ and $w_{\ell+1}$ are
algebraically independent and in particular distinct. ∎

## Acknowledgements

I would like to thank Pavle Blagojević for proposing to study [3], for many
valuable discussions and for providing relevant references. I would further
thank Florian Frick for his help and questions that helped to shape this
paper.

## References

  * [1] Avis, D.: Non-partitionable point sets. Inform. Process. Letters 19, 125–129 (1984).
  * [2] Beyer, W., Zardecki, A.: The early history of the Ham Sandwich Theorem. Amer. Monthly 111, 58–61 (2004).
  * [3] Blagojević, P., Frick, F., Haase, A., Ziegler, G.: Hyperplane mass partitions via relative equivariant obstruction theory. Doc. Math. 21, 735–771 (2016).
  * [4] Hadwiger, H.: Simultane Vierteilung zweier Körper. Arch. Math. 17, 274–278 (1966).
  * [5] Knuth, D.: The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1. Addison-Wesley Professional, 2011.
  * [6] Mani-Levitska, P., Vrećica, S., Živaljević, R.: Topology and combinatorics of partitions of masses by hyperplanes. Adv. Math. 207, 266–296 (2006).
  * [7] Moll, V.: Numbers and functions: From a classical-experimental mathematician’s point of view. Student Mathematical Library, vol. 65, American Mathematical Society, 2012.
  * [8] Ramos, E.: Equipartition of mass distributions by hyperplanes. Discrete Comput. Geom. 15, 147–167 (1996).


$\Delta\\_{arm}=200$. Since with this approximation, the final linking length
should be equivalent to twice the chosen arm length, the equivalent linking
lengths would be $l_{f_{mean}}\sim 250$ kpc and $l_{f_{median}}\sim 200$ kpc,
respectively. The produced FoF halo-mass function (Fig. 10) shows they do not
match well with the expected halo-mass function of Sheth-Tormen 2001 nor
Tinker et al. 2010. The number of clusters identified by FoF was also found to
be considerably less (2347 and 6119, respectively) in comparison to MITRO
(i.e., 18821). This lead us to do a further iterative test that resulted in a
linking length of $l_{f}=450$ kpc, where FoF could roughly able to produce a
match with the halo-mass function of Tinker 2010 and Sheth-Tormen 2001. This
certainly indicates the real strength of the MITRO algorithm as MITRO produces
desired results without the need for an iterative trial. In addition, our
algorithm also provides a tentative over-density threshold for the identified
structure i.e., $\gtrsim\Delta_{arm}/2\sim 100$.

Figure 10:  Halo mass function of FoF galaxy clusters identified from SAG
catalogue. Different coloured dashed lines correspond to different linking
lengths as mentioned in the legend.

##  Appendix B Structures in SDSS selected wedge near Saraswati region

###  B.1 Choice of over-density search in SDSS sample

We carried out an iterative test (each step of 10% increment) to understand
the sensitivity of the $\Delta_{SDSS}$ parameter. We found, even if we
increase the value just by 10% (i.e., $\Delta_{SDSS}$ = 2.60) of the chosen
value (i.e., $\Delta_{SDSS}$ = 2.36), the Saraswati supercluster splits into
two parts. The wing-like structure on the right side of the supercluster gets
detached (see Fig. 11a). Likewise, when we increase the value by 60% of 2.36
($\Delta_{SDSS}$ 3.78), the structure splits into three parts (left wing,
centre part + lower tail, right-wing, see Fig. 11b). However, similar to at
$\Delta_{SDSS}=2.36$, the density threshold of identified two splitted parts
at $\Delta_{SDSS}=2.60$ are also not above the theoretical limit of
gravitationally bound structures. It is only at $\Delta_{SDSS}=3.78$ that all
three splitted structures become gravitationally bound.

Figure 11: Panel a: MITRO identified structure at the position of Saraswati supercluster splits into two parts with the choice of $\Delta_{SDSS}=2.60$. Part 2 (red) parts away from the main structure. Panel b: The same structure when searched at $\Delta_{SDSS}=3.78$, splits into three distinct parts. Table 3: List of 53 MITRO identified structures from JB17-Saraswati-region (‘LEGACY’, ‘BOSS’, and ‘SOUTHERN’ program of SDSS-III DR12) CID | Mass | $z$ | R.A. | dec. | Richness | $\frac{\overline{\rho}}{\rho_{c}}$ | $R_{max}$ | LLS | $\bar{l}$
---|---|---|---|---|---|---|---|---|---
| $(10^{15}M_{\odot})$ |  | (deg.) | (deg.) |  |  | (Mpc) | (Mpc) | (Mpc)
1* | 185.602 | 0.277 | 354.409 | 0.267 | 384 | 1.619 | 124.976 | 226.306 | 9.397
2 | 75.259 | 0.3 | 352.942 | 0.069 | 92 | 3.074 | 71.179 | 112.115 | 11.204
3 | 59.827 | 0.302 | 1.018 | -0.009 | 53 | 4.111 | 83.19 | 138.652 | 12.276
4 | 54.982 | 0.33 | 336.031 | -1.002 | 40 | 7.168 | 96.034 | 96.034 | 13.122
5 | 50.929 | 0.318 | 4.709 | -0.094 | 41 | 3.546 | 74.782 | 111.204 | 12.518
6 | 49.397 | 0.307 | 6.554 | 0.084 | 48 | 2.756 | 78.104 | 116.96 | 11.992
7 | 43.796 | 0.253 | 359.801 | 0.251 | 99 | 1.951 | 77.811 | 116.988 | 9.256
8 | 34.449 | 0.235 | 340.97 | -0.123 | 42 | 5.085 | 45.148 | 75.611 | 11.802
9 | 30.623 | 0.321 | 351.079 | -0.029 | 20 | 8.501 | 44.268 | 66.634 | 13.792
10 | 30.237 | 0.232 | 6.156 | -0.026 | 45 | 3.134 | 53.775 | 95.147 | 10.786
11 | 28.789 | 0.293 | 5.959 | 0.568 | 43 | 4.131 | 51.605 | 75.443 | 10.554
12 | 24.32 | 0.317 | 339.708 | 1.078 | 19 | 10.283 | 52.201 | 56.732 | 12.956
13 | 23.894 | 0.274 | 343.581 | 0.688 | 36 | 4.242 | 62.194 | 106.461 | 10.583
14 | 23.052 | 0.312 | 347.422 | 1.104 | 18 | 10.979 | 50.394 | 57.309 | 13.238
15 | 22.924 | 0.292 | 347.077 | -0.187 | 32 | 3.93 | 47.131 | 78.523 | 10.811
16 | 22.121 | 0.259 | 5.826 | 0.192 | 46 | 3.777 | 36.005 | 65.306 | 9.781
17 | 20.143 | 0.31 | 345.581 | 0.603 | 17 | 13.993 | 54.98 | 71.364 | 12.427
18 | 19.328 | 0.315 | 337.573 | 0.68 | 18 | 10.365 | 35.539 | 48.453 | 12.229
19 | 18.484 | 0.268 | 345.786 | -0.92 | 26 | 7.053 | 39.845 | 50.391 | 10.955
20 | 17.365 | 0.294 | 336.605 | 0.893 | 21 | 11.901 | 24.156 | 42.673 | 11.278
21 | 16.918 | 0.259 | 9.098 | 0.533 | 30 | 4.835 | 48.112 | 69.719 | 10.01
22 | 16.135 | 0.29 | 344.517 | 0.357 | 20 | 8.2 | 51.161 | 66.819 | 11.507
23 | 15.632 | 0.323 | 342.799 | 0.704 | 13 | 7.809 | 45.396 | 53.84 | 12.134
24 | 15.374 | 0.3 | 338.308 | 0.131 | 18 | 12.646 | 38.697 | 57.745 | 11.413
25 | 15.336 | 0.287 | 350.167 | -1.014 | 20 | 7.672 | 42.938 | 56.825 | 11.433
26 | 14.433 | 0.302 | 11.856 | 0.679 | 17 | 7.564 | 32.612 | 49.729 | 11.026
27 | 12.616 | 0.298 | 338.9 | -1.077 | 11 | 14.114 | 47.898 | 47.898 | 12.767
28 | 12.294 | 0.313 | 10.766 | 0.049 | 10 | 8.875 | 33.85 | 49.348 | 12.517
29 | 12.009 | 0.326 | 358.504 | -0.58 | 12 | 11.029 | 26.54 | 46.212 | 11.748
30 | 11.338 | 0.259 | 15.307 | -0.719 | 18 | 11.907 | 35.758 | 47.51 | 11.163
31 | 10.43 | 0.243 | 15.626 | -0.848 | 15 | 21.887 | 30.041 | 54.335 | 10.623
32 | 10.419 | 0.285 | 14.34 | 0.849 | 18 | 16.294 | 27.533 | 40.096 | 10.094
33 | 10.063 | 0.256 | 354.491 | 0.776 | 22 | 9.457 | 41.29 | 51.335 | 9.616
34 | 9.454 | 0.301 | 348.575 | 0.191 | 13 | 4.96 | 39.153 | 54.615 | 11.118
35 | 8.573 | 0.235 | 7.463 | 0.754 | 14 | 10.082 | 42.411 | 55.644 | 10.124
36 | 7.844 | 0.242 | 337.205 | -0.95 | 13 | 19.201 | 38.987 | 51.832 | 10.01
37 | 7.786 | 0.275 | 339.645 | 0.743 | 11 | 15.349 | 21.771 | 35.626 | 10.918
38 | 7.099 | 0.247 | 343.733 | -1.042 | 10 | 22.859 | 21.615 | 27.324 | 11.731
39 | 6.968 | 0.254 | 337.905 | 0.259 | 11 | 17.649 | 23.835 | 40.85 | 10.882
40 | 6.926 | 0.283 | 341.549 | 0.494 | 11 | 10.472 | 25.987 | 39.252 | 9.881
41 | 6.734 | 0.264 | 353.209 | -1.041 | 10 | 15.8 | 24.33 | 34.619 | 10.722
42 | 6.578 | 0.272 | 347.447 | 0.055 | 10 | 16.784 | 39.977 | 51.357 | 10.245
43 | 6.554 | 0.26 | 344.632 | -0.111 | 13 | 6.266 | 29.5 | 47.385 | 9.828
44 | 6.223 | 0.268 | 10.69 | 0.473 | 14 | 6.655 | 27.749 | 47.466 | 9.581
45 | 6.075 | 0.251 | 351.795 | -1.139 | 11 | 14.511 | 30.705 | 32.196 | 10.283
46 | 5.217 | 0.262 | 13.614 | 0.461 | 10 | 22.064 | 21.594 | 24.636 | 9.712
47 | 4.789 | 0.276 | 9.9 | -0.494 | 12 | 10.204 | 24.451 | 34.629 | 8.939
48 | 4.751 | 0.252 | 339.093 | -0.514 | 10 | 24.829 | 21.683 | 33.723 | 9.555
49 | 4.278 | 0.254 | 339.177 | -0.275 | 12 | 7.481 | 24.232 | 37.156 | 8.338
50 | 4.274 | 0.268 | 12.31 | -0.158 | 10 | 7.884 | 22.534 | 37.062 | 8.97
51 | 4.271 | 0.252 | 353.213 | 1.174 | 10 | 7.958 | 35.918 | 39.626 | 9.821
52 | 3.823 | 0.248 | 3.508 | 0.794 | 12 | 15.294 | 34.126 | 41.634 | 8.599
53 | 3.239 | 0.232 | 358.556 | -0.321 | 11 | 7.416 | 25.007 | 36.919 | 8.273
Table 4: WHL clusters in MITRO identified galaxy clusters and superclusters CID | WHL clusters | $z$ | R.A. (deg.) | dec. (deg.) | CID | WHL clusters | $z$ | R.A. (deg.) | dec. (deg.)
---|---|---|---|---|---|---|---|---|---
1* | WHL J232229.7-002223 | 0.2728 | 350.62363 | -0.37294 |  | WHL J002648.8-001610 | 0.2311 | 6.70325 | -0.26937
| WHL J232406.1+002147 | 0.278 | 351.02548 | +0.36311 |  | WHL J003130.2-010303 | 0.232 | 7.87587 | -1.05083
| WHL J232517.4+001639 | 0.2767 | 351.32236 | +0.27741 |  |  |  |  |
| WHL J232643.3+010803 | 0.2774 | 351.68036 | +1.13423 | 11 | WHL J002400.3+003007 | 0.2923 | 6.00129 | +0.50202
| WHL J232731.9+005634 | 0.2788 | 351.88278 | +0.94281 |  | WHL J002813.8-002154 | 0.2912 | 7.05762 | -0.36497
| WHL J232809.2+001109 | 0.2768 | 352.03851 | +0.18593 |  |  |  |  |
| WHL J232842.6+004939 | 0.2784 | 352.17755 | +0.82753 | 12 | WHL J223818.0+005559 | 0.3173 | 339.57483 | +0.9331
| WHL J233040.2+003634 | 0.2757 | 352.66733 | +0.6094 |  |  |  |  |
| WHL J233116.7+004336 | 0.2812 | 352.81964 | +0.72678 | 13 | WHL J224807.9+002040 | 0.2657 | 342.03287 | +0.34439
| WHL J233126.4+003655 | 0.2739 | 352.8602 | +0.61525 |  | WHL J225424.4+004101 | 0.2738 | 343.60159 | +0.68368
| WHL J233430.3-005906 | 0.2772 | 353.62631 | -0.98501 |  | WHL J225729.0+003102 | 0.273 | 344.37103 | +0.51717
| WHL J233530.9-003152 | 0.2764 | 353.87866 | -0.53116 |  |  |  |  |
| WHL J233558.5-002904 | 0.27 | 353.99393 | -0.48434 | 16 | WHL J002314.8+001158 | 0.2597 | 5.81154 | +0.19944
| WHL J233739.7+001617 | 0.2772 | 354.41553 | +0.27137 |  | WHL J002358.9-003039 | 0.2575 | 5.99551 | -0.51081
| WHL J233816.0+000700 | 0.2711 | 354.56683 | +0.1168 |  | WHL J002427.4+002450 | 0.2598 | 6.11418 | +0.41389
| WHL J233955.0-002558 | 0.2769 | 354.97913 | -0.43282 |  |  |  |  |
| WHL J234024.5-000535 | 0.2761 | 355.10202 | -0.09299 | 17 | WHL J230408.4-010642 | 0.3147 | 346.03494 | -1.11163
| WHL J234106.9+001833 | 0.2768 | 355.27872 | +0.30925 |  |  |  |  |
| WHL J234144.1-004031 | 0.268 | 355.43375 | -0.67539 | 19 | WHL J230227.7-002516 | 0.2647 | 345.61533 | -0.42125
| WHL J234233.1-001719 | 0.277 | 355.63776 | -0.28874 |  | WHL J230316.7-005114 | 0.2685 | 345.81976 | -0.85396
| WHL J234335.7+001951 | 0.2694 | 355.89862 | +0.33093 |  | WHL J230348.8-004024 | 0.2662 | 345.95328 | -0.67331
| WHL J234403.1+001335 | 0.2714 | 356.01273 | +0.22644 |  |  |  |  |
| WHL J234446.7-000520 | 0.2674 | 356.19455 | -0.0888 | 20 | WHL J222426.4+002737 | 0.2938 | 336.1102 | +0.46019
| WHL J234519.2-000312 | 0.2665 | 356.3298 | -0.05331 |  | WHL J222542.8+010339 | 0.2952 | 336.42816 | +1.06081
| WHL J234548.2-010740 | 0.2797 | 356.45068 | -1.12771 |  | WHL J222627.3+005329 | 0.294 | 336.61365 | +0.89143
| WHL J234604.7-001109 | 0.2665 | 356.5195 | -0.18573 |  |  |  |  |
| WHL J234623.9+004458 | 0.2746 | 356.59955 | +0.74942 | 21 | WHL J003200.8+001353 | 0.2532 | 8.00329 | +0.23131
| WHL J234643.2+005003 | 0.2637 | 356.67987 | +0.83429 |  | WHL J003511.8+004348 | 0.262 | 8.79937 | +0.73002
| WHL J234703.0-000508 | 0.2655 | 356.76236 | -0.08544 |  | WHL J003554.5+000924 | 0.2597 | 8.97729 | +0.15671
| WHL J234727.0+003140 | 0.2729 | 356.86264 | +0.52773 |  | WHL J003555.1+002123 | 0.2592 | 8.97972 | +0.35646
| WHL J234727.6-000914 | 0.2639 | 356.86499 | -0.15381 |  | WHL J003614.8+000940 | 0.2569 | 9.06173 | +0.16113
| WHL J234856.4-005328 | 0.2793 | 357.2352 | -0.8912 |  | WHL J003619.8+002418 | 0.2576 | 9.08261 | +0.40494
| WHL J234946.6+003846 | 0.2705 | 357.44406 | +0.64598 |  | WHL J003629.8+003520 | 0.2607 | 9.1241 | +0.58899
| WHL J235016.5+005301 | 0.2767 | 357.56891 | +0.88373 |  | WHL J003711.8+000546 | 0.2588 | 9.29924 | +0.09603
| WHL J235121.3+003701 | 0.2775 | 357.83884 | +0.61691 |  |  |  |  |
| WHL J235143.9+003339 | 0.2703 | 357.93292 | +0.56078 | 22 | WHL J225211.0+003531 | 0.2877 | 343.04565 | +0.59185
| WHL J235229.5+003622 | 0.2698 | 358.12292 | +0.60623 |  |  |  |  |
| WHL J235301.3-001351 | 0.2779 | 358.25555 | -0.23089 | 24 | WHL J223304.8+000818 | 0.2989 | 338.2699 | +0.13823
| WHL J235347.5+004859 | 0.2787 | 358.448 | +0.81641 |  |  |  |  |
| WHL J235424.8+004838 | 0.2805 | 358.60349 | +0.81066 | 25 | WHL J231948.3-010932 | 0.2909 | 349.95108 | -1.15889
| WHL J235508.9+004512 | 0.2789 | 358.7869 | +0.75341 |  |  |  |  |
| WHL J235543.8-000014 | 0.2752 | 358.93271 | -0.00393 | 28 | WHL J004051.3+000858 | 0.3132 | 10.2137 | +0.14932
2 | WHL J232420.2-003229 | 0.293 | 351.08423 | -0.54147 | 29 | WHL J235424.3-001451 | 0.3277 | 358.60135 | -0.24737
| WHL J232902.7-003847 | 0.2988 | 352.26129 | -0.6463 |  |  |  |  |
| WHL J232923.7-004856 | 0.3007 | 352.34872 | -0.81546 | 30 | WHL J010154.7-005216 | 0.2616 | 15.47781 | -0.87118
| WHL J233317.2+000741 | 0.3025 | 353.32187 | +0.12797 |  | WHL J010214.5-010851 | 0.2611 | 15.56023 | -1.14737
| WHL J233531.2-005320 | 0.2974 | 353.87994 | -0.88882 |  | WHL J010331.1-005932 | 0.2565 | 15.87962 | -0.99216
|  |  |  |  |  | WHL J010339.2-004508 | 0.2646 | 15.91314 | -0.75235
3 | WHL J000201.8+002049 | 0.3007 | 0.50762 | +0.34701 |  |  |  |  |
| WHL J000224.7-003253 | 0.2895 | 0.60304 | -0.54798 | 31 | WHL J010229.8-004848 | 0.2375 | 15.62412 | -0.81343
| WHL J000242.3-001320 | 0.2989 | 0.67632 | -0.22217 |  |  |  |  |
| WHL J000455.1+003415 | 0.3197 | 1.22949 | +0.57073 | 32 | WHL J005338.1+005515 | 0.2851 | 13.40886 | +0.92088
| WHL J000825.6+004152 | 0.3124 | 2.10683 | +0.69786 |  | WHL J005434.9+004819 | 0.2847 | 13.6454 | +0.80528
|  |  |  |  |  | WHL J005526.0+004731 | 0.2846 | 13.85833 | +0.79205
4 | WHL J222456.1-002115 | 0.3154 | 336.23373 | -0.35415 |  |  |  |  |
| WHL J222523.4-005410 | 0.3204 | 336.3476 | -0.90289 | 33 | WHL J234145.8+010728 | 0.2507 | 355.44077 | +1.12444
| WHL J222731.7-003945 | 0.3226 | 336.88223 | -0.66258 |  |  |  |  |
| WHL J222810.8-004848 | 0.327 | 337.04517 | -0.8134 | 34 | WHL J231308.1-010632 | 0.3007 | 348.28375 | -1.10895
5 | WHL J001848.6-000430 | 0.3184 | 4.70257 | -0.07488 | 35 | WHL J003407.0-001147 | 0.2365 | 8.52936 | -0.19632
| WHL J002020.7-001925 | 0.3251 | 5.08638 | -0.32355 |  |  |  |  |
| WHL J002025.3-001216 | 0.3267 | 5.10561 | -0.20444 | 36 | WHL J222652.2-005607 | 0.2483 | 336.71762 | -0.93536
| WHL J002105.0+001406 | 0.3291 | 5.27082 | +0.23505 |  | WHL J222720.8-005610 | 0.2468 | 336.83655 | -0.93623
6 | WHL J002200.7-001839 | 0.304 | 5.50288 | -0.31075 | 39 | WHL J223126.2+001743 | 0.2547 | 337.85898 | +0.29532
| WHL J002334.8+001907 | 0.3084 | 5.89512 | +0.31861 |  |  |  |  |
| WHL J002557.8+000734 | 0.3076 | 6.49081 | +0.12619 | 40 | WHL J224641.8+002837 | 0.2846 | 341.67407 | +0.47684
| WHL J002920.4-004032 | 0.3085 | 7.3349 | -0.67557 |  |  |  |  |
| WHL J003401.5-003314 | 0.3076 | 8.50633 | -0.55392 | 41 | WHL J233608.8-011100 | 0.265 | 354.03653 | -1.18347
| WHL J003508.9-010803 | 0.3055 | 8.78714 | -1.13429 |  |  |  |  |
| WHL J003542.0+000158 | 0.3102 | 8.92503 | +0.03278 | 43 | WHL J225830.5-000608 | 0.2587 | 344.62704 | -0.1021
| WHL J003744.4-011424 | 0.3106 | 9.43509 | -1.23996 |  |  |  |  |
|  |  |  |  | 44 | WHL J004249.9+001255 | 0.2691 | 10.70773 | +0.21535
7 | WHL J235751.3+002133 | 0.2596 | 359.46381 | +0.3593 |  | WHL J004252.7+004306 | 0.2699 | 10.71962 | +0.71844
| WHL J235914.3+001405 | 0.2533 | 359.8096 | +0.23484 |  |  |  |  |
| WHL J235952.8+004155 | 0.2674 | 359.96988 | +0.69856 | 45 | WHL J232708.4-011250 | 0.2517 | 351.7851 | -1.21393
| WHL J000024.3+003804 | 0.2522 | 0.1013 | +0.63434 |  |  |  |  |
| WHL J000050.5+004705 | 0.2634 | 0.21052 | +0.78478 | 48 | WHL J223556.1-001411 | 0.2488 | 338.98395 | -0.2364
| WHL J000117.8-005612 | 0.2473 | 0.32417 | -0.93668 |  | WHL J223622.8-003049 | 0.2488 | 339.09488 | -0.51352
| WHL J000126.3-000143 | 0.2479 | 0.35969 | -0.02867 |  |  |  |  |
|  |  |  |  | 49 | WHL J223629.1+000118 | 0.2539 | 339.12143 | +0.02159
8 | WHL J224249.1-001648 | 0.2363 | 340.70477 | -0.2801 |  |  |  |  |
10 | WHL J001637.4-002739 | 0.2325 | 4.15592 | -0.46084 | 52 | WHL J001229.8+003911 | 0.2505 | 3.12436 | +0.65318
| WHL J002435.2-001307 | 0.2308 | 6.14679 | -0.21867 |  | WHL J001357.8+004706 | 0.2465 | 3.49083 | +0.78508