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\begin{center}
\vskip 1cm{\Large\bf 
Counting Self-Dual Monotone Boolean Functions
}
\vskip 1cm
\large

Bart\l{}omiej Pawelski and Andrzej Szepietowski\\
Institute of Informatics\\
University of Gda\'nsk\\
Poland\\
\href{mailto:bartlomiej.pawelski@ug.edu.pl}{\tt bartlomiej.pawelski@ug.edu.pl} \\
\href{mailto:andrzej.szepietowski@ug.edu.pl}{\tt andrzej.szepietowski@ug.edu.pl} 
\end{center}

\vskip .2in

\begin{abstract}
Let $D_n$ denote the set of monotone Boolean functions with $n$ variables. The cardinality of $D_n$, denoted by $d_n$, is known as the $n$-th Dedekind number. Elements of $D_n$ can be represented as strings of bits of length $2^n$. 
For each $f\in D_n$, we have the {\it dual} function $f^*\in D_n $ which is obtained by reversing and negating all bits.
An element $f\in D_n$ is {\it self-dual} if $f=f^*$.
Let $\Lambda_n$ denote the set of all self-dual monotone Boolean functions of $n$ variables
and let $\lambda_n$ denote $|\Lambda_n|$. There is a natural action of the permutation group $S_n$ on the set of Boolean functions by permutation of variables. The sets $D_n$ and $\Lambda_{n}$ are closed under this action. We let $R_n$ and $Q_n$ denote the sets of all equivalence classes in $D_n$ and $\Lambda_n$, respectively.
In this paper, we derive several algorithms for counting self-dual monotone Boolean functions and confirm the known result that $\lambda_9$ equals 423,295,099,074,735,261,880. Furthermore, we calculate $|Q_8|$ to be 6,001,501.
\end{abstract}
\section{Introduction}\label{sec:intro}
Let $B$ denote the set $\{0,1\}$ and $B^n$ the set of $n$-element sequences of $B$. A Boolean function with $n$ variables is a function from $B^n$ into $B$. There is the order relation in $B$ (namely: $0\le 0$, $\;0\le1$, $\;1\le1$) and the partial order in $B^n$: for any two elements: $x=(x_1,\dots,x_n)$, $\;y=(y_1,\dots,y_n)$ in $B^n$, $x\le y$ if and only if $x_i\le y_i$ for all $1\le i\le n$. A function $h:B^n\to B$ is monotone if $x\le y$ implies $h(x)\le h(y)$. Let $D_n$ denote the set of monotone functions with $n$ variables. The cardinality of $D_n$, denoted by $d_n$, is known as the $n$-th Dedekind number. We have the partial order in $D_n$ defined by
$$g\le h\quad\hbox{if and only if}\quad g(x)\le h(x)\quad
\hbox{for all } x\in B^n.$$
We represent an element $f \in D_n$ as a string of bits of length $2^n$ where the values of $f(x)$ are listed in lexicographical order of $x$. The two elements of $D_0$ are represented as 0 and 1. For every $n \geq 1$, the set $D_n$ can be represented as the set of all concatenations $g_0\cdot g_1$, where $g_0, g_1\in D_{n-1}$ and $g_0\le g_1$. For example, $D_1=\{00, 01, 11\}$ and $D_2=\{0000, 0001, 0011, 0101, 0111, 1111\}$. 

For each $f\in D_n$, we have the {\it dual} $f^*\in D_n$, which is obtained by reversing and negating all bits. Formally, the dual of $f \in D_n$ is the function $f^* : B^n \to B$ defined by
$$f^*(x) = (f(x^c))^c,$$
where $x^c$ is the negation of $x \in B^n$ and $(f(x^c))^c$ is the negation of $f(x^c)\in B$. For example, $1111^*=0000$ and $0001^*=0111$. An element $f\in D_n$ is {\it self-dual} if $f=f^*$. For example, $0101$ and $0011$ are self-dual in $D_2$.

Let $\Lambda_n$ be the set of all self-dual monotone Boolean functions of $n$ variables, and let $\lambda_n$ denote the cardinality of this set. The value $\lambda_n$ is also known as the $n$-th Hosten-Morris number (\seqnum{A001206} in the \textit{On-Line Encyclopedia of Integer Sequences}).

The first attempt to determine the values of $\lambda_n$ was made in 1968 by Riviere \cite{riviere}, who found all values up to $\lambda_5$. In 1972, Brouwer and Verbeek provided the values up to $\lambda_7$ \cite{brouwer7}. The value of $\lambda_8$ was determined by Mills and Mills \cite{mills} in 1978.

The most recent known term, $\lambda_9$, was obtained by Brouwer, Mills, Mills, and Verbeek \cite{brouwer} in 2013. The value of $\lambda_n$ corresponds to the number of non-dominated coteries on $n$ members \cite[Section 1]{bioch}, and also corresponds to the number of maximal linked systems (see Section~\ref{mls} and \cite[Section 1]{brouwer}).

Any two Boolean functions are said to be \textit{equivalent} if one can be transformed into the other by a permutation of the input variables (see Section \ref{perm}). Let $Q_n$ denote the set of all equivalence classes in $\Lambda_n$ and let $q_n$ denote $|Q_n|$. The values of $q_n$ are described by the \seqnum{A008840} OEIS sequence.

In this paper, we derive several algorithms for counting self-dual monotone Boolean functions and we confirm the result of \cite{brouwer} that $\lambda_9$ equals 423,295,099,074,735,261,880. Furthermore, employing Burnside's lemma and techniques discussed in \cite{pawelski, szepietowski}, we calculate $q_8$ to be 6,001,501, which is the first-ever calculation of the value $q_8$. We also show that the two sequences $d_n$ and $\lambda_n$ intertwine, or more precisely, we have $$\lambda_1 \le d_0\le\lambda_2\le d_1\le  \cdots \le\lambda_{n+1} \le d_n\le\lambda_{n+2}\le  \cdots$$ and we have the following estimates for the unknown value of $\lambda_{10}$ $$d_8\le\lambda_{10} \le d_9.$$
\begin{table}[H]
\begin{center}
\begin{tabular}[h]{|l|l|l|}
\hline
 $n$ & $\lambda_n$ & $q_n$\\ 
\hline
0 & 0 & 0\\
1 & 1 & 1\\
2 & 2 & 1\\
3 & 4 & 2\\
4 & 12 & 3\\
5 & 81 & 7\\
6 & 2,646 & 30\\
7 & 1,422,564 & 716 \\
8 & 229,809,982,112 & 6,001,501 \\
9 & 423,295,099,074,735,261,880 & -- \\
\hline
\end{tabular}
\end{center}
\caption{Known values of $\lambda_n$ (\seqnum{A001206}) and $q_n$ (\seqnum{A008840}).}
\label{tab:SD}
\end{table}
\section{Preliminaries}
A {\it poset (partially ordered set)} $(S,\le)$ consists of a set $S$ (called the carrier) together with a binary relation (partial order) $\le$ which is reflexive, transitive, and antisymmetric.
For example, $B$, $B^n$, and $D_n$ are posets. 
Given two posets $(S,\le)$ and $(T,\le)$, a function $f:S \to T$ is {\it monotone}, if $x \le y$ implies $f(x)\le f(y)$.
We let $T^S$ denote the poset of all monotone functions from $S$ to $T$ with the partial order defined by
$$f\le g\quad\hbox{ if and only if}\quad f(x)\le g(x)\hbox{ for all }x\in S.$$
Notice that $D_n=B^{B^n}$ and $B^n=B^{A_n}$, where $A_n$ is the antichain with the carrier $\{1,\ldots,n\}$.
A set $A$ is an antichain if any two distinct elements in $A$ are incomparable.
In this paper we use the following well-known lemma \cite{szepietowski}:
\begin{lem}\label{L1} The poset $D_{n+k}$ is isomorphic to the poset $D_n^{B^k}$---the poset of monotone functions from $B^k$ to $D_n$.
\end{lem}
By $\top$ we denote the maximal element in $D_n$, that is $\top=(1\ldots 1)$, and by $\bot$ the minimal element in $D_n$, that is $\bot=(0\ldots 0)$. For two elements $f,g\in D_n$, by $f|g$ we denote the \textit{bitwise or}; and by $f\&g$ the \textit{bitwise and}. Furthermore, let $\re(f,g)$ denote the number of elements in the interval $[f,g]$, that is $|\{h\in D_n : f\le h\le g\}|$. Note that $\re(f,\top)=|\{h\in D_n : f\le h\}|$ and $\re(\bot,g)=|\{h\in D_n : h\le g\}|$.
For $f\in D_n$, by $\ell(f)$ we denote the number of ones in $f$, also known as its Hamming weight. For example,
$\ell(0000)=0$ and $\ell(0101)=2$. The next result is straightforward.
\begin{lem}\label{L0}
For each $f,g\in D_n$, we have
\begin{enumerate}
\item $f^{**}=f$
\item if $f\le g$ then $g^*\le f^*$
\item $(f|g)^*=f^*\&g^*$
\item $(f\&g)^*=f^*|g^*$
\end{enumerate}
\end{lem}
\subsection{Maximal linked system}\label{mls}
Let $X=\{1,\dots,n\}$ and ${\cal P}(X)$ be the power set of $X$. A family ${\cal W}\subseteq {\cal P}(X)$ is {\it linked} if for 
 all $A$ and $B$ in ${\cal W}$, $A\cap B$ is not empty.
 A family ${\cal U}\subseteq {\cal P}(X)$ is a {\it maximal linked system (mls)} on $X$ if ${\cal U}$ is linked and for all ${\cal W}$ with ${\cal U}\subseteq{\cal W}\subseteq{\cal P}(X)$, either ${\cal W}={\cal U}$ or ${\cal W}$ is not linked.
 
 If ${\cal U}$ is linked, then $\emptyset\notin {\cal U}$
 and for each set $A\in {\cal P}(X)$,
 it is not possible that both $A$ and its complement $A^c=X-A$ belong to ${\cal U}$.
 
 For $n=0$, $X=\emptyset$, ${\cal P}(X)=\{\emptyset\}$ and we have one mls; namely, the empty family.
 Notice that the set of self-dual functions $\Lambda_0$ is empty and $\lambda_0=0$.
\begin{lem}
If $n\ge 1$ and a family ${\cal U}\subset{\cal P}(X)$ is a mls, then
\begin{itemize}
\item[(L1)] ${\cal U}$ is an upset, i.e., if $A\subseteq B$ and $A\in {\cal U}$, then $B\in {\cal U}$.
\item[(L2)] For every subset $A\in{\cal P}(X)$ exactly one of the two subsets $A$ or $A^c$ is in ${\cal U}$.
\end{itemize}
\end{lem}
 \begin{proof} (L1) For each $C\in {\cal U}$, $A\cap C\ne \emptyset$ and $B\cap C\ne \emptyset$. Hence, either $B\in {\cal U}$ or ${\cal U}$ is not maximal.
 
 (L2) We have two cases:
 
 Case 1. For every $B\in {\cal U}$, $A\cap B\ne \emptyset$. Then either $A\in {\cal U}$ or ${\cal U}$
 is not maximal.
 
 Case 2. There is $B\in{\cal U}$ such that $B\cap A=\emptyset$. In this case we have that $B \subseteq A^c$ and, by (L1), $A^c \in {\cal U}$.
  \end{proof}
 \begin{lem}
 If $n\ge 1$ and a family ${\cal U}\subseteq {\cal P}(X)$ satisfies conditions (L1) and (L2), then ${\cal U}$ is an mls. 
 \end{lem}
 \begin{proof} First, we prove that ${\cal U}$ is linked.
 Suppose, for a contradiction, that there are two subsets
 $A,B\in {\cal U}$ with $A\cap B=\emptyset$.
 Then $B\subseteq A^c$ and $A^c\in {\cal U}$, a contradiction.
 
 If ${\cal U}$ satisfies condition (L2), then exactly half of the elements in ${\cal P}(X)$ belong to ${\cal U}$. On the other hand, if $|{\cal U}| > 2^{n-1}$ then ${\cal U}$ contains a pair $A, A^c$ and ${\cal U}$ is not linked.
 \end{proof}
 For $n=1$, the family $\{\{1\}\}$ is the only mls on $X=\{1\}$.
 
 For $n=2$, we have two mls: $\{\{1\},\{1,2\}\}$ and $\{\{2\},\{1,2\}\}$. 
 
 Recall that ${\cal P}(X)$ is isomorphic to $B^n$ by identifying a subset of $X$ with its characteristic vector and any subset of ${\cal P}(X)$ can be represented by a function from $B^n$ to $B$. For $n=1$, the mls $\{\{1\}\}$ can be represented as the string $01$, which is the only self-dual function in $\Lambda_1$.
 For $n=2$, the two mls can be represented as $0101$ and $0011$, and they form the set $\Lambda_2$. For $n\ge 1$, the set of mls on $\{1,\dots,n\}$ can be represented as the set of self-dual functions
 $\Lambda_n$  \cite[Section 1.1]{brouwer}.
\subsection{Permutations and equivalence relation}\label{perm}
Let $S_n$ denote the set of permutations on $\{1,\ldots,n\}$.
Every permutation $\pi\in S_n$ defines a permutation on $B^n$ 
as follows:
for every $x = (x_1, \ldots, x_n) \in B^n$ let $\pi(x) = (x_{\pi(1)}, \ldots, x_{\pi(n)})$.

The permutation $\pi$ also defines the function
$f \circ \pi:B^n\to B$ for every function $f:B^n\to B$ by
$$(f \circ \pi)(x) = f(\pi(x))\; \text{ for }\; x \in B^n.$$
By $\sim$ we denote an equivalence relation on $D_n$. Namely,
two functions $f, g\in D_n$ are {\it equivalent}, $f\sim g$, if there is a
permutation $\pi\in S_n$ such that $f=g \circ \pi$.
For a function $f\in D_n$, its {\it equivalence class}
is the set $[f]=\{g\in D_n : g \sim f\}$. By $R_n$ we denote the set of equivalence classes in $D_n$ and by $r_n$ we denote $|R_n|$. By $\gamma(f)$ we denote $|[f]|$.
For the class $[f]$, its {\it representative} is its minimal element
(according to the lexicographical order in $D_n$). Sometimes, we identify the class $[f]\in R_n$ with
its representative and treat $[f]$ as an element in $D_n$.
Let $Q_n$ denote the set of all equivalence classes in $\Lambda_n$, and let $q_n$ denote $|Q_n|$.

The following lemma is straightforward.
\begin{lem}\label{L2} For every function $f:B^n\to B$ and for every permutation $\pi\in S_n$
\begin{enumerate}
\item The function $f \circ \pi$ is monotone if $f$ is monotone.
\item If $f \in D_n$, then $f^* \circ \pi = (f \circ \pi)^*.$
\item If $f \in \Lambda_n$, then for every equivalent $g\in[f]$ we have $g \in \Lambda_n$.
\end{enumerate}
\end{lem}
For $n=2$, we have $d_2=6$ monotone functions, $$D_2=\{0000,0001,0011,0101,0111,1111\}$$ and $r_2=5$
equivalence classes in $D_2$; namely
	$$R_2=\{\{0000\},\{0001\},\{0011,0101\},\{0111\},\{1111\}\}.$$ Furthermore, there are two self-dual functions
	$\Lambda_2=\{0011,0101\}$ and they form one equivalence class. Hence $\lambda_2=2$ and $q_2=1$.

For $n=3$, we have $d_3=20$ monotone functions and $r_3=10$
equivalence classes in $D_3$. There are four self-dual functions
	$$\Lambda_3=\{01010101,00110011, 00001111,00010111\}$$
	 and they form two equivalence classes:
$$Q_3=\{\{01010101,00110011, 00001111\},\{00010111\}\}.$$
Hence, $\lambda_3=4$ and $q_3=2$.
\section{Counting functions from \texorpdfstring{$B$}{B} to \texorpdfstring{$D_n$}{D(n)}}
Let $n\ge 0$.
By Lemma~\ref{L1}, the poset $D_{n+1}$ is isomorphic to the poset $D_n^{B}$---the poset of monotone functions from $B=\{0,1\}$ to $D_n$. 
Consider a monotone function $H:B\to D_n$. It can be represented as the concatenation
$$H=H(0)\cdot H(1)$$
with $H(0),H(1)\in D_n$ and $H(0)\le H(1)$.
The dual of $H$ is
$$H^*=H(1)^*\cdot H(0)^*.$$
Recall that we identify each function in $D_n$ with the sequence of bits.
If $H\in D_{n+1}$ is self-dual then it is of the form $b\cdot b^*$ with $b\in D_n$ and $b\le b^*$.
And vice versa, if $b\in D_n$ and $b\le b^*$, then the concatenation $b\cdot b^*$ is self-dual in $D_{n+1}$.
Therefore, we have proved the following theorem.
\begin{theorem}\label{L3} 
For every $n\ge 0$, the number of self-dual functions
$\lambda_{n+1}$ is equal to the number of elements $b\in D_n$ that satisfy $b\le b^*$. In other words
$$\lambda_{n+1}=\sum_{\substack{b\in D_n\\b\le b^*}}1.$$
Furthermore, 
$$\lambda_{n+1}=\sum_{\substack{b\in R_n\\b\le b^*}}\gamma(b).$$
Here we identify each class $b\in R_n$ with its representative.
\end{theorem}
The following corollary is presented, in a different form, in \cite{timo}
\begin{corollary}\label{c1}
For every $n\ge 0$, we have $\lambda_{n+1} \le d_{n}$.
\end{corollary}
The next result is straightforward.
\begin{lem}\label{Lem8} Let $b$ be a function in $D_n$. Then
\begin{itemize}
\item $b\le b^*$, only if $\ell(b)\le 2^{n-1}$.
\item if $\ell(b)=2^{n-1}$ and $b\le b^*$, then $b=b^*$, and $b$ is self-dual in $D_n$.
\item if $b$ is self-dual, then $\ell(b)=2^{n-1}$.
\item $\ell(b^*)=2^n-\ell(b)$, hence,
 $$|\{f\in D_n:\ell(f)<2^{n-1}\}|=|\{f\in D_n:\ell(f)>2^{n-1}\}|.$$
\end{itemize}
\end{lem}
As a corollary we have the following theorem.
\begin{theorem}
$$\lambda_{n+1}=\lambda_n+\sum_{\substack{b\in R_n\\\ell(b)<2^{n-1}\\b\le b^*}}\gamma(b).$$
$$\lambda_{n+1}\le \lambda_n+\frac{1}{2}(d_n-\lambda_n)=\frac{1}{2}(d_n+\lambda_n).$$
\end{theorem}
The second part of the theorem is presented, in a different form, in \cite{timo}. Notice that 
$\lambda_{n+1}=\tfrac{1}{2}(d_n+\lambda_n)$ for each $n\le 3$.
\section{Counting functions from \texorpdfstring{$B^2$}{B2} to \texorpdfstring{$D_n$}{D(n)}}\label{sec:s4}
Let $n\ge 0$.
By Lemma~\ref{L1}, the poset $D_{n+2}$ is isomorphic to the poset $D_n^{B^2}$---the poset of monotone functions from $B^2=\{00,01,10,11\}$ to $D_n$. 
Consider a monotone function $H:B^2\to D_n$. It can be represented as the concatenation
$$H(00)\cdot H(01)\cdot H(10) \cdot H(11)$$
and as the graph below.
\def\drawconnectionsa{\draw (0,0) -- (-2,2);
 \draw (0,0) -- (2,2);
 \draw (-2,2) -- (0,4);
 \draw (2,2) -- (0,4);}
\def\drawconnectionsb{\draw (0,0) -- (-2,2);
 \draw (0,0) -- (0,2);
 \draw (0,0) -- (2,2);
 \draw (-2,2) -- (-2,4);
 \draw (-2,2) -- (0,4);
 \draw (0,2) -- (-2,4);
 \draw (0,2) -- (2,4);
 \draw (2,2) -- (0,4);
 \draw (2,2) -- (2,4);
 \draw (-2,4) -- (0,6);
 \draw (0,4) -- (0,6);
 \draw (2,4) -- (0,6);}
\begin{center}
\begin{tikzpicture}[scale=0.6]
 \drawconnectionsa{}
 \draw[black, fill=white] (0,0) circle (.1cm) node[right] {$H(00)$};
 \draw[black, fill=white] (-2,2) circle (.1cm) node[left] {$H(01)$};
 \draw[black, fill=white] (2,2) circle (.1cm) node[right] {$H(10)$};
 \draw[black, fill=white] (0,4) circle (.1cm) node[right] {$H(11)$};
 \end{tikzpicture}
\end{center}
where the vertices represent the set of values $H(B^2)$ and the edges represent the partial order: $$H(00) \leq H(01) \leq H(11) \text{ and } H(00) \leq H(10) \leq H(11).$$
Recall that we identify each function in $D_n$ with the sequence of bits.
The dual of $H$ is represented by
$$H(11)^*\cdot H(10)^*\cdot H(01)^* \cdot H(00 )^*.$$
If $H$ is self-dual then $H(00)=H(11)^*$, $H(01)=H(10)^*$, $H(10)=H(01)^*$, and $H(11)=H(00)^*$ and we can represent $H$ as the graph
\begin{center}
\begin{tikzpicture}[scale=0.7]
 \drawconnectionsa{}
 \draw[black, fill=white] (0,0) circle (.1cm) node[right] {$d^*$};
 \draw[black, fill=white] (-2,2) circle (.1cm) node[left] {$b$};
 \draw[black, fill=white] (2,2) circle (.1cm) node[right] {$b^*$};
 \draw[black, fill=white] (0,4) circle (.1cm) node[right] {$d$};
 \end{tikzpicture}
\end{center}
where $d$ denotes $H(11)$ and $b$ denotes $H(01)$. Furthermore, $d^*$ denotes $H(00)$ and $b^*$ denotes $H(10)$.
If $H$ is self-dual then it is of the form $d^*\cdot b\cdot b^*\cdot d$ with $b,d\in D_n$ and $d\ge b|b^*$.
And vice versa, if $b,d\in D_n$, and $d\ge b|b^*$,
then, by Lemma \ref{L0}, $d^* \le b \& b^*$ and the concatenation 
$d^*\cdot b\cdot b^*\cdot d$
is self-dual in $D_{n+2}$. Hence, we have proved the following theorem.
\begin{theorem}\label{L31} For every $n\ge0$, the number of self-dual functions
$\lambda_{n+2}$ is equal to the number of pairs $b,d\in D_n$ which satisfy condition $d\ge b|b^*$. In other words
$$\lambda_{n+2}=\sum_{b\in D_n}\re(b|b^*,\top).$$
Furthermore, Lemma~\ref{L2} implies
$$\lambda_{n+2}=\sum_{b\in R_n}\gamma(b)\cdot \re(b|b^*,\top).$$
Here we identify each class $b\in R_n$ with its representative.
\end{theorem}
\begin{corollary}\label{c2}
For every $n\ge 0$, we have $\lambda_{n+2} \ge d_{n}$.
\end{corollary}
By Colloraries \ref{c1} and \ref{c2}, we have that $$\lambda_1 \le d_0\le\lambda_2\le d_1\le  \cdots \le\lambda_{n+1} \le d_n\le\lambda_{n+2}\le  \cdots$$ and we have the following estimates for the unknown value of $\lambda_{10}$ $$d_8\le\lambda_{10} \le d_9.$$
\section{Counting functions from \texorpdfstring{$B^4$}{B4} to \texorpdfstring{$D_n$}{D(n)}}
Let $n\ge 0$.
By Lemma~\ref{L1}, the poset $D_{n+4}$ is isomorphic to the poset $D_n^{B^4}$---the set of monotone functions from $B^4=\{0000, 0001, \ldots, 1111\}$ to $D_n$.
Consider a monotone function $H \colon B^4 \to D_n$. It can be represented as the concatenation
\begin{align*}
&H(0000) \cdot H(0001) \cdot H(0010) \cdot H(0011) \cdot H(0100) \cdot H(0101) \cdot H(0110) \cdot H(0111) \cdot {}\\
&H(1000) \cdot H(1001) \cdot H(1010) \cdot H(1011) \cdot H(1100) \cdot H(1101) \cdot H(1110) \cdot H(1111).
\end{align*}
and its dual as
\begin{align*}
&H(1111)^* \cdot H(1110)^* \cdot H(1101)^* \cdot H(1100)^* \cdot H(1011)^* \cdot H(1010)^* \cdot H(1001)^* \cdot H(1000)^* \cdot {}\\
&H(0111)^* \cdot H(0110)^* \cdot H(0101)^* \cdot H(0100)^* \cdot H(0011)^* \cdot H(0010)^* \cdot H(0001)^* \cdot H(0000)^*.
\end{align*}
If $H$ is self-dual, then
\begin{align*}
H(0000) &= H(1111)^*, & H(0001) &= H(1110)^*, \\
H(0010) &= H(1101)^*, & H(0011) &= H(1100)^*, \\
H(0100) &= H(1011)^*, & H(0101) &= H(1010)^*, \\
H(0110) &= H(1001)^*, & H(0111) &= H(1000)^*.
\end{align*}
Recall that we identify each function in $D_n$ with the sequence of bits. Similarly, as in Section \ref{sec:s4} we can represent $H$ as a graph. See Figure \ref{fig3} where $h$ represents $H(1111)$, $d$ represents $H(0111)$, $e$ represents $H(1011)$, $f$ represents $H(1101)$, $g$ represents $H(1110)$, \,$\ldots$ \,, and $h^*$ represents $H(0000)$.


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 \draw (#1-2,#2+2) -- (#1,#2+4);
 \draw (#1,#2+2) -- (#1-2,#2+4);
 \draw (#1,#2+2) -- (#1+2,#2+4);
 \draw (#1+2,#2+2) -- (#1,#2+4);
 \draw (#1+2,#2+2) -- (#1+2,#2+4);
 \draw (#1-2,#2+4) -- (#1,#2+6);
 \draw (#1,#2+4) -- (#1,#2+6);
 \draw (#1+2,#2+4) -- (#1,#2+6);}
\def\drawconnectionsd{
\draw (0,0) -- (5,2);
\draw (-2,2) -- (3,4);
\draw (0,2) -- (5,4);
\draw (2,2) -- (7,4);
\draw (-2,4) -- (3,6);
\draw (0,4) -- (5,6);
\draw (2,4) -- (7,6);
\draw (0,6) -- (5,8);}
\begin{center}
\begin{tikzpicture}[scale=1.1]
\drawconnections{0}{0}{0}
\drawconnectionsd
\draw[black, fill=white] (0,0) circle (.1cm) node[right] {$h^*$};
\draw[black, fill=white] (-2,2) circle (.1cm) node[left] {$g^*$};
\draw[black, fill=white] (0,2) circle (.1cm) node[left] {$f^*$};
\draw[black, fill=white] (2,2) circle (.1cm) node[left] {$e^*$};
\draw[black, fill=white] (-2,4) circle (.1cm) node[left] {$a$};
\draw[black, fill=white] (0,4) circle (.1cm) node[left] {$b$};
\draw[black, fill=white] (2,4) circle (.1cm) node[left] {$c$};
\draw[black, fill=white] (0,6) circle (.1cm) node[right] {$d$};
\drawconnections{5}{2}{0}
\draw[black, fill=white] (5,2) circle (.1cm) node[right] {$d^*$};
\draw[black, fill=white] (3,4) circle (.1cm) node[right] {$c^*$};
\draw[black, fill=white] (5,4) circle (.1cm) node[right] {$b^*$};
\draw[black, fill=white] (7,4) circle (.1cm) node[right] {$a^*$};
\draw[black, fill=white] (3,6) circle (.1cm) node[left] {$e$};
\draw[black, fill=white] (5,6) circle (.1cm) node[right] {$f$};
\draw[black, fill=white] (7,6) circle (.1cm) node[right] {$g$};
\draw[black, fill=white] (5,8) circle (.1cm) node[right] {$h$};
\end{tikzpicture}
\captionof{figure}{Structure of $H:B^4\to D_n$ if $H$ is self-dual. The graph represents the hypercube $B^4$.
The label at a vertex $x$ represents the value $H(x)\in D_n$.}\label{fig3}
\end{center}

\begin{theorem}\label{L41}
For each $a,b,c\in D_n$,

for each $h\in D_n$ such that $h\geq a|b|c|a^*|b^*|c^*$,

for each $d,e,f,g\in D_n$ such that 
\begin{align*}
a|b|c &\leq d\leq h, \\
a|b^*|c^* &\leq e\leq h, \\
b|a^*|c^* &\leq f\leq h, \\
c|a^*|b^* &\leq g\leq h,
\end{align*}
the concatenation
\begin{equation*}
h^*\cdot g^*\cdot f^*\cdot a\cdot e^*\cdot b\cdot c\cdot d\cdot d^*\cdot c^*\cdot b^*\cdot e\cdot a^*\cdot f\cdot g\cdot h
\end{equation*}
represents a self-dual function in $D_{n+4}$; see Figure~\ref{fig3}. And vice versa, each self-dual function in $D_{n+4}$ is of the above form.
\end{theorem}
The proof is straightforward.
\begin{theorem}\label{L51} The number of self-dual functions
$$\lambda_{n+4}=\sum_{a,b,c\in D_n}\sum_{\substack{h\in D_n \\ h \ge (a|b|c|a^*|b^*|c^*)}} \re(a|b|c,h)\cdot \re(a|b^*|c^*,h)\cdot \re(b|a^*|c^*,h)\cdot \re(c|a^*|b^*,h).$$
\end{theorem}
\begin{theorem}\label{L42}
For each $h\in D_n$ such that $h\geq h^*$,

for each $a,b,c\in D_n$, $h^*\leq a,b,c \leq h$,

for each $d,e,f,g\in D_n$ such that 
\begin{align*}
a|b|c &\leq d\leq h, \\
a|b^*|c^* &\leq e\leq h, \\
b|a^*|c^* &\leq f\leq h, \\
c|a^*|b^* &\leq g\leq h,
\end{align*}
the concatenation
\begin{equation*}
h^*\cdot g^*\cdot f^*\cdot a\cdot e^*\cdot b\cdot c\cdot d\cdot d^*\cdot c^*\cdot b^*\cdot e\cdot a^*\cdot f\cdot g\cdot h
\end{equation*}
represents a self-dual function in $D_{n+4}$; see Figure~\ref{fig3}. And vice versa, each self-dual function in $D_{n+4}$ is of the above form.
\end{theorem}
Let $$F(h)= \sum_{\substack{a, b, c \in D_n \\ h \ge a,b,c \ge h^*}} \re(a|b|c,h)\cdot \re(a|b^*|c^*,h)\cdot \re(b|a^*|c^*,h)\cdot \re(c|a^*|b^*,h).$$
Observe that $F(h)$ is the number of self-dual functions $H\in D_n^{B^4}$, with $H(0000)=h^*$ and $H(1111)=h$. Therefore,
$$\lambda_{n+4}=\sum_{\substack{h\in D_n\\h^*\le h}}F(h).$$
Furthermore, Lemma~\ref{L2} implies that for any two elements $h_1\sim h_2$ we have $F(h_1)=F(h_2)$.
Hence, we have
$$\lambda_{n+4}=\sum_{\substack {h\in R_n\\h\ge h^*}}\gamma(h)\cdot F(h).$$
Here again we identify the class $h\in R_n$ with its representative.
Similarly as in Lemma~\ref{Lem8} we can observe that $h\ge h^*$, only if $\ell(h)\ge 2^{n-1}$.
Furthermore, if $\ell(h)=2^{n-1}$ and $h\ge h^*$, then $h=h^*$, and
we have only one self-dual function
$H\in D_n^{B^4}$ with $H(0000)=H(1111)=h$. Hence,
\begin{equation}\label{alg3}
\lambda_{n+4}=\lambda_n+\sum_{\substack {h\in R_n\\h^*\le h\\\ell(h)> 2^{n-1}}}\gamma(h)\cdot F(h). 
\end{equation}
\section{Implementation}
In this section we present three algorithms based on results from the previous sections. We implemented the algorithms in Rust and ran them on a 32-thread Xeon CPU.
\begin{algorithm}[H]
\caption{Calculation of $\lambda_{n+2}$}
\label{alg:a1}
 \vspace{1mm}
 \hspace*{\algorithmicindent} \textbf{Input:} $R_n$ (each class is represented by its minimal element) with $re(x, \top)$ for all $x \in R_n$ \\
 \hspace*{\algorithmicindent} \textbf{Output:} $s = \lambda_{n+2}$
\begin{algorithmic}[1]
\State Initialize $s = 0$,
\ForAll {$b \in R_n $}
\State $s = s + \re(b|b^*,\top) \cdot \gamma(b)$
\EndFor
\end{algorithmic}
\end{algorithm}
Algorithm 1 is based on Theorem \ref{L31}. After loading the preprocessed data into main memory, $\lambda_9$ was computed in 15 seconds. However, preprocessing (the calculation of $R_7$ and its intervals) took approximately 2.5 hours.
\begin{algorithm}[H]
\caption{Calculation of $\lambda_{n+4}$ based on Theorem \ref{L51}}
 \vspace{1mm}
 \hspace*{\algorithmicindent} {\textbf{Input:} $D_n$; $R_n$ (each class is represented by its minimal element); $re(x, y)$ for all $(x, y) \in D_n \times D_n$} \\
 \hspace*{\algorithmicindent} \textbf{Output:} $s = \lambda_{n+4}$
\begin{algorithmic}[1]
\State Initialize $s = 0$,
\ForAll {$a \in R_n $}
\ForAll {$b \in D_n $}
\ForAll {$c \in D_n $}
\ForAll {$h \in D_n, h \ge (a | b | c | a^* | b^* | c^*)$}
\State $s = s + \re(a|b|c,h)\cdot \re(a|b^*|c^*,h)\cdot \re(b|a^*|c^*,h)\cdot \re(c|a^*|b^*,h) \cdot \gamma(a)$
\EndFor
\EndFor
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}
Using our implementation of the algorithm, we calculated $\lambda_9$ in 76 seconds, and the preprocessing was almost instantaneous.
The calculation of $\lambda_9$ using our implementation of the algorithm lasted approximately 25 minutes. In all cases, we have obtained the following value:
$$ \lambda_9 = 423295099074735261880,$$
which confirms the result of Brouwer et al.~\cite{brouwer}.

\begin{algorithm}[H]
\caption{Calculation of $\lambda_{n+4}$ based on Equation \ref{alg3}}
 \vspace{1mm}
 \hspace*{\algorithmicindent} {\textbf{Input:} $D_n$; $R_n$ (each class is represented by its minimal element); $\re(x, y)$ for all $(x, y) \in D_n \times D_n$} \\
 \hspace*{\algorithmicindent} \textbf{Output:} $s = \lambda_{n+4}$
\begin{algorithmic}[1]
\State Initialize $s = \lambda_n$,
\ForAll {$h \in R_n, h^*\le h, \ell(h)> 2^{n-1} $}
\ForAll {$a \in D_n, h^*\le a \le h$}
\ForAll {$b \in D_n, h^*\le b \le h$}
\ForAll {$c \in D_n, h^*\le c \le h$}
\State $s = s + \re(a|b|c,h)\cdot \re(a|b^*|c^*,h)\cdot \re(b|a^*|c^*,h)\cdot \re(c|a^*|b^*,h) \cdot \gamma(h)$
\EndFor
\EndFor
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}

Algorithms 2 and 3 differ in the order in which values of the functions are chosen. It should be noted that Algorithm 2 works much faster than Algorithm 3 due to a significantly smaller number of iterations, requiring only 76 seconds compared to 25 minutes of computation time.
\section{Calculation of \texorpdfstring{$q_n$}{q(n)}}
The number of inequivalent self-dual monotone Boolean functions ($q_n$) is listed on the OEIS \seqnum{A008840} sequence. In order to calculate $q_n$ for $n\le 7$, we can use the following simple approach:
$$q_{n}=\sum_{\substack{a\in R_n \\ a = a^*}} 1.$$
For $n=8$, this direct approach becomes computationally infeasible. Instead, we apply Burnside's lemma to our specific problem.
Recall that each permutation $\pi \in S_n$ can be represented as a product of disjoint cycles. The {\it cycle type} of $\pi$ is defined as the tuple of lengths of its disjoint cycles arranged in increasing order. For example, the type of permutation $\pi =(12)(34)(567)$ is $(2,2,3)$, and its total length is 7. In the sequel, we use a list of cycle types in $S_n$ and we represent each cycle type by its index in the list, see the tables in Section \ref{sec:72}.

To calculate $q_8$ we use the approach developed in \cite{pawelski, pawelski24, szepietowski}. 
\begin{equation}
q_n = \frac{1}{n!} \sum_{i=1}^{k} \mu_i\cdot |\Phi(\pi_i, \Lambda_n)|
\end{equation}
where
\begin{itemize}
 \item $q_n$ is the number of equivalence classes in $\Lambda_n$,
 \item $k$ is the number of different cycle types in $S_n$,
 \item $\pi_i$ is a representative permutation of cycle type $i$,
 \item $\Phi(\pi_i, \Lambda_n)$ is the set of all elements in $\Lambda_n$ which are fixed under $\pi_i$,
 \item $\mu_i$ number of permutations $\pi \in S_n$ with cycle type $i$.
\end{itemize}
For $n=1$, we have $\lambda_1=q_1=1$. For $n=2$, we have two permutations:
the identity $e$ with $|\Phi(e,\Lambda_2)|=|\Lambda_2|=2$,
and the inversion $\pi = (12)$ with three cycles when acting on $B^2$, namely: 
$C_1=(00)$, $C_2=(01,10)$, and $C_3=(11)$.
The two elements $01$ and $10$ form
a cycle, hence, if a function $f\in D_n$ is a fixed point of $\pi$, then $f(01)=f(10)$.
On the other hand, if $f$ is self-dual, then it represents an mls on $\{1,2\}$, and $f(01)\ne f(10)$, because $01$ and $10$ represent subsets $\{1\}$ and $\{2\}$ in $\{1,2\}$ which are the complements of each other.
Thus, the set of fixed points $\Phi((12),\Lambda_2)=\emptyset$.
By Burnside's lemma, we have
$$q_2=\frac{1}{2}(|\Phi(e,\Lambda_2)|+|\Phi((12),\Lambda_2)|)=\frac{1}{2}(2+0)=1.$$
Indeed, there is one equivalence class in $\Lambda_2$; namely, $\{0101,0011\}$.
We have just shown that $\Phi((12),\Lambda_2)=\emptyset$. Similarly, we can show that
$\Phi(\pi,\Lambda_8)=\emptyset$, if $\pi=(12)(34)(56)(78).$
Indeed, for $\pi$, the two elements $01010101, \; 10101010\in B^8$ form a cycle.
Hence, if $f\in D_8$ is a fixed point of $\pi$, then $f(01010101)=f(10101010)$. On the other side, if $f$ is self-dual, then $f(01010101)\ne f(10101010)$, because $101010101$ and $01010101$ represent subsets of $\{1,\dots,8\}$ which are the complements of each other.
\begin{lem} 
\label{even}
Suppose that $n$ is even, and a permutation $\pi$, when acting on $\{1,\dots,n\}$, is a product of disjoint cycles of even length. Then $\Phi(\pi,\Lambda_n)=\emptyset$. 
\end{lem}
\begin{proof} If the permutation $\pi$ is a product of disjoint cycles of even length, then there exist two elements
$x, y\in B^n$ such that
\begin{itemize}
\item $x$ and $y$ represent subsets of $\{1,\dots,n\}$ which are the complements of each other.
\item $\pi(x)=y$ and $\pi(y)=x$, so $x,y$ form a cycle in $B^n$.
\end{itemize}
 Hence, if $f\in D_n$ is a fixed point of $\pi$, then
 $f(x)=f(y)$. On the other side, if $f$ is self-dual, then
 $f(x)\ne f(y)$. 
 \end{proof}
\begin{corollary}
We have $\Phi(\pi,\Lambda_8)=\emptyset$ for each of the following permutations:
$(12345678)$, $(12)(345678)$, $(1234)(5678)$, $(12)(34)(5678)$, and $(12)(34)(56)(78)$.
\end{corollary}
For $n=3$, we have three cycle types:
\begin{itemize}
\item the identity $e$ with $|\Phi(e,\Lambda_3)|=|\Lambda_3|=4$;
\item three inversions, with $|\Phi((12),\Lambda_3)|=2$; and
\item two cycles of length 3 with $|\Phi((123),\Lambda_3)|=1$.
\end{itemize}
By Burnside's lemma, we have
$$q_3=\frac{1}{6}(4+3\cdot 2+2\cdot 1)=2.$$
Notice, that the element
$\maj=00010111\in D_3$ 
is self-dual, and is a fixed point for every permutation $\pi\in S_3$. Hence, for every $\pi\in S_3$, 
$\Phi(\pi,\Lambda_3)\ne\emptyset$.
Similarly, we can show the following lemma:
\begin{lem}
For each odd $n$ and each permutation $\pi\in S_n$, we have $\Phi(\pi,\Lambda_n)\ne\emptyset$. 
\end{lem}
\begin{proof}
Consider the function $\maj\in D_n$, which returns $\maj(x)=1$
if and only if $\ell(x)>n/2$. The function $\maj$ is self-dual, and is a fixed point for every permutation $\pi\in S_n$. 
\end{proof}
\subsection{Algorithms counting fixed points in \texorpdfstring{$\Lambda_n$}{Lambda(n)}}
In order to count or generate fixed points of permutations in $\Lambda_n$ we use
Lemma~\ref{even} and two algorithms.
\begin{algorithm}[H]
\caption{Generation of $\Phi(\pi, \Lambda_n)$}
\label{alg:a4}
 \vspace{1mm}
 \hspace*{\algorithmicindent} \textbf{Input:} $\Phi(\pi, D_n)$ \\
 \hspace*{\algorithmicindent} \textbf{Output:} $S = \Phi(\pi, \Lambda_n)$
\begin{algorithmic}[1]
\State Initialize $S=\emptyset$
\ForAll {$b \in \Phi(\pi, D_n)$}
\If {$b = b^*$}
\State Add $b$ to $S$
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
Algorithm \ref{alg:a4} simply runs through the set of fixed points $\Phi(\pi, D_n)$ and selects self-dual functions.
For example, there are five fixed points in
$$\Phi((123),D_3)=\{00000000,00000001,00010111,01111111,11111111\}$$
and only one of them is self-dual; namely, $00010111$,
so $|\Phi((123), \Lambda_3)|=1$.
\begin{algorithm}[H]
\caption{Calculation of $|\Phi(\pi, \Lambda_{n+2})|$}
\label{alg:a5}
 \vspace{1mm}
 \hspace*{\algorithmicindent} \textbf{Input:} $\Phi(\pi, D_n)$ \\
 \hspace*{\algorithmicindent} \textbf{Output:} $s = |\Phi(\pi, \Lambda_{n+2})|$
\begin{algorithmic}[1]
\State Initialize $s = 0$,
\ForAll {$b \in \Phi(\pi, D_n)$}
\State Calculate $up = |\{h \in \Phi(\pi, D_n): h \ge (b | b^*)\}|$
\State $s = s + up$
\EndFor
\end{algorithmic}
\end{algorithm}
Algorithm \ref{alg:a5} is based on the following facts.
Consider a permutation $\pi$ acting on $B^n$ and on $D_n$. We can say that $\pi$ also acts on 
$B^{n+2}$ and on $D_{n+2}$.
By \cite[Lemma 6]{szepietowski}, $\Phi(\pi,D_{n+2})=\Phi(\pi,D_n)^{B^2}$.
Every function $F\in \Phi(\pi,D_n)^{B^2}$ can be represented as the concatenation 
$$F=F(00)\cdot F(01)\cdot F(10)\cdot F(11),$$
where 
$F(00)$, $F(01)$, $F(10)$, $F(11)\in \Phi(\pi, D_n)$,
and
$$F(00)\le F(01), F(10)\le F(11).$$
The dual of $F$ can be represented as
$$F^*=F(11)^*\cdot F(10)^*\cdot F(01)^*\cdot F(00)^*.$$
If $F$ is self-dual, then it is of the form
$$d b b^* d^*,$$
where $b,d\in \Phi(\pi,D_n)$ and $d^*\ge b|b^*$. Notice that this implies that
$d\le b\&b^*$. On the other hand, if $b,d\in \Phi(\pi, D_n)$ and $d^*\ge b|b^*$, then
$dbb^*d^*\in \Phi(\pi, \Lambda_{n+2})$.
\subsection{Result tables}\label{sec:72}
In this section we present three tables which contain the values of $|\Phi(\pi_i, \Lambda_n)|$, for $n \in \{6,7,8\}$ and all permutations.
\begin{table}[htb]
\centering
\label{tab:lambda6}
 {\renewcommand{\arraystretch}{0.9}
 \begin{tabular}{r*{3}{r}r}
 $i$  & $\pi_i$ & $\mu_i$ & $|\Phi(\pi_i, \Lambda_6)|$ \\
 \hline
 1 & (1)  & 1 & 2646 \\
 2 & (12)  & 15 & 372 \\
 3 & (123)  & 40 & 54 \\
 4 & (1234)  & 90 & 18 \\
 5 & (12345)  & 144 & 6 \\
 6 & (123456)  & 120 & 0 \\
 7 & (12)(34)  & 45 & 130 \\
 8 & (12)(345) & 120 & 18 \\
 9 & (12)(3456) & 90 & 0 \\
 10 & (123)(456) & 40 & 18 \\
 11 & (12)(34)(56) & 15 & 0 \\
 \hline
 \end{tabular}}
 \[ q_6 = \frac{1}{720} \sum_{i=1}^{11} \mu_i \cdot |\Phi(\pi_i, \Lambda_6)| = \frac{21600}{720} = 30. \]
\vspace{-2mm}
\caption{Values of $|\Phi(\pi_i, \Lambda_6)|$ and calculation of $q_6$.}
\end{table}
\begin{table}[H]
\begin{center}
% {\renewcommand{\arraystretch}{0.9}
\scalebox{0.7}{
 \begin{tabular}{r*{3}{r}r}
 $i$  & $\pi_i$ & $\mu_i$ & $|\Phi(\pi_i, \Lambda_7)|$ \\
 \hline
 1 & (1)  & 1 & 1422564 \\
 2 & (12)  & 21 & 43556 \\
 3 & (123)  & 70 & 1332 \\
 4 & (1234)  & 210 & 216 \\
 5 & (12345)  & 504 & 34 \\
 6 & (123456)  & 840 & 12 \\
 7 & (1234567) & 720 & 3 \\
 8 & (12)(34)  & 105 & 7212 \\
 9 & (12)(345) & 420 & 218 \\
 10 & (12)(3456) & 630 & 76 \\
 11 & (12)(34567) & 504 & 6 \\
 12 & (123)(456) & 280 & 210 \\
 13 & (123)(4567) & 420 & 6 \\
 14 & (12)(34)(56) & 105 & 1284 \\
 15 & (12)(34)(567) & 210 & 36 \\
 \hline
 \end{tabular}
}
 %}
 \[ q_7 = \frac{1}{5040} \sum_{i=1}^{15} \mu_i \cdot |\Phi(\pi_i, \Lambda_7)| = \frac{3608640}{5040} = 716. \]
\end{center}
\label{tab:lambda7}
\caption{Values of $|\Phi(\pi_i, \Lambda_7)|$ and calculation of $q_7$.}
\end{table}
\begin{table}[H]
\begin{center}
 %{\renewcommand{\arraystretch}{0.8}
\scalebox{0.7}{
 \begin{tabular}{r*{3}{r}r}
 $i$  & $\pi_i$ & $\mu_i$ & $|\Phi(\pi_i, \Lambda_8)|$ \\
 \hline
 1 & (1)  & 1 & 229809982112 \\
 2 & (12)  & 28 & 300991356 \\
 3 & (123)  & 112 & 476120 \\
 4 & (1234)  & 420 & 18984 \\
 5 & (12345)  & 1344 & 662 \\
 6 & (123456)  & 3360 & 296 \\
 7 & (1234567) & 5760 & 46 \\
 8 & (12345678) & 5040 & 0 \\
 9 & (12)(34)  & 210 & 12716048 \\
 10 & (12)(345) & 1120 & 18384 \\
 11 & (12)(3456) & 2520 & 7952 \\
 12 & (12)(34567) & 4032 & 116 \\
 13 & (12)(345678) & 3360 & 0 \\
 14 & (123)(456) & 1120 & 21020 \\
 15 & (123)(4567) & 3360 & 120 \\
 16 & (123)(45678) & 2688 & 20 \\
 17 & (1234)(5678) & 1260 & 0 \\
 18 & (12)(34)(56) & 420 & 2230724 \\
 19 & (12)(34)(567) & 1680 & 3152 \\
 20 & (12)(34)(5678) & 1260 & 0 \\
 21 & (12)(345)(678) & 1120 & 1488 \\
 22 & (12)(34)(56)(78) & 105 & 0 \\
 \hline
 \end{tabular}
 %}
}
 \[ q_8 = \frac{1}{40320} \sum_{i=1}^{22} \mu_i \cdot |\Phi(\pi_i, \Lambda_8)| = \frac{241980137280}{40320} = 6001501. \]
\end{center}
\label{tab:lambda8}
\caption{Values of $|\Phi(\pi_i, \Lambda_8)|$ and calculation of $q_8$.}
 \end{table}
\begin{thebibliography}{99}
\bibitem{bioch}
J. C. Bioch and T. Ibaraki, Generating and approximating nondominated coteries, \textit{IEEE Trans. Parallel Distrib. Syst.} \textbf{6} (1995), 905--914.

\bibitem{brouwer}
A. E. Brouwer, C. F. Mills, W. H. Mills, and A. Verbeek, Counting families of mutually intersecting sets, \textit{Electron. J. Combin.} \textbf{20} (2013), Article P8.

\bibitem{brouwer7}
A. E. Brouwer and A. Verbeek, Counting families of mutually intersecting sets, Report ZN 41, Mathematical Centre, Amsterdam, 1972.

\bibitem{loeb}
D. E. Loeb and A. R. Conway, Voting fairly: transitive maximal intersecting families of sets, \textit{J. Combin. Theory Ser. A} \textbf{91} (2000), 386--410.

\bibitem{mills}
C. F. Mills and W. H. Mills, The calculation of $\lambda(8)$, preprint, 1979.

\bibitem{riviere}
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\bibitem{oeis}
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\bibitem{pawelski}
B. Pawelski, On the number of inequivalent monotone Boolean functions of 8 variables, \textit{J. Integer Sequences} \textbf{25} (2022), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL25/Pawelski/pawelski7.html}{Article 22.7.7}.

\bibitem{pawelski24}
B. Pawelski, On the number of inequivalent monotone Boolean functions of 9 variables, \textit{IEEE Trans. Inform. Theory} \textbf{70} (2024), 5358--5364.

\bibitem{szepietowski}
A. Szepietowski, Fixes of permutations acting on monotone Boolean functions, \textit{J. Integer Sequences} \textbf{25} (2022), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL25/Szepietowski/szep7.html}{Article 22.9.6}.

\bibitem{timo}
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\end{thebibliography}
 
 \bigskip
 \hrule
 \bigskip
 
 \noindent 
 2020 \emph{Mathematics Subject Classification}:~Primary 06E30.
 
 \medskip
 
 \noindent 
 \emph{Keywords}:~
 monotone Boolean function,
 self-dual monotone Boolean function,
 Dedekind number,
 Hosten-Morris number.
 
 \bigskip
 \hrule
 \bigskip
 
\noindent (Concerned with sequences \seqnum{A001206} and \seqnum{A008840}.)
 
\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received February 26 2024;
revised versions received  December 11 2024; June 12 2025; October 22 2025.
Published in {\it Journal of Integer Sequences}, October 31 2025.

\bigskip
\hrule
\bigskip

\noindent
Return to \href{https://cs.uwaterloo.ca/journals/JIS/}{Journal of Integer Sequences home page}.
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