\documentclass[12pt,reqno]{article}
\usepackage
[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}
\usepackage{fullpage}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}
\usepackage{epsf}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\topmargin}{-.2in}
\setlength{\textheight}{8.0in}
\newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}}
\begin{document}
\makeatletter
\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}
\begin{center}
\vskip 1cm {\LARGE\bf Sequences realized as Parker vectors of oligomorphic permutation
groups}
\vskip 1cm
\large Daniele A. Gewurz and Francesca Merola\\
\vskip .5cm
Dipartimento di Matematica\\
Universit\`a di Roma ``La Sapienza''\\
Piazzale Aldo Moro, 2 -- 00185 Roma \\
Italy\\
\href{mailto:gewurz@mat.uniroma1.it}{\tt gewurz@mat.uniroma1.it}\\
\href{mailto:merola@mat.uniroma1.it}{\tt merola@mat.uniroma1.it}\\
\end{center}
\date{}
\pagestyle{myheadings}
\newcommand{\no}{\noindent}
\newcommand{\pf}[1]{\no{\bf Proof.} #1\diam\medbreak}
\newcommand{\ndiv}{/\hskip-2.52mm\mid}
\newcommand{\bin}[2]{\left(\hskip-2mm \begin{array}{c} #1\\#2
\end{array} \hskip-2mm\right)}
\newcommand{\qbin}[2]{\left[\hskip-2mm \begin{array}{c} #1\\#2
\end{array} \hskip-2mm\right]_q}
\newcommand{\etalchar}[1]{$^{#1}$}
\newcommand{\comment}[1]{}
\newcommand{\diam}{\hfill$\diamondsuit$}
\newcommand{\G}{$GL(n,q)$}
\newcommand{\lcm}{{\rm lcm}}
\newtheorem{tm}{Theorem}[section]
\newtheorem{pr}[tm]{Proposition}
\newtheorem{lm}[tm]{Lemma}
\newtheorem{cl}[tm]{Corollary}
\newtheorem{cj}[tm]{Conjecture}
%\usepackage{eufrak}
%\usepackage{times}
%\usepackage{mathptm}
\hoffset-1truecm
\textwidth 14cm
\thispagestyle{empty}
\null
\addtolength{\textheight}{2cm}
\begin{abstract}
The purpose of this paper is to study the Parker vectors (in fact,
sequences) of several known classes of oligomorphic groups. The Parker
sequence of a group $G$ is the sequence that counts the number of
$G$-orbits on cycles appearing in elements of $G$. This work was inspired
by Cameron's paper on the sequences realized by counting orbits on
$k$-sets and $k$-tuples.
\end{abstract}
\bigskip
\section{Introduction}
In a recent paper \cite{Cam2000}, P. J. Cameron describes several
``classical" sequences (in the sense of appearing in the
\emph{Encyclopedia of Integer Sequences} \cite{njas}) obtainable as
U- or L-sequences of oligomorphic groups, that is as sequences of
numbers counting the orbits of such groups on $k$-subsets and on
ordered $k$-tuples, respectively.
Oligomorphic permutation groups \cite{Cam90} constitute a class of
infinite groups to which it is meaningful to extend the concept of Parker
vector, originally defined for finite groups (see
\vfill\eject
\noindent\cite{GeMe01}).
So it is natural to study which integer sequences are obtained as Parker
sequence, that is, by counting orbits on $k$-cycles.
Recall that the \emph{Parker sequence}, or \emph{Parker vector}, of an
oligomorphic permutation group $G$ is the sequence ${\mathbf p}(G) =
(p_1,p_2,p_3,\dots)$, where $p_k$ is the number of orbits of $G$ on the
set of $k$-cycles appearing in elements of $G$, with $G$ acting by
conjugation. For instance, for the symmetric group $S$ acting on
a countable set, the Parker sequence is just $(1,1,1,\dots)$. A less
trivial example is the group $C$ preserving a circular order on a
countable set; for the Parker sequence one has $p_k=\varphi(k)$
Let us fix the notation for some sequences needed in this paper:
$\varphi(k)$ is the Euler (totient) function (A000010 in Sloane's
\emph{Encyclopedia} \cite{njas}), $d(k)$ is the number of divisors of $k$
(A000005), and $\sigma(k)$ is the sum of the divisors of $k$ (A000203).
\section{Operators on sequences}\label{newold}
Cameron \cite{Cam2000} describes how obtaining ``new
groups from old" (mainly by taking direct and wreath product, and by
taking the stabilizer) corresponds to operators on and transforms of their
U- and L-sequences (in the sense of Sloane \cite{transf}).
Analogously, it is possible to study how the Parker sequences of ``new"
groups are related to those of ``old" ones. The general effect on Parker
sequences of taking direct and wreath products of groups is studied in the
authors' papers \cite{Gew02} and \cite{GeMe01}.
Let $G$ and $H$ be permutation groups acting on the sets $X$ and $Y$,
respectively. Recall that, if we consider the direct product $G\times H$
acting on the disjoint union of $X$ and $Y$, the U-sequence for $G\times
H$ is obtained as CONV of the U-sequences of the factors (we are
multiplying the ordinary generating functions of the sequences); on the
other hand, the L-sequence of the direct product is obtained as EXPCONV
(here one considers the exponential generating functions).
For the Parker sequences the corresponding operation is simply the sum
(element by element):
$$p_k(G\times H) = p_k(G) + p_k(H).$$
Forming the direct product of $G$ with the countable symmetric group $S$
gives, as U-sequence, PSUM of the L-sequence of $G$; as L-sequence,
BINOMIAL of its L-sequence. For the Parker sequence, it simply yields
$$p_k(G\times S) = p_k(G) + 1.$$
One may also consider the product action of $G\times H$ on the cartesian
product $X\times Y$. For this action one has:
$$p_k(G\times H) = \sum_{\stackrel{\scriptstyle i,j}{\lcm(i,j)=k}}
p_i(G) p_j(H).$$
What happens for wreath products is more interesting. Recall \cite{Gew02,GeMe01}
that for the Parker sequences of the wreath product of
$G$ and $H$ the following holds:
$$p_k(G\wr H) = \sum_{d|k} p_d(G) p_{k/d}(G).$$
This is the Dirichlet convolution, which in the terminology of Sloane
\cite{transf} is the DIRICHLET transform of the two sequences.
We may now study, for a given oligomorphic group $H$, the operator mapping
the Parker sequence of any group $G$ to that of $G\wr H$. For U-sequences,
this procedure gives rise to the operators EULER, INVERT, and CIK,
respectively for $H=S$, $H=A$, and $H=C$. For Parker sequences we get, for
$H=S$, the MOBIUSi operator
$$p_k(G\wr S) = \sum_{d|k} p_d(G);$$
and, for $H=A$, the identity operator
$$p_k(G\wr A) = p_k(G).$$
For $H=C$ we get
$$p_k(G\wr C) = \sum_{d|k} p_d(G) \varphi(k/d);$$
in particular note that for square-free $k$'s (that is, the values of $k$
such that $\mu(k)\neq 0$) one has $p_k(G\wr\nolinebreak C) = \varphi(k)
\sum_{d|k} p_d(G)/\varphi(d)$.
Notice that, while in general $G\wr H$ and $H\wr G$ may be different
groups, they have the same Parker sequence; so these operators are also
those mapping ${\bf p}(G)$ to ${\bf p}(H\wr G)$.
\section{Parker sequences and circulant relational
structures}\label{circulant}
Recall \cite{GeMe01} that, if we are dealing with a group $G$ defined as
the automorphism group of the limit of a Fra\"{\i}ss\'e class $\cal F$ of
relational structures, the Parker sequence of $G$ has an alternative
interpretation as the sequence enumerating the finite circulant structures
in that class. More precisely, the $k$th component of the Parker sequence
counts the relational structures in (the age of) $\cal F$ on the set
$\{1,2,\dots,k\}$ admitting as an automorphism the permutation $(1\ 2\
\dots\ k)$ (note that this is different than just requesting that the
structure admits a circular symmetry). In what follows we shall use
``circulant [structure]'' to mean ``circulant [structure] on the set
$\{1,2,\dots,k\}$ admitting the automorphism $(1\ 2\ \dots\ k)$''. All of
the Parker sequences listed in the ``Fra\"{\i}ss\'e class'' table were
obtained by counting these circulant structures.
This mirrors what happens with the L-sequence $(F_k)$ of the same group,
which is defined as the number of orbits on $k$-tuples of distinct
elements, and is equal to the number of labelled structures on $k$ points.
The same holds for the U-sequence $f_k$ of the number of orbits on
$k$-sets, giving the number of unlabelled structures. The theory behind
this can be found in Cameron's book \cite{Cam90}.
In order to give an idea of the techniques involved in deriving Parker
sequences, let us first briefly recall \cite{GeMe01} what happens for
graphs.
To describe a circulant graph $\Gamma$ on the vertex set
$\{0,1,2,\dots,k-1\}$, it is sufficient to give the neighbours of a fixed
vertex (say 0); this subset, which has the property that it contains a
vertex $i$ if and only if it contains $k-i$, is called \emph{symbol} of
$\Gamma$. On the other hand any subset $S$ of $\{1,2,\dots,k-1\}$ such
that $i\in S$ implies $k-i\in S$ is a possible symbol for a graph. So the
$k$th entry of the Parker sequence of the automorphism group of the limit
of the Fra\"{\i}ss\'e class of graphs (that is the well-known random, or
Erd\H{o}s-R\'enyi, or Rado, graph) is $2^{\lfloor k/2\rfloor}$.
Several variations to this method yield the Parker sequences for
other relational structures.
For instance, if we consider the symbol for a digraph (a structure with a
relation $\rightarrow$ in which for each pair of distinct vertices $a$,
$b$, any of $a\rightarrow b$, $b\rightarrow a$, both, or none may hold) we
choose whether or not to join, by putting a directed edge, 0 with any
other vertex. So we get $p_k = 4^{(k-1)/2} = 2^{k-1}$. Similarly, if we do
not allow a double orientation on an edge, we get the class of oriented
graphs, for which $p_k = 3^{\lfloor k/2\rfloor}$.
Of course, this kind of argument holds also for the class of $n$-ary
relations, for $n\geq 2$. The symbol for a circulant $n$-relation on $k$
points can be any possible set of $(n-1)$-tuples (admitting repetitions)
of the points. For instance, for a ternary relation, we may have $(0,0)$
(meaning that $(0,0,0)$ holds), $(0,1)$, $(1,0)$, $(1,1)$, \dots\ So we
have $k^{n-1}$ such $(n-1)$-tuples, and $2^{k^{n-1}}$ possible symbols
(sets of such tuples).
More examples in same vein appear in the tables.
The same techniques can be applied to the class of two-graphs; this case,
however, requires some care.
Recall that a \emph{two-graph} is defined as a pair $(X, T)$, where $X$ is
a set of points, and $T$ a set of 3-subsets of $X$ with the property that
any 4-subset of $X$ contains an even number of members of $T$.
Two-graphs on $k$ vertices are in bijection with switching classes of
graphs on $k$ vertices. Recall that switching a graph $\Gamma = (V,E)$
with respect to $S\subseteq V$ gives a graph $(V,E')$ such that
$\{v,w\}\in E'$ if and only if either $v$ and $w$ are both in $S$ or both
in $V\setminus S$ and $\{v,w\}\in E$, or one is in $S$ and the other is in
$V\setminus S$, and $\{v,w\}\not\in E$ (see \cite{Sei76}, also for the
description of the correspondence between two-graphs and switching
classes).
Note that a two-graph $(X,T)$ is circulant if and only if at least one
graph in the corresponding switching class is. In fact, assume that
$\alpha$ is a permutation of $X$ inducing an automorphism of $(X,T)$; then
$\alpha$ induces an automorphism of at least one graph in the
corresponding switching class (as proved by Mallows and Sloane
\cite{MaSlo75}; see also Cameron \cite{cam77}).
The following result relates circulant two-graphs to circulant graphs.
\begin{tm}\label{twographs}
Let $\Gamma$ be a circulant $k$-vertex graph. If $k$ is odd, then $\Gamma$
is the only circulant graph in its switching class; if $k$ is even, there
are exactly two circulant graphs in its switching class.
\end{tm}
In order to prove this, let us first show in some detail what happens
switching circulant and regular graphs.
\begin{pr}
For $k$ odd, in each switching class of graphs on $k$ vertices there is at
most one regular graph.
\end{pr}
\pf{Let $\Gamma$ be a regular graph of valency $r$ on $k$ vertices. Let
us switch it with respect to the set $S\subseteq V(\Gamma)$, $0<|S|=m0$
let
$$\delta_i(k) := \sum_{d|k} \delta_{i-1}(d),$$
that is, $\delta_i$ is the Dirichlet convolution $\delta_{i-1} *
\delta_0$. Thus, $\delta_i(k) = p_k(S^{\wr i+1})$.
All the functions $\delta_i$ are multiplicative, because $\delta_0$ is,
and the Dirichlet convolution preserves multiplicativity. Thus, it
suffices to compute the value of $\delta_i$ on prime powers.
We claim that
$$\delta_i(p^j) = \bin{i+j}{i}.$$
To obtain a different description of the $\delta_i$s, note that
$\delta_1(k)$ gives the number of divisors of $k$, including $1$ and $k$;
so it is equal to $d(k)$. Next, $\delta_2(k)$ is the sum over the divisors
of $k$ of the number of their divisors; in other words, it gives the
number of pairs $(h,d)$ with $h|d$ and $d|k$ (observe that $h$ and $d$ may
well coincide). In general, we see that $\delta_i(k)$ gives the number of
$i$-ples $(d_1,d_2,\dots,d_i)$ with $d_1|d_2$, \dots, $d_{i-1}|d_i$,
$d_i|k$. We call such a sequence a \emph{generalised gozinta chain},
recalling that a gozinta (``goes into'') chain for $k$ is a sequence of
divisors of $k$ each of which strictly divides the next one.
When $k=p^j$, a sequence of divisors of $k$ each of which divides the next
one corresponds to a nondecreasing sequence of exponents of $p$, that is
to a nondecreasing sequence of numbers in $[j]=\{0,1,\dots,j\}$, which in
turn can be seen as a multiset of elements of $[j]$.
So, it is enough to enumerate the multisubsets of $\{0,1,\dots,j\}$ of
size $i$. It is well known (see for instance \cite{stanley}) that their
number is given by $\bin{i+j}{i}$, as claimed.
\smallskip
\no{\bf Remark 3}\quad If $k$ is square-free, $p_k$ is equal to
$\sum_{d|k} \varphi(k) = d(k) \varphi(k)$.
\bigskip\bigskip
\begin{center}
*\qquad*\qquad*
\end{center}
\bigskip\bigskip
The following groups arise as automorphism groups of Fra\"{\i}ss\'e
classes (see section \ref{circulant}).
The calculation of Parker sequences for ``treelike objects'' and related
structures is carried out in detail in the forthcoming paper
\cite{GeMe??}.
The letters R and L mean ``shifted right'' and ``shifted left''
respectively.
\medskip
\vfill\eject
\centerline{Automorphism Groups of Homogeneous Structures}
\medskip
{
\footnotesize
\begin{center}
\begin{tabular}{|p{5cm}|p{5.7cm}|l|l|}
\hline
{\bf Fra\"{\i}ss\'e class} & {\bf Parker sequence} & {\bf EIS entry} &
{\bf Notes}\\
\hline
&&&\\
Graphs & $2^{\lfloor k/2\rfloor}$ & A016116 & See \cite{GeMe01}\\
Graphs up to complement & $p_1=1$, $p_k=2^{\lfloor k/2\rfloor -1}$ for
$k>1$ & A016116RR & See rem.\ 4\\
$K_3$-free graphs & 1,2,1,3,3,4,4,8,4,14,11,14,\dots & A083041 &
See rem.\ 5\\
Graphs with bipartite block & 2,2,2,\dots & A007395 & See rem.\ 6\\
Graphs with loops & $2^{\lfloor k/2\rfloor +1}$ & A016116LL &
See rem.\ 7\\
Digraphs & $2^{k-1} (= 4^{(k-1)/2})$ & A000079R & \\
Digraphs with loops (or binary relations) & $2^k$ & A000079 & \\
Oriented graphs & $3^{\lfloor k/2\rfloor}$ & [missing] & \\
Topologies & $d(k)$ & A000005 & See rem.\ 8 \\
Posets & 1,1,1,\dots & A000012 & See rem.\ 9 \\
Tournaments & $k$ odd: $2^{\lfloor k/2\rfloor}$, $k$ even: 0 & [missing] &
See \cite{GeMe01}\\
Local orders & $k$ odd: $\varphi(k)$, $k$ even: 0 & [missing] & See
\cite{GeMe??}\\
Two-graphs & $2^{\lceil k/2\rceil}$ & A016116L & See
Thm.~\ref{twographs}\\
Oriented two-graphs & $2^{\lceil k/2\rceil}$ & A016116L & See
Thm.~\ref{twographs}\\
Total orders with subset & 2,0,0,\dots & A000038 &\\
Total orders with 2-partition & 1,0,0,\dots & A000007 &\\
$C$-structures with subset & $2\varphi(k)$ & [missing] &
See rem.\ 10 \\
$D$-structures with subset & $\varphi(k)$ & A000010 & See rem.\ 10\\
2 total orders (distinguished) & 1,0,0,\dots & A000007 &\\
2 total orders (not distinguished) & 1,1,0,0,\dots & A019590 &\\
2 betweennesses (not distinguished) & 1,1,0,0,\dots & A019590 &\\
Boron trees (leaves) (or $T_3$) &characteristic fn. of $\{3^a
2^b\}_{a\in\{0,1\}, b\geq0}$ & [missing] & See \cite{GeMe??}\\
HI trees (leaves) (or $T$) & nr. of ordered factorisations of $k$ &
A002033R & See \cite{GeMe??}\\
R(Boron trees (leaves)) (or $\partial T_3$) & characteristic fn. of powers
of 2 &A036987& See \cite{GeMe??} \\
R(HI trees (leaves)) (or $\partial T$) & nr. of ordered factorisations of
$k$ & A002033R & See \cite{GeMe??} \\
Trees (edges) &1,1,1,\dots & A000012 & See \cite{GeMe??}\\
Covington structures (or $\partial T_3(2)$) & $p_{2^i} = 2^i$, 0
otherwise & A048298 & See \cite{GeMe??} \\
Binary trees (or $\partial PT_3$) & 1,0,0,0,\dots & A000007& See
\cite{GeMe??}\\
Binary trees up to reflection (or $\partial P^*T_3$) & 1,1,0,0,\dots
& A019590 & See \cite{GeMe??}\\
Plane trees (or $PT$)$\dag$ & $\varphi(k)$ & A000010 & See
\cite{GeMe??}\\
Plane trees up to reflection (or $P^*T$)$\dag$ & $\sim\varphi(k)/2$
& A023022 & See \cite{GeMe??}\\
Plane boron trees (or $PT_3$) & 1,1,2,0,0,\dots & [missing] & See
\cite{GeMe??}\\
Plane boron trees up to reflection (or $P^*T_3$) & 1,1,1,0,0,\dots &
[missing] &
See \cite{GeMe??}\\
3-hypergraphs & $2^{f(k,3)}$, where $f(k,3) =
0,0,1,1,4,4,5,7,10,12,15,19,\dots$
& [missing] & See \cite{GeMe01}\\
$t$-hypergraphs$\dag$ & $2^{f(k,t)}$ && See \cite{GeMe01}\\
Ternary relations & $2^{k^2}$ & A002416 &\\
Quaternary relations & $2^{k^3}$ & [missing] &\\
%&&&\\
\hline
\multicolumn{4}{l}{}\\
\multicolumn{4}{l}{$\dag$ Not in \cite{Cam2000}.}\\
\end{tabular}
\end{center}
}
%\medskip
\vfill\eject
\no{\bf Remark 4}\quad Each (symbol for a) circulant graph represents also
its complement, so (for $k>1$) each term is one half of the corresponding
term for graphs. For instance, $p_2 = 1$ because the graphs $K_2$ and
$N_2$ are now identified.
\medskip
\no{\bf Remark 5}\quad This is the number of symmetric sum-free subsets of
${\bf Z}/(k)^*$ (see \cite{cam87}): if the symbol contains $a$ and $b$, it
cannot contain $a+b$, and (as for generic graphs) if it contains $a$, it
must contain $k-a$.
\medskip
\no{\bf Remark 6}\quad We cannot exchange ``black" and ``white" vertices,
so a circulant structure is an all-black or all-white null graph.
\medskip
\no{\bf Remark 7}\quad Reason as in section \ref{circulant}, but take in
addition to ``basic" circulant graphs (those with symbol of the form
$\{i,k-i\}$) also that with $k$ vertices, each with a loop attached, and
no other edges. In other words, in the symbol (set of ``neighbours" of 0)
for a circulant graph with loops, also 0 may appear.
\medskip
\no{\bf Remark 8}\quad The ``basic" graphs do not work as they are; the
request for the relation to be transitive forces any $k$-gon to ``become"
a complete directed graph (that is, $K_k$ where all edges are bidirected):
by transitivity, connect vertices at distance 2, then at distance 3 and so
on. The superposition of $d$ copies of $K_{k/d}$ and $l$ copies of
$K_{k/l}$ becomes by transitivity the superposition of GCD$(d,l)$ copies
of $K_{{\rm lcm}(k/d, k/l)}$. So the lattice of divisors of $k$ describes
all the possible circulant transitive digraphs, that is topologies.
In other words, a topology is the transitive closure of a union of cyclic
graphs; its incidence matrix can be seen as the $k$th power of the
incidence matrix of the starting graph with, as its entries, boolean
variables $0$ and $1$ (so that $1+1=1$).
\medskip
\no{\bf Remark 9}\quad By acyclicity, for each $n$ the only circulant
poset is the one with $n$ incomparable elements.
\medskip
\no{\bf Remark 10}\quad The only possible distinguished sets are the empty
and the full ones.
\medskip
\centerline{One Last Example}
\medskip
{
\footnotesize
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
{\bf Group} & {\bf Parker sequence} & {\bf EIS entry} & {\bf Notes}\\
\hline
&&&\\
$S^2$ (product action) & $d(k^2)$ & A048691 & See rem.\ 11\\
&&&\\
\hline
\end{tabular}
\end{center}
}
\medskip
\no{\bf Remark 11}\quad The result follows from Section~\ref{newold},
keeping in mind that $d(k^2)$ is equal to the number of
pairs $(i,j)$ such that $\lcm(i,j)=k$.
\comment{
It is known that $d(k^2)$ is equal to the number of
ordered factorisations of $k$ in 3 factors: $k = f_1 f_2 f_3$ ($1\leq f_i
\leq k$), such that $(f_1, f_2) = 1$.
A $k$-cycle of $S^2$ corresponds to such a factorisation as follows:
\begin{flushleft}
$( (1\ \ f_1+1\ \ 2f_1+1\ \dots\ (f_3-1)f_1+1\ \ 2\ \ f_1+2\ \dots\
f_1 f_3),$
\end{flushleft}
\begin{flushright}
$(1\ \ f_2+1\ \ 2f_2+1\ \dots\ (f_3-1)f_2+1\ \ 2\ \ f_2+2\ \dots\ f_2 f_3))
=$
\end{flushright}
\begin{flushleft}
$= ((1,1)\ (f_1+1,f_2+1)\ (2f_1+1,2f_2+1)\ \dots\ (f_1 f_3,f_2 f_3))$
\end{flushleft}
which is indeed a cycle of length $k = f_1f_2f_3$ (that is, $\lcm(f_1
f_3,f_2 f_3)$).
Each cycle of $S^2$ is of the above form; and two cycles
$\gamma_1=(\alpha_1, \beta_1)$ and $\gamma_2=(\alpha_2, \beta_2)$ of $S^2$
are conjugate if and only if the length of $\alpha_1$ is equal to the
length of $\alpha_2$ and the length of $\beta_1$ is equal to the length of
$\beta_2$.}
\section*{Acknowledgements}
We thank Dina Ghinelli and Peter J.~Cameron for help and encouragement.
We also thank the referee for useful comments.
\begin{thebibliography}{ABC}
\addcontentsline{toc}{chapter}{Bibliography}
\bibitem{cam76} Peter J. Cameron, Transitivity of permutation groups on
unordered sets, {\it Math.~Z.} {\bf 148} (1976), 127--139.
\bibitem{cam77} Peter J. Cameron, Cohomological aspects of two-graphs,
{\it Math.~Z.} {\bf 157} (1977), 101--119.
\bibitem{cam87} Peter J. Cameron, Portrait of a typical sum-free set, {\it
Surveys in Combinatorics} (C. Whitehead, ed.), 13--42, LMS Lecture Notes
{\bf 123}, Cambridge Univ. Press, Cambridge, 1987.
\bibitem{treelike} Peter J. Cameron, Some treelike objects, {\it Quart.\
J. Math.\ Oxford Ser.\ (2)} {\bf 38} (1987), 155--183.
\bibitem{Cam90} Peter J. Cameron, {\it Oligomorphic Permutation Groups},
LMS Lecture Notes {\bf 152}, Cambridge Univ. Press, Cambridge, 1990.
\bibitem{Cam2000} Peter J. Cameron, Sequences realized by oligomorphic
permutation groups, {\it J. Integer Seq.} {\bf 3} (2000), 00.1.5
[\href{http://www.math.uwaterloo.ca/JIS/VOL3/groups.html}{\tt
http://www.math.uwaterloo.ca/JIS/VOL3/groups.html}].
\bibitem{Gew02} Daniele A. Gewurz, Parker vectors and cycle indices of
permutation groups, {\it Quaderni Elettronici del Seminario di Geometria
Combinatoria} {\bf 4E} (2002)
[\href{http://www.mat.uniroma1.it/~combinat/quaderni}{\tt
http://www.mat.uniroma1.it/\~{}combinat/quaderni}].
\bibitem{GeMe01} Daniele A.~Gewurz and Francesca Merola, Parker vectors
for infinite groups, {\it European J. Combin.} {\bf 22} (2001),
1065--1073.
\bibitem{GeMe??} Daniele A.~Gewurz and Francesca Merola, Cycle action on
treelike structures, preprint.
\bibitem{MaSlo75} C.L.~Mallows and N.J.A.~Sloane, Two-graphs, switching
classes and Euler graphs are equal in number, {\it SIAM J.~Appl.~Math.}
{\bf 28} (1975), 876--880.
\bibitem{Sei76} J.~J.~Seidel, A survey of two-graphs, {\it Colloquio
Internazionale sulle Teorie Combinatorie (Rome, 1973)}, Tomo I, Atti dei
Convegni Lincei, No.~17, Accad.~Naz.~Lincei, Rome, 1976, pp.~481--511.
\bibitem{njas} N.J.A. Sloane, ed., The On-Line Encyclopedia of Integer
Sequences, published electronically at
\href{http://www.research.att.com/~njas/sequences/}{\tt
http://www.research.att.com/\~{}njas/sequences/}.
\bibitem{transf} N.J.A. Sloane, ed., Transformations of Integer Sequences,
published electronically at
\href{http://www.research.att.com/~njas/sequences/transforms.html}{\tt
http://www.research.att.com/\~{}njas/sequences/transforms.html}.
\bibitem{stanley} Richard P. Stanley, {\it Enumerative Combinatorics},
Vol.\ 1, Wadsworth, 1986 (Cambridge University Press, 1997).
\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2000 {\it Mathematics Subject Classification}:
Primary 20B07; Secondary 05A15.
\noindent \emph{Keywords:
Oligomorphic permutation groups, action on cycles,
Parker vectors, circulant relational structures}
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A000010},
\seqnum{A000005},
\seqnum{A000203},
\seqnum{A000012},
\seqnum{A000007},
\seqnum{A019590},
\seqnum{A023022},
\seqnum{A007395},
\seqnum{A054977},
\seqnum{A000038},
\seqnum{A010701},
\seqnum{A000027},
\seqnum{A000034},
\seqnum{A083039}, %
\seqnum{A083040}, %
\seqnum{A007425},
%\seqnum{A023022},
\seqnum{A029935}, %
\seqnum{A016166},
\seqnum{A083041}, %
\seqnum{A000079},
\seqnum{A002033},
\seqnum{A036987},
\seqnum{A048298},
\seqnum{A002416},
\seqnum{A048691}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in} \noindent Received October 1, 2002;
revised version received April 4, 2003.
Published in {\it Journal of Integer Sequences}
April 15, 2003. Slight revisions, June 11, 2003.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}.
\end{document}