Stories about Groups and Sequences

The main theme of this article is that counting orbits of an in nite permutation group on nite subsets or tuples is very closely related to combinatorial enumeration this point of view ties together various dis parate stories Two graphs and even graphs The rst story originated with Neil Sloane when he was compiling the rst edition of his dictionary of integer sequences He observed that certain counting sequences appeared to agree The rst sequence enumerates even graphs those in which any vertex has even valency so that the graph is a disjoint union of Eulerian graphs These graphs were enumerated by Robinson and Liskovec The second sequence counts switching classes of graphs If is a graph on the vertex set X and Y is a subset of X the result of switching with respect to Y is obtained by deleting all edges between Y and its complement putting in all edges between Y and its complement which didn t exist before and leaving the rest unaltered Switching is an equivalence relation on the graphs with vertex set X the equivalence classes are called switching classes This concept was introduced by Seidel for studying strongly regular graphs The nal sequence counts two graphs A two graph on a set X consists of a set T of triples or element subsets of X with the property that any element subset of T contains an even number of elements of T Two graphs were introduced by G Higman in a construction of Conway s third sporadic group The theory has been developed in many directions Seidel has written several surveys They also link several themes in combinatorics including equiangular lines in Euclidean space and double covers of complete graphs It was already known that switching classes and two graphs are equinumer ous There is a map from graphs on the set X to two graphs on X as follows the triples of the two graph are all sets which contain an odd number of edges of the graph Every two graph is obtained in this way and graphs and give the same two graph if and only if they lie in the same switching class So there is a natural bijection from switching classes to two graphs It was also known that switching classes and even graphs on an odd number of vertices are equinumerous Any switching class on an odd number of ver tices contains a unique even graph obtained by taking any graph in the class and switching with respect to the set of vertices of odd degree But no such correspondence exists if the number of vertices is even Mallows and Sloane proved that the numbers were equal by deriving a formula for the number of switching classes and observing that it coincides with the Robinson Liskovec formula for the number of even graphs The right explanation actually shows that the classes are dual Let X be a set of n points and V the set of all graphs on the vertex set X Each graph can be represented by a binary vector of length n n whose ones give the positions of the edges So V is a vector space over GF of dimension n n The addition in V corresponds to taking the symmetric di erence of the edge sets of the two graphs We consider two subsets of V U the set of complete bipartite graphs W the set of even graphs It is easy to see that U is a subspace of V spanned by the stars Now a graph is even if and only if it is orthogonal to all stars so W U and W is also a subspace The cosets of U in V are precisely the switching classes of graphs So V U is the set of switching classes Since W U this quotient V U is isomorphic to the dual space W of W not just as vector space but as module for the symmetric group on X Now a group acting on a nite vector space has equally many orbits on the space and on its dual by Brauer s lemma and the orbits of the symmetric group are the isomorphism classes So the numbers of switching classes and even graphs are equal Recently I noticed another feature which may be related in some way to this duality As noted above an even graph is the disjoint union of Eulerian graphs A similar looking decomposition holds for two graphs We de ne a relation on the point set of a two graph by the rule that x y if and only if either x y or no triple contains x and y From the de nition of a two graph it is easy to see that this is an equivalence relation and is even a congruence that is membership of a triple in T is una ected if we replace some of its points by equivalent ones Thus a two graph is described by a partition of X with no structure on the parts of the partition and the structure of a reduced two graph one in which all classes are singletons on the set of parts By contrast for even graphs we have an Eulerian graph on each part of the partition and no structure on the set of parts this is in some vague sense dual to the preceding The numbers of Eulerian graphs and of reduced two graphs on n points agree for n but di er for n