Let r, n and n _1,..., n _r be positive integers with n=n _1+?+n _r. Let X be a connected graph with n vertices. For 1≤i≤r, let P _i be the i-th color class of n _i distinct pebbles. A configuration of the set of pebbles P=P _1∪?∪P _r on X is defined as a bijection from the set of vertices of X to P. A move of pebbles is defined as exchanging two pebbles with distinct colors on the two endvertices of a common edge. For a pair of configurations f and g, we write f~g if f can be transformed into g by a sequence of finite moves. The relation ~ is an equivalence relation on the set of all the configurations of P on X. We study the number c(X, n _1,..., n _r) of the equivalence classes. A tuple (X, n _1,..., n _r) is called transitive if for any configuration f and for any vertex u, a pebble f(u) can be moved to any other vertex by a sequence of finite moves. We determine c(X, n _1,..., n _r) for an arbitrary transitive tuple (X, n _1,..., n _r).
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