left one. In symbols, this means that the action is now a right action, while the action is now a left action. Example We let be the 3-cycle and the transposition , as before. Since we now write maps on the right of their arguments we have Using the first interpretation we have , the result of which agrees with the action of on . So . On the other hand, if we use the second interpretation, we have , the result of which agrees with the action of on . So . Summary In conclusion, we summarize the four possibilities considered in this article. Here are the four variations: Although there are four variations, there are still only two different ways of acting; the four variations arise from the choice of writing maps on the left or right, a choice which is purely a matter of convention. Notes References Tom Halverson and Arun Ram, "Partition algebras", European J. Combin. 26 (2005), no. 6, 869–921. Thomas Hungerford, Algebra. Springer Lecture Notes 73, Springer-Verlag 1974. Gordon D. James, The Representation Theory of the Symmetric Groups. Lecture Notes in Math. 682 (1978), Springer. Hermann Weyl, The Classical Groups: Their Invariants and Representations. Princeton University Press, Princeton, N.J., 1939. Permutations