In mathematics, there are two natural interpretations of the place-permutation action of symmetric groups, in which the group elements act on positions or places. Each may be regarded as either a left or a right action, depending on the order in which one chooses to compose permutations. There are just two interpretations of the meaning of "acting by a permutation " but these lead to four variations, depending whether maps are written on the left or right of their arguments. The presence of so many variations often leads to confusion. When regarding the group algebra of a symmetric group as a diagram algebra it is natural to write maps on the right so as to compute compositions of diagrams from left to right. Maps written on the left First we assume that maps are written on the left of their arguments, so that compositions take place from right to left. Let be the symmetric group on letters, with compositions computed from right to left. Imagine a situation in which elements of act on the “places” (i.e., positions) of something. The places could be vertices of a regular polygon of sides, the tensor positions of a simple tensor, or even the inputs of a polynomial of variables. So we have places, numbered in order from 1 to , occupied by objects that we can number . In short, we can regard our items as a word of length in which the position of each element is significant. Now what does it mean to act by “place-permutation” on ? There are two possible answers: an element can move the item in the th place to the th place, or it can do the opposite, moving an item from the th place to the th place. Each of these interpretations of the meaning of an “action” by (on the places) is equally natural, and both are widely used by mathematicians. Thus, when encountering an instance of a "place-permutation" action one must take care to determine from the context which interpretation is intended, if the author does not give specific formulas. Consider the first interpretation. The following descriptions are all equivalent ways to describe the rule for the first interpretation of the action: For each , move the item in the th place to the th place. For each , move the item in the th place to the th place. For each ,