Twelvefold Way - Selection, Labelling, Grouping

Selection, Labelling, Grouping

A function ƒ : N → X can be considered from a the perspective of X or of N. This leads to different views:

• the function ƒ labels each element of N by an element of X.
• the function ƒ selects (chooses) an element of the set X for each element of N, a total of n choices.
• the function ƒ groups the elements of N together that are mapped to the same element of X.

These points of view are not equally suited to all cases. The labelling and selection points of view are not well compatible with permutation of the elements of X, since this changes the labels or the selection; on the other hand the grouping point of view does not give complete information about the configuration unless the elements of X may be freely permuted. The labelling and selection points of view are more or less equivalent when N is not permuted, but when it is, the selection point of view is more suited. The selection can then be viewed as an unordered selection: a single choice of a (multi-)set of n elements from X is made.